ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 2 - 2016
A Test of Symmetry Based on the Kernel Kullback-Leibler Information with Application to Base Deficit Data
Hani M Samawi* and Robert Vogel
Department of Biostatistics, Georgia Southern University, USA
Received: January 07, 2016 | Published: January 27, 2016
*Corresponding author: Hani M Samawi, Department of Biostatistics, Jiann Ping Hsu College of Public Health, PO Box 8015, Georgia Southern University Statesboro, GA 30460, USA, Email: ,
Citation: Samawi, Vogel R (2016) A Test of Symmetry Based on the Kernel Kullback-Leibler Information with Application to Base Deficit Data. Biom Biostat Int J 3(2): 00060. DOI: 10.15406/bbij.2016.03.00060

Abstract

The assumption of the symmetry of the underlying distribution is important to many statistical inference and modeling procedures. This paper provides a test of symmetry using kernel density estimation and the Kullback-Leibler information. Based on simulation studies, the new test procedure outperforms other tests of symmetry found in the literature, including the Runs Test of Symmetry. We illustrate our new procedure using real data.

Keywords: Test of symmetry; Power of the test; Overlap coefficients; Kernel Density estimation; Kullback-Leibler information

Introduction

Many statistical applications and inferences rely on the validity of the underlying distributional assumption. Symmetry of the underlying distribution is essential in many statistical inference and modeling procedures. There are several tests of symmetry in the literature; however most of these tests suffer from low statistical power. Tests have been suggested by Butler [1], Rothman & Woodroofe [2], Hill & Roa [3], Baklizi [4], and McWilliams [5]. McWilliams [5] showed, using simulation, that his runs test of symmetry is more powerful than those provided by Butler [1], Rothman & Woodroofe [2], and Hill & Roa [3] for various asymmetric alternatives. However, Tajuddin [6] introduced a distribution-free test for symmetry based on Wilcoxon two-sample test which is more powerful than the runs test.

Moreover, Modarres & Gastwirth [7] modified McWilliams [5] runs test by using Wilcoxon scores to weight the runs. The new test improved the power for testing for symmetry about a known center but did not perform well when the asymmetry is focused in regions close to the median for a given distribution. Mira [8], introduced a distribution free test for symmetry based on Boferroni’s Measure. She showed that her test outperform tests introduced by Modarres & Gastwirth [7] and Tajauddin [6]. Recently, Samawi et al. [9] provided a test of symmetry based on a nonparametric overlap measure. They demonstrated that the test of symmetry based on an overlap measure outperformed other tests of symmetry in the literature, including the runs test. Samawi & Helu [10] introduced a runs test of conditional symmetry which is reasonably powerful to detect even small asymmetry in the shape of the conditional distribution. In addition, the Samawi & Helu [10] test does not need any approximation nor extra computations such as kernel estimation of the density function as in the other tests that are found in the literature.

This paper uses the Kullback-Leibler information to test for the symmetry of the underlying distribution. Let f 1 (x) and  f 2 (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIhacaGGPaGaaeii aiaabggacaqGUbGaaeizaiaabccacaWGMbWaaSbaaKqbGeaacaaIYa aajuaGbeaacaGGOaGaamiEaiaacMcaaaa@4439@ be two probability density functions. Assume samples of observations are drawn from continuous distributions. The Kullback-Leibler discrimination information function is given by

D( f 1 , f 2 )= f 1 (x)ln( f 1 (x) f 2 (x) )dx ,= f 1 (x)ln( f 1 (x) )dx f 1 (x)ln( f 2 (x) )dx, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaiabg2da9maapehaba GaamOzamaaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIhacaGG PaGaciiBaiaac6gadaqadaqaamaalaaabaGaamOzamaaBaaajuaiba GaaGymaaqcfayabaGaaiikaiaadIhacaGGPaaabaGaamOzamaaBaaa juaibaGaaGOmaaqcfayabaGaaiikaiaadIhacaGGPaaaaaGaayjkai aawMcaaiaadsgacaWG4baajuaibaGaeyOeI0IaeyOhIukabaGaeyOh IukajuaGcqGHRiI8aiaacYcacqGH9aqpdaWdXbqaaiaadAgadaWgaa qcfasaaiaaigdaaKqbagqaaiaacIcacaWG4bGaaiykaiGacYgacaGG UbWaaeWaaeaacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGOa GaamiEaiaacMcaaiaawIcacaGLPaaacaWGKbGaamiEaiabgkHiTaqc fasaaiabgkHiTiabg6HiLcqaaiabg6HiLcqcfaOaey4kIipadaWdXb qaaiaadAgadaWgaaqcfasaaiaaigdaaKqbagqaaiaacIcacaWG4bGa aiykaiGacYgacaGGUbWaaeWaaeaacaWGMbWaaSbaaKqbGeaacaaIYa aajuaGbeaacaGGOaGaamiEaiaacMcaaiaawIcacaGLPaaacaWGKbGa amiEaiaacYcaaKqbGeaacqGHsislcqGHEisPaeaacqGHEisPaKqbak abgUIiYdaaaa@8944@  (1)

as defined by Kullback & Leibler [11]. For simplicity we will write (1) as

D( f 1 , f 2 )= D 11 ( f 1 , f 1 ) D 12 ( f 1 , f 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaiabg2da9iaadseada WgaaqcfasaaiaaigdacaaIXaaajuaGbeaacaGGOaGaamOzamaaBaaa juaibaGaaGymaaqcfayabaGaaiilaiaadAgadaWgaaqcfasaaiaaig daaKqbagqaaiaacMcacqGHsislcaWGebWaaSbaaKqbGeaacaaIXaGa aGOmaaqcfayabaGaaiikaiaadAgadaWgaaqcfasaaiaaigdaaKqbag qaaiaacYcacaWGMbWaaSbaaKqbGeaacaaIYaaajuaGbeaacaGGPaGa aiilaaaa@5558@ where D 11 ( f 1 , f 1 )= f 1 (x)ln( f 1 (x) )dx and  D 12 ( f 1 , f 2 )= f 1 (x)ln( f 2 (x) )dx. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaWGMbWaaSba aKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBaaajuaibaGaaG ymaaqcfayabaGaaiykaiabg2da9maapehabaGaamOzamaaBaaajuai baGaaGymaaqcfayabaGaaiikaiaadIhacaGGPaGaciiBaiaac6gada qadaqaaiaadAgadaWgaaqcfasaaiaaigdaaKqbagqaaiaacIcacaWG 4bGaaiykaaGaayjkaiaawMcaaiaadsgacaWG4bGaaeiiaiaabggaca qGUbGaaeizaiaabccacaWGebWaaSbaaKqbGeaacaaIXaGaaGOmaaqc fayabaGaaiikaiaadAgadaWgaaqcfasaaiaaigdaaKqbagqaaiaacY cacaWGMbWaaSbaaKqbGeaacaaIYaaajuaGbeaacaGGPaGaeyypa0da juaibaGaeyOeI0IaeyOhIukabaGaeyOhIukajuaGcqGHRiI8amaape habaGaamOzamaaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIha caGGPaGaciiBaiaac6gadaqadaqaaiaadAgadaWgaaqcfasaaiaaik daaKqbagqaaiaacIcacaWG4bGaaiykaaGaayjkaiaawMcaaiaadsga caWG4bGaaiOlaaqcfasaaiabgkHiTiabg6HiLcqaaiabg6HiLcqcfa Oaey4kIipaaaa@7D5F@

This measure can be directly applied to discrete distributions by replacing the integrals with summations. It is well known that D( f 1 , f 2 )0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaiabgwMiZkaaicdaca GGSaaaaa@418D@ and the equality holds if and only if f 1 (x)= f 2 (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIhacaGGPaGaeyyp a0JaamOzamaaBaaajuaibaGaaGOmaaqcfayabaGaaiikaiaadIhaca GGPaaaaa@413D@ almost everywhere. The discrimination function D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ measures the disparity between f 1  and  f 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaGaaeiiaiaabggacaqGUbGaaeiz aiaabccacaWGMbWaaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@3F8D@ .

Many authors used the discrimination function D(.,.) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaGGUaGaaiilaiaac6cacaGGPaaaaa@3ABA@ for testing goodness of fit of some distributions. For example see Alizadeh & Arghami [12,13].

In this paper we consider testing the null hypothesis of symmetry for an underlying absolutely continuous distribution F(.) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOrai aacIcacaGGUaGaaiykaaaa@395A@ with known location parameter and density denoted by f(.) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzai aacIcacaGGUaGaaiykaaaa@397A@ H 0 :f(x)=f(x)  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaGaaiOoaiaadAgacaGGOaGaamiE aiaacMcacqGH9aqpcaWGMbGaaiikaiabgkHiTiaadIhacaGGPaGaae iiaaaa@42BE@ versus  H a :f(x)f(x);for some x. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeODai aabwgacaqGYbGaae4CaiaabwhacaqGZbGaaeiiaiaadIeadaWgaaqc fasaaiaadggaaKqbagqaaiaacQdacaWGMbGaaiikaiaadIhacaGGPa GaeyiyIKRaamOzaiaacIcacqGHsislcaWG4bGaaiykaiaacUdacaqG MbGaae4BaiaabkhacaqGGaGaae4Caiaab+gacaqGTbGaaeyzaiaabc cacaWG4bGaaiOlaaaa@53A9@ Under the null hypothesis of symmetry, if we let f 1 (x)=f(x) and   f 2 (x)=f(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIhacaGGPaGaeyyp a0JaamOzaiaacIcacaWG4bGaaiykaiaabccacaqGHbGaaeOBaiaabs gacaqGGaGaaeiiaiaadAgadaWgaaqcfasaaiaaikdaaKqbagqaaiaa cIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaeyOeI0IaamiEai aacMcaaaa@4E57@ then D( f 1 , f 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaiabg2da9iaaicdaaa a@401D@ .

Since kernel density estimation procedures are readily available in various statistical software packages such as SAS, STATA, S-Plus and R, we were interested in exploring the development of a new test of symmetry using kernel density estimation of D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ . This paper will introduce a powerful test of symmetry based on Kullback-Leibler discrimination information function. The Kullback-Leibler information test of symmetry and its asymptotic properties are introduced in Section 2. A simulation study is provided in Section 3. Illustrations of the test using base deficit score data and final comments are given in Section 4.

Test of Symmetry Based on the Kullback-Leibler Discrimination Information Function

Assume that a random sample, X 1 , X 2 ...., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGyb WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiwamaaBaaajuai baGaaGOmaaqcfayabaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcaca WGybWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@4261@ , is drawn from absolutely continuous distribution having known median, assumed to be 0. In the case of an unknown median, or if the center of the distribution is not known, then the data can be centered by a consistent estimate of the median. However, the implications of centering the data around a consistent estimator of the median on the asymptotic properties are not straightforward. Therefore, further investigations are needed to study the robustness of the proposed test of symmetry and compare it with other available tests of symmetry when the median is unknown. In this paper we will discuss only the case where the median of the underlying distribution is assumed known.

Consider testing for symmetry H 0 :f(x)=f(x)  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaGaaiOoaiaadAgacaGGOaGaamiE aiaacMcacqGH9aqpcaWGMbGaaiikaiabgkHiTiaadIhacaGGPaGaae iiaaaa@42BE@ versus  H a :f(x)f(x);for some x. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeODai aabwgacaqGYbGaae4CaiaabwhacaqGZbGaaeiiaiaadIeadaWgaaqc fasaaiaadggaaKqbagqaaiaacQdacaWGMbGaaiikaiaadIhacaGGPa GaeyiyIKRaamOzaiaacIcacqGHsislcaWG4bGaaiykaiaacUdacaqG MbGaae4BaiaabkhacaqGGaGaae4Caiaab+gacaqGTbGaaeyzaiaabc cacaWG4bGaaiOlaaaa@53A9@ Let f 1 (x)=f(x) and   f 2 (x)=f(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIhacaGGPaGaeyyp a0JaamOzaiaacIcacaWG4bGaaiykaiaabccacaqGHbGaaeOBaiaabs gacaqGGaGaaeiiaiaadAgadaWgaaqcfasaaiaaikdaaKqbagqaaiaa cIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaeyOeI0IaamiEai aacMcaaaa@4E57@ . Under the null hypothesis, D( f 1 , f 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaiabg2da9iaaicdaaa a@401D@ . An equivalent hypothesis for testing the symmetry is H 0 :D( f 1 , f 2 )=0  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaGaaiOoaiaadseacaGGOaGaamOz amaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadAgadaWgaaqcfa saaiaaikdaaKqbagqaaiaacMcacqGH9aqpcaaIWaGaaeiiaaaa@43E2@ versus  H a :D( f 1 , f 2 )>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaeODai aabwgacaqGYbGaae4CaiaabwhacaqGZbGaaeiiaiaadIeadaWgaaqc fasaaiaadggaaKqbagqaaiaacQdacaWGebGaaiikaiaadAgadaWgaa qcfasaaiaaigdaaKqbagqaaiaacYcacaWGMbWaaSbaaKqbGeaacaaI YaaajuaGbeaacaGGPaGaeyOpa4JaaGimaaaa@49CA@ let D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaaaaa@375D@ be a consistent nonparametric estimator of D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ . Under the null hypothesis of symmetry and some regularity assumptions, which will be discussed later in this paper, we propose the following test of symmetry:

z 0 = D ^ 0 σ ^ D ^ L N(0,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaaGimaaqcfayabaGaeyypa0ZaaSaaaeaaceWGebGb aKaacqGHsislcaaIWaaabaGafq4WdmNbaKaadaWgaaqaaiqadseaga qcaaqabaaaamaaoqcabaGaamitaaqabiaawkziaiaad6eacaGGOaGa aGimaiaacYcacaaIXaGaaiykaaaa@4622@

For large n, where σ ^ D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafq4Wdm NbaKaadaWgaaqcfasaaiqadseagaqcaaqcfayabaaaaa@3A0D@ is a consistent estimator of the standard error of D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaaaaa@375D@ . An asymptotic significant test procedure at level α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ is to reject H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamisam aaBaaajuaibaGaaGimaaqcfayabaaaaa@38E8@ if z 0 > z α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaaGimaaqcfayabaGaeyOpa4JaamOEamaaBaaajuai baGaeqySdegajuaGbeaaaaa@3D9D@ , where z α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacqaHXoqyaeqaaaaa@38C0@ is the upper α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ percentile of the standard normal distribution.

Kernel estimation of D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@

For the i.i.d. sample X 1 , X 2 ...., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGyb WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiwamaaBaaajuai baGaaGOmaaqcfayabaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcaca WGybWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@4261@ , let D ^ 11 ( f 1 , f 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaWaaSbaaKqbGeaacaaIXaGaaGymaaqcfayabaGaaiikaiaadAga daWgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWGMbWaaSbaaKqbGe aacaaIXaaajuaGbeaacaGGPaaaaa@40BF@ be an estimate of D 11 ( f 1 , f 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaWGMbWaaSba aKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBaaajuaibaGaaG ymaaqcfayabaGaaiykaaaa@40AF@ . To address which estimator of D 11 ( f 1 , f 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaWGMbWaaSba aKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBaaajuaibaGaaG ymaaqcfayabaGaaiykaaaa@40AF@ will be appropriate to our inference procedure we need to state some necessary conditions: C1: f is continuous. (Smoothness conditions) C2: f is k times differentiable. (Smoothness conditions) C3: D 11 ([X],[X])<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiram aaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaGGBbGaamiw aiaac2facaGGSaGaai4waiaadIfacaGGDbGaaiykaiabgYda8iaaig daaaa@42A2@ , where [X] is the integer part of X. (Tail condition) C4: In f f(x)>0 f(x)>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamysai aad6gacaWGMbWaaSbaaKqbGeaacaWGMbGaaiikaiaadIhacaGGPaGa eyOpa4JaaGimaaqcfayabaGaamOzaiaacIcacaWG4bGaaiykaiabg6 da+iaaicdaaaa@4413@ (Tail condition) C5: f (lnf) 2 < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qaae aacaWGMbGaaiikaiGacYgacaGGUbGaamOzaiaacMcadaahaaqabKqb GeaacaaIYaaaaaqcfayabeqacqGHRiI8aiabgYda8iabg6HiLcaa@4189@ (Peak condition) (Note that, this is also a mild tail condition.) C6: f is bounded. (Peak condition)

Some suggested estimators for D 11 ( f 1 , f 1 )= f 1 (x)ln( f 1 (x) )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaamiramaaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaWG MbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBaaaju aibaGaaGymaaqcfayabaGaaiykaiabg2da9maapehabaGaamOzamaa BaaajuaibaGaaGymaaqcfayabaGaaiikaiaadIhacaGGPaGaciiBai aac6gadaqadaqaaiaadAgadaWgaaqcfasaaiaaigdaaKqbagqaaiaa cIcacaWG4bGaaiykaaGaayjkaiaawMcaaiaadsgacaWG4baajuaiba GaeyOeI0IaeyOhIukabaGaeyOhIukajuaGcqGHRiI8aaaa@5883@ may be found in the literature. These include the plug-in estimates of entropy which are based on a consistent density estimate f n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamOBaaqcfayabaaaaa@393F@ of f. For example, the integral estimate of entropy introduced by Dmitriev & Tarasenko [14]. Joe [15] considers estimating D 11 ( f 1 , f 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaamiramaaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaWG MbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBaaaju aibaGaaGymaaqcfayabaGaaiykaaaa@419C@ when f 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajqwba+FaaiaaigdaaKqbagqaaaaa@3ACA@ is a multivariate pdf, but he points out that the calculation when f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmOzay aajaWaaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@3917@ is a kernel estimator gets more difficult when the dimension of the integral is more than two. He therefore excludes the integral estimate from further study. The integral estimator can however be easily calculated if, for example, f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmOzay aajaWaaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@3917@ is a histogram.

The re-substitution estimate is proposed by Ahmad & Lin [16] as follows:

D ^ 11 ( f ^ 1 , f ^ 1 )= 1 n i=1 n ln f ^ 1 ( X i ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IabmirayaajaWaaSbaaKqbGeaacaaIXaGaaGymaaqcfayabaGaaiik aiqadAgagaqcamaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiqadA gagaqcamaaBaaajuaibaGaaGymaaqcfayabaGaaiykaiabg2da9iab gkHiTmaalaaabaGaaGymaaqaaiaad6gaaaWaaabCaeaaciGGSbGaai OBaiqadAgagaqcamaaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaa dIfadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMcacaGGSaaajuaiba GaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5aaaa@552D@ (3)

Where f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmOzay aajaWaaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@3917@ is a kernel density estimator? They showed the mean square consistency of (3), such that n lim  E{ ( D ^ 11 ( f ^ 1 , f ^ 1 ) D 11 ( f 1 , f 1 )) 2 }=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGda Wgaaqcfasaaiaad6gaaKqbagqaamaaFyaabaGaciiBaiaacMgacaGG TbaacaGLxdcadaWgaaqcfasaaiabg6HiLcqcfayabaGaaeiiaiaadw eacaGG7bGaaiikaiqadseagaqcamaaBaaajuaibaGaaGymaiaaigda aKqbagqaaiaacIcaceWGMbGbaKaadaWgaaqcfasaaiaaigdaaKqbag qaaiaacYcaceWGMbGbaKaadaWgaaqcfasaaiaaigdaaKqbagqaaiaa cMcacqGHsislcaWGebWaaSbaaKqbGeaacaaIXaGaaGymaaqcfayaba GaaiikaiaadAgadaWgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWG MbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGPaGaaiykamaaCaaabe qcfasaaiaaikdaaaqcfaOaaiyFaiabg2da9iaaicdaaaa@5E72@ Joe [15] considers the estimation of D 11 ( f 1 , f 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaamiramaaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaWG MbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBaaaju aibaGaaGymaaqcfayabaGaaiykaaaa@419C@ for multivariate pdfs by an entropy estimate of the re-substitution type (3), also based on a kernel density estimate. He obtained asymptotic bias and variance terms, and showed that non-unimodal kernels satisfying certain conditions can reduce the mean square error. His analysis and simulations suggest that the sample size needed for good estimates increases rapidly when the dimension of the multivariate density increases. His results rely heavily on conditions C4 and C6. Hall & Morton [17] investigated the properties of an estimator of the type (3) both when f n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamOBaaqcfayabaaaaa@393F@ is a histogram density estimator and when it is a kernel estimator. For the histogram estimation they showed that n lim   n 1/2 ( D ^ 11 ( f ^ 1 , f 1 ) D 11 ( f 1 , f 1 ))N(0, σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGda Wgaaqcfasaaiaad6gaaKqbagqaamaaFyaabaGaciiBaiaacMgacaGG TbaacaGLxdcadaWgaaqcfasaaiabg6HiLcqcfayabaGaaeiiaiaad6 gadaahaaqabKqbGeaacaaIXaGaai4laiaaikdaaaqcfaOaaiikaiqa dseagaqcamaaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcace WGMbGbaKaadaWgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWGMbWa aSbaaKqbGeaacaaIXaaajuaGbeaacaGGPaGaeyOeI0IaamiramaaBa aajuaibaGaaGymaiaaigdaaKqbagqaaiaacIcacaWGMbWaaSbaaKqb GeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBaaajuaibaGaaGymaa qcfayabaGaaiykaiaacMcacqWI8iIocaWGobGaaiikaiaaicdacaGG SaGaeq4Wdm3aaWbaaeqajuaibaGaaGOmaaaajuaGcaGGPaaaaa@6455@ under certain tail and smoothness conditions with σ 2 =Var(ln(f(X)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH9aqpcaWGwbGaamyyaiaa dkhacaGGOaGaciiBaiaac6gacaGGOaGaamOzaiaacIcacaWGybGaai ykaiaacMcaaaa@44A9@ .(4)

Other estimators using sampling-spacing are investigated by Tarasenko [18], Beirlant & van Zuijlen [19], Hall [20], Cressie [21], Dudewicz & van der Meulen [22], and Beirlant [23]. Finally, other nonparametric estimator has been discussed by many authors including Vasicek [24], Dudewicz & Van der Meulen [22], Bowman [25] and Alizadeh [26]. Among these various entropy estimators, Vasicek’s sample entropy has been most widely used in developing entropy based statistical procedures. However, deriving the asymptotic distribution for there is hard to establish. Therefore, in this paper we will adopt the kernel re-substitution estimate which is proposed by Ahmad & Lin [16].

We will adopt the notation of Samawi et al. [9]. Our proposed test of symmetry is as follow: Let X 1 , X 2 ...., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGyb WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiwamaaBaaajuai baGaaGOmaaqcfayabaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcaca WGybWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@4261@ be a random sample from absolutely continuous distribution F(.) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGgb Gaaiikaiaac6cacaGGPaaaaa@39C2@ which is continuously differentiable with uniformly bounded derivatives and having known median.

Let K be a kernel function satisfying the condition K(x)dx=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWdXa qaaiaadUeacaGGOaGaamiEaiaacMcacaWGKbGaamiEaiabg2da9iaa igdaaKqbGeaacqGHsislcqGHEisPaeaacqGHEisPaKqbakabgUIiYd aaaa@440A@

For simplicity, the kernel K will be assumed to be a symmetric density function with mean 0 and finite variance; an example is the standard normal density. The kernel estimators for f( w i ) and f( w i ),i=1,2,...,C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGMb GaaiikaiaadEhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMcacaqG GaGaaeyyaiaab6gacaqGKbGaaeiiaiaadAgacaGGOaGaeyOeI0Iaam 4DamaaBaaajuaibaGaamyAaaqcfayabaGaaiykaiaacYcacaWGPbGa eyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlai aacYcacaWGdbaaaa@4EAF@ , are:

f ^ K ( w i )= 1 nh   j=1 n K( w i x j h ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGceWGMb GbaKaadaWgaaqaaiaadUeaaeqaaiaacIcacqGHsislcaWG3bWaaSba aKqbGeaacaWGPbaajuaGbeaacaGGPaGaeyypa0ZaaSaaaeaacaaIXa aabaGaamOBaiaadIgaaaGaaeiiamaaqahabaGaam4samaabmaabaWa aSaaaeaacqGHsislcaWG3bWaaSbaaKqbGeaacaWGPbaajuaGbeaacq GHsislcaWG4bWaaSbaaKqbGeaacaWGQbaajuaGbeaaaeaacaWGObaa aaGaayjkaiaawMcaaaqcfasaaiaadQgacqGH9aqpcaaIXaaabaGaam OBaaqcfaOaeyyeIuoaaaa@5337@ (6)

and

f ^ K ( w i )= 1 nh   j=1 n K( w i x j h ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGceWGMb GbaKaadaWgaaqcfasaaiaadUeaaKqbagqaaiaacIcacaWG3bWaaSba aKqbGeaacaWGPbaajuaGbeaacaGGPaGaeyypa0ZaaSaaaeaacaaIXa aabaGaamOBaiaadIgaaaGaaeiiamaaqahabaGaam4samaabmaabaWa aSaaaeaacaWG3bWaaSbaaKqbGeaacaWGPbaajuaGbeaacqGHsislca WG4bWaaSbaaKqbGeaacaWGQbaajuaGbeaaaeaacaWGObaaaaGaayjk aiaawMcaaiaacYcaaKqbGeaacaWGQbGaeyypa0JaaGymaaqaaiaad6 gaaKqbakabggHiLdaaaa@52C9@

(7)

Respectively, where C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdb aaaa@3769@ is the number of bins and depends on the sample size. As in Samawi et al. [9], we suggest to take the integer of C= n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdb Gaeyypa0ZaaOaaaeaacaWGUbaabeaaaaa@3972@ . In addition, h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@3700@ is the bandwidths of the kernel estimators satisfying the conditions that h>0,h0 and (nh) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGOb GaeyOpa4JaaGimaiaacYcacaWGObGaeyOKH4QaaGimaiaabccacaqG HbGaaeOBaiaabsgacaqGGaGaaeikaiaad6gacaWGObGaeyOKH4Qaey OhIuQaaiykaaaa@482C@ as n  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUb GaeyOKH4QaeyOhIuQaaeiiaaaa@3B95@ . There are many choices of the bandwidths ( h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGOb aaaa@378E@ ). In our procedure we use the method suggested by Silverman [27] Using the normal distribution as the parametric family, the bandwidths of the kernel estimators are h=0.9A (n) 1/5   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGOb Gaeyypa0JaaGimaiaac6cacaaI5aGaamyqaiaacIcacaWGUbGaaiyk amaaCaaabeqcfasaaiabgkHiTiaaigdacaGGVaGaaGynaaaajuaGca qGGaaaaa@4270@ , (8)

Where A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbaaaa@36D9@ =min{standard deviation of ( x 1 , x 2 ...., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWG4b WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiEamaaBaaajuai baGaaGOmaaqcfayabaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcaca WG4bWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@42C1@ ), interquantile range of ( x 1 , x 2 ...., x n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWG4b WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiEamaaBaaajuai baGaaGOmaaqcfayabaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcaca WG4bWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@42C1@ )/1.349}. This form of (8) is found to be adequate choices of the bandwidth for many purposes which minimizes the integrated mean squared error (IMSE),

IMSE= E [ f ^ K (x)f(x)] 2 dx. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGjb GaamytaiaadofacaWGfbGaeyypa0Zaa8qaaeaacaWGfbGaai4waiqa dAgagaqcamaaBaaajuaibaGaam4saaqcfayabaGaaiikaiaadIhaca GGPaGaeyOeI0IaamOzaiaacIcacaWG4bGaaiykaiaac2fadaahaaqa bKqbGeaacaaIYaaaaKqbakaadsgacaWG4bGaaiOlaaqabeqacqGHRi I8aaaa@4CB4@

We will use the Samawi et al. [9] suggestion to calculate the bins as follows: Let R=range( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGsb Gaeyypa0JaamOCaiaadggacaWGUbGaam4zaiaadwgacaGGOaGaamiE amaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadIhadaWgaaqcfa saaiaaikdaaKqbagqaaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGa amiEamaaBaaajuaibaGaamOBaaqcfayabaGaaiykaaaa@4A9B@ , then bins will be selected as w i = w i1 + δ x , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWG3b WaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH9aqpcaWG3bWaaSbaaKqb GeaacaWGPbGaeyOeI0IaaGymaaqcfayabaGaey4kaSIaeqiTdq2aaS baaKqbGeaacaWG4baajuaGbeaacaGGSaaaaa@43EE@ where i=2,...,C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGPb Gaeyypa0JaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam4q aaaa@3D8F@ , w 1 =min( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Dam aaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaciyBaiaacMgacaGG UbGaaiikaiaadIhadaWgaaqcfasaaiaaigdaaKqbagqaaiaacYcaca WG4bWaaSbaaKqbGeaacaaIYaaajuaGbeaacaGGSaGaaiOlaiaac6ca caGGUaGaaiilaiaadIhadaWgaaqcfauaaiaad6gaaKqbagqaaiaacM caaaa@4A87@ and δ x = R C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH0o azdaWgaaqcfasaaiaadIhaaKqbagqaaiabg2da9maalaaabaGaamOu aaqaaiaadoeaaaaaaa@3CD5@ .

Using the above kernel estimator, the nonparametric kernel estimator of D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaadjuaGca WGebGaaiikaiaadAgadaWgaaqcfasaaiaaigdaaKqbagqaaiaacYca caWGMbWaaSbaaKqbGeaacaaIYaaajuaGbeaacaGGPaaaaa@3E6C@ under the null hypothesis is given by

D ^ = f ^ K (x)ln( f ^ K (x) f ^ K (x) )dx , =  D ^ 11 ( f ^ K (x), f ^ K (x)) D ^ 12 ( f ^ K (x), f ^ K (x)), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaGaeyypa0Zaa8qCaeaaceWGMbGbaKaadaWgaaqcfasaaiaadUea aKqbagqaaiaacIcacaWG4bGaaiykaiGacYgacaGGUbWaaeWaaeaada WcaaqaaiqadAgagaqcamaaBaaajuaibaGaam4saaqcfayabaGaaiik aiaadIhacaGGPaaabaGabmOzayaajaWaaSbaaKqbGeaacaWGlbaaju aGbeaacaGGOaGaeyOeI0IaamiEaiaacMcaaaaacaGLOaGaayzkaaGa amizaiaadIhaaeaaaeaaaiabgUIiYdGaaiilaiaabccacaqG9aGaae iiaiqadseagaqcamaaBaaajuaibaGaaGymaiaaigdaaKqbagqaaiaa cIcaceWGMbGbaKaadaWgaaqcfasaaiaadUeaaKqbagqaaiaacIcaca WG4bGaaiykaiaacYcaceWGMbGbaKaadaWgaaqcfasaaiaadUeaaKqb agqaaiaacIcacaWG4bGaaiykaiaacMcacqGHsislceWGebGbaKaada WgaaqcfasaaiaaigdacaaIYaaajuaGbeaacaGGOaGabmOzayaajaWa aSbaaKqbGeaacaWGlbaajuaGbeaacaGGOaGaamiEaiaacMcacaGGSa GabmOzayaajaWaaSbaaKqbGeaacaWGlbaajuaGbeaacaGGOaGaeyOe I0IaamiEaiaacMcacaGGPaGaaiilaaaa@73A1@

Which can be approximated by?

D ^ = 1 C i=1 C ln f ^ K ( w i ) 1 C i=1 C ln f ^ K ( w i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaGaeyypa0ZaaSaaaeaacaaIXaaabaGaam4qaaaadaaeWbqaaiGa cYgacaGGUbGabmOzayaajaWaaSbaaKqbGeaacaWGlbaajuaGbeaaca GGOaGaam4DamaaBaaajuaibaGaamyAaaqcfayabaGaaiykaiabgkHi TmaalaaabaGaaGymaaqaaiaadoeaaaWaaabCaeaaciGGSbGaaiOBai qadAgagaqcamaaBaaajuaibaGaam4saaqcfayabaGaaiikaiabgkHi TiaadEhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMcaaKqbGeaaca WGPbGaeyypa0JaaGymaaqaaiaadoeaaKqbakabggHiLdaajuaibaGa amyAaiabg2da9iaaigdaaeaacaWGdbaajuaGcqGHris5aaaa@5B75@

The approximate variance of D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaaaaa@375D@ is given by

Var( D ^ )= Var( i=1 C ln f ^ K ( w i )) C 2 + Var( i=1 C ln f ^ K ( w i )) C 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOvai aadggacaWGYbGaaiikaiqadseagaqcaiaacMcacqGH9aqpdaWcaaqa aiaadAfacaWGHbGaamOCaiaacIcadaaeWbqaaiGacYgacaGGUbGabm OzayaajaWaaSbaaKqbGeaacaWGlbaajuaGbeaacaGGOaGaam4Damaa BaaajuaibaGaamyAaaqcfayabaGaaiykaiaacMcaaKqbGeaacaWGPb Gaeyypa0JaaGymaaqaaiaadoeaaKqbakabggHiLdaabaGaam4qamaa CaaabeqcfasaaiaaikdaaaaaaKqbakabgUcaRmaalaaabaGaamOvai aadggacaWGYbGaaiikamaaqahabaGaciiBaiaac6gaceWGMbGbaKaa daWgaaqcfasaaiaadUeaaKqbagqaaiaacIcacqGHsislcaWG3bWaaS baaKqbGeaacaWGPbaajuaGbeaacaGGPaGaaiykaaqcfasaaiaadMga cqGH9aqpcaaIXaaabaGaam4qaaqcfaOaeyyeIuoaaeaacaWGdbWaaW baaeqajuaibaGaaGOmaaaaaaqcfaOaaiOlaaaa@6A0D@

Asymptotic properties of D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaaaaa@375D@

The nonparametric kernel estimator of D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ ( D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaaaaa@375D@ ) is based on the univariate kernel for density estimation, K: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8cspy0xbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak aadUeacaGG6aGaeSyhHeQaeyOKH4QaeSyhHekaaa@3EB1@ . The necessary regularity conditions imposed on the univariate kernel for density estimation are:

I. R K(z)dz=1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWdra qaaiaadUeacaGGOaGaamOEaiaacMcacaWGKbGaamOEaiabg2da9iaa igdacaGGUaaajuaibaGaamOuaaqcfayabiabgUIiYdaaaa@41A9@

II. R z β K(z)dz=0 for any β=1,..., r 1, and  R |z | r K(z)dz<.   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWdra qaaiaadQhadaahaaqabKqbGeaacqaHYoGyaaqcfaOaam4saiaacIca caWG6bGaaiykaiaadsgacaWG6bGaeyypa0JaaGimaiaabccacaqGMb Gaae4BaiaabkhacaqGGaGaaeyyaiaab6gacaqG5bGaaeiiaiabek7a Ijabg2da9iaaigdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaabc cacaWGYbaajuaibaGaamOuaaqcfayabiabgUIiYdGaeyOeI0IaaGym aiaabYcacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiamaapebabaGaai iFaiaadQhacaGG8bWaaWbaaeqajuaibaGaamOCaaaajuaGcaWGlbGa aiikaiaadQhacaGGPaGaamizaiaadQhacqGH8aapcqGHEisPcaGGUa aajuaibaGaamOuaaqcfayabiabgUIiYdGaaeiiaaaa@6BFC@

III. R= R K 2 (z)dz<. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGsb Gaeyypa0Zaa8qeaeaacaWGlbWaaWbaaeqajuaibaGaaGOmaaaajuaG caGGOaGaamOEaiaacMcacaWGKbGaamOEaiabgYda8iabg6HiLkaac6 caaKqbGeaacaWGsbaajuaGbeGaey4kIipaaaa@45D4@

IV. h>0,h0 , (nh) and ( nh logn ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGOb GaeyOpa4JaaGimaiaacYcacaWGObGaeyOKH4QaaGimaiaabccacaqG SaGaaeiiaiaabIcacaWGUbGaamiAaiabgkziUkabg6HiLkaacMcaca qGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaabIcadaWcaaqaaiaad6ga caWGObaabaGaciiBaiaac+gacaGGNbGaamOBaaaacqGHsgIRcqGHEi sPcaGGPaaaaa@548A@

These conditions may be found in Silverman [27] (Chapter 3) or Wand & Jones [28] (Chapter 2).

To show consistency of , apply the kernel density asymptotic properties found in Silverman [27], (Chapter 3) or Wand & Jones [28], (Chapter 2). Under assumptions 1-4 and assuming that the density f: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8cspy0xbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXx e9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbak aadAgacaGG6aGaeSyhHeQaeyOKH4QaeSyhHekaaa@3ECC@ is continuous at each w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWG3b WaaSbaaKqbGeaacaWGPbaajuaGbeaaaaa@3968@ , i=1, 2,… C,

Bias( f ^ K ( w i ))=o (1)  and  Bias( f ^ K ( w i ))=o (1) + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGcb GaamyAaiaadggacaWGZbGaaiikaiqadAgagaqcamaaBaaajuaibaGa am4saaqcfayabaGaaiikaiabgkHiTiaadEhadaWgaaqcfasaaiaadM gaaKqbagqaaiaacMcacaGGPaGaeyypa0Jaam4BaiaacIcacaaIXaGa aiykamaaBaaabaGaeyOeI0cabeaacaqGGaGaaeyyaiaab6gacaqGKb GaaeiiaiaabccacaWGcbGaamyAaiaadggacaWGZbGaaiikaiqadAga gaqcamaaBaaajuaibaGaam4saaqcfayabaGaaiikaiaadEhadaWgaa qcfasaaiaadMgaaKqbagqaaiaacMcacaGGPaGaeyypa0Jaam4Baiaa cIcacaaIXaGaaiykamaaBaaabaGaey4kaScabeaaaaa@5DC8@ (12)

Var( f ^ K ( w i ))= f( w i ) nh K 2 (z)dz+o( 1 n h ) and Var( f ^ K ( w i ))= f( w i ) nh K 2 (z)dz+o( 1 nh ),  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGwb GaamyyaiaadkhacaGGOaGabmOzayaajaWaaSbaaKqbGeaacaWGlbaa juaGbeaacaGGOaGaeyOeI0Iaam4DamaaBaaajuaibaGaamyAaaqcfa yabaGaaiykaiaacMcacqGH9aqpdaWcaaqaaiaadAgacaGGOaGaeyOe I0Iaam4DamaaBaaajuaibaGaamyAaaqcfayabaGaaiykaaqaaiaad6 gacaWGObaaamaapebabaGaam4samaaCaaabeqcfasaaiaaikdaaaqc faOaaiikaiaadQhacaGGPaGaamizaiaadQhacqGHRaWkcaWGVbGaai ikamaalaaabaGaaGymaaqaaiaad6gacaWGObWaaSbaaeaaaeqaaaaa aKqbGeaacqWIDesOaKqbagqacqGHRiI8aiaabMcacaqGGaGaaeyyai aab6gacaqGKbGaaeiiaiaadAfacaWGHbGaamOCaiaacIcaceWGMbGb aKaadaWgaaqcfasaaiaadUeaaKqbagqaaiaacIcacaWG3bWaaSbaaK qbGeaacaWGPbaajuaGbeaacaGGPaGaaiykaiabg2da9maalaaabaGa amOzaiaacIcacaWG3bWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGPa aabaGaamOBaiaadIgaaaWaa8qeaeaacaWGlbWaaWbaaeqajuaibaGa aGOmaaaajuaGcaGGOaGaamOEaiaacMcacaWGKbGaamOEaiabgUcaRi aad+gacaGGOaWaaSaaaeaacaaIXaaabaGaamOBaiaadIgaaaaajuai baGaeSyhHekajuaGbeGaey4kIipacaqGPaGaaeilaiaabccaaaa@84E6@ (13)

and for h>0,h0 and (nh) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGOb GaeyOpa4JaaGimaiaacYcacaWGObGaeyOKH4QaaGimaiaabccacaqG HbGaaeOBaiaabsgacaqGGaGaaeikaiaad6gacaWGObGaeyOKH4Qaey OhIuQaaiykaaaa@482C@ as n  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUb GaeyOKH4QaeyOhIuQaaeiiaaaa@3B95@

f ^ K ( w i ) P f( w i ) and  f ^ K ( w i ) P f( w i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGceWGMb GbaKaadaWgaaqcfasaaiaadUeaaKqbagqaaiaacIcacqGHsislcaWG 3bWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGPaGaeyOKH46aaWbaae qabaGaamiuaaaacaWGMbGaaiikaiabgkHiTiaadEhadaWgaaqcfasa aiaadMgaaKqbagqaaiaacMcacaqGGaGaaeyyaiaab6gacaqGKbGaae iiaiqadAgagaqcamaaBaaajuaibaGaam4saaqcfayabaGaaiikaiaa dEhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMcacqGHsgIRdaahaa qabeaacaWGqbaaaiaadAgacaGGOaGaam4DamaaBaaajuaibaGaamyA aaqcfayabaGaaiykaaaa@59EB@ If f(.) uniformly continuous, then the kernel density estimate is strongly consistent. Moreover, as in Ahmad & Lin [16], C lim  E{ ( D ^ 11 ( f ^ K (x), f ^ K (x)) D 11 ( f K (x), f K (x))) 2 }=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGda WgaaqcfasaaiaadoeaaKqbagqaamaaFyaabaGaciiBaiaacMgacaGG TbaacaGLxdcadaWgaaqcfasaaiabg6HiLcqcfayabaGaaeiiaiaadw eacaGG7bGaaiikaiqadseagaqcamaaBaaajuaibaGaaGymaiaaigda aKqbagqaaiaacIcaceWGMbGbaKaadaWgaaqcfasaaiaadUeaaKqbag qaaiaacIcacaWG4bGaaiykaiaacYcaceWGMbGbaKaadaWgaaqcfasa aiaadUeaaKqbagqaaiaacIcacaWG4bGaaiykaiaacMcacqGHsislca WGebWaaSbaaKqbGeaacaaIXaGaaGymaaqcfayabaGaaiikaiaadAga daWgaaqcfasaaiaadUeaaKqbagqaaiaacIcacaWG4bGaaiykaiaacY cacaWGMbWaaSbaaKqbGeaacaWGlbaajuaGbeaacaGGOaGaamiEaiaa cMcacaGGPaGaaiykamaaCaaabeqcfasaaiaaikdaaaqcfaOaaiyFai abg2da9iaaicdacaGGSaaaaa@68A3@ and hence D ^ 11 ( f ^ K (x), f ^ K (x)) P D 11 ( f K (x), f K (x)), as C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaWaaSbaaKqbGeaacaaIXaGaaGymaaqcfayabaGaaiikaiqadAga gaqcamaaBaaajuaibaGaam4saaqcfayabaGaaiikaiaadIhacaGGPa GaaiilaiqadAgagaqcamaaBaaajuaibaGaam4saaqcfayabaGaaiik aiaadIhacaGGPaGaaiykamaaoqcabaGaamiuaaqabiaawkziaiaads eadaWgaaqcfasaaiaaigdacaaIXaaajuaGbeaacaGGOaGaamOzamaa BaaajuaibaGaam4saaqcfayabaGaaiikaiaadIhacaGGPaGaaiilai aadAgadaWgaaqcfasaaiaadUeaaKqbagqaaiaacIcacaWG4bGaaiyk aiaacMcacaGGSaGaaeiiaiaabggacaqGZbGaaeiiaiaadoeacqGHsg IRcqGHEisPaaa@5F04@ and . However, since D ^ = D ^ 11 ( f ^ K (x), f ^ K (x)) D ^ 12 ( f ^ K (x), f ^ K (x)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaGaeyypa0JabmirayaajaWaaSbaaKqbGeaacaaIXaGaaGymaaqc fayabaGaaiikaiqadAgagaqcamaaBaaajuaibaGaam4saaqcfayaba GaaiikaiaadIhacaGGPaGaaiilaiqadAgagaqcamaaBaaajuaibaGa am4saaqcfayabaGaaiikaiaadIhacaGGPaGaaiykaiabgkHiTiqads eagaqcamaaBaaajuaibaGaaGymaiaaikdaaKqbagqaaiaacIcaceWG MbGbaKaadaWgaaqcfasaaiaadUeaaKqbagqaaiaacIcacaWG4bGaai ykaiaacYcaceWGMbGbaKaadaWgaaqcfasaaiaadUeaaKqbagqaaiaa cIcacqGHsislcaWG4bGaaiykaiaacMcaaaa@58A0@ therefore D ^ p D(f(w),f(w)), as C. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaWaa4ajaeaacaWGWbaabeGaayPKHaGaamiraiaacIcacaWGMbGa aiikaiaadEhacaGGPaGaaiilaiaadAgacaGGOaGaeyOeI0Iaam4Dai aacMcacaGGPaGaaiilaiaabccacaqGHbGaae4CaiaabccacaWGdbGa eyOKH4QaeyOhIuQaaiOlaaaa@4CBC@

To drive the asymptotic distribution of D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaaaaa@375D@ , we will define D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ as a functional

D( f 1 , f 2 )= f 1 (w)ln( f 1 (w))dw f 1 (w)ln f 2 (w)dw= ln( f 1 (w))d F 1 ln f 2 (w)d F 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaiabg2da9maapehaba GaamOzamaaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadEhacaGG PaGaciiBaiaac6gacaGGOaaajuaibaGaeyOeI0IaeyOhIukabaGaey OhIukajuaGcqGHRiI8aiaadAgadaWgaaqcfasaaiaaigdaaKqbagqa aiaacIcacaWG3bGaaiykaiaacMcacaWGKbGaam4DaiabgkHiTmaape habaGaamOzamaaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadEha caGGPaGaciiBaiaac6gaaKqbGeaacqGHsislcqGHEisPaeaacqGHEi sPaKqbakabgUIiYdGaamOzamaaBaaajuaibaGaaGOmaaqcfayabaGa aiikaiaadEhacaGGPaGaamizaiaadEhacqGH9aqpdaWdXbqaaiGacY gacaGGUbGaaiikaaqcfasaaiabgkHiTiabg6HiLcqaaiabg6HiLcqc faOaey4kIipacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGOa Gaam4DaiaacMcacaGGPaGaamizaiaadAeadaWgaaqcfasaaiaaigda aKqbagqaaiabgkHiTmaapehabaGaciiBaiaac6gaaKqbGeaacqGHsi slcqGHEisPaeaacqGHEisPaKqbakabgUIiYdGaamOzamaaBaaajuai baGaaGOmaaqcfayabaGaaiikaiaadEhacaGGPaGaamizaiaadAeada WgaaqcfasaaiaaigdaaKqbagqaaaaa@8F69@

Using the previously stated regularity conditions, some regularity conditions given by Serfing [29] and assuming that the Gteuax derivatives of the functional D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ exist, we can show that the partial influence function of the functional D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ [30] are as follows:

L 1 (w; F 1 , F 1 )=ln f 1 (w) f 1 (w)ln f 1 (w)dw, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadEhacaGG7aGaamOr amaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadAeadaWgaaqcfa saaiaaigdaaKqbagqaaiaacMcacqGH9aqpciGGSbGaaiOBaiaadAga daWgaaqcfasaaiaaigdaaKqbagqaaiaacIcacaWG3bGaaiykaiabgk HiTmaapehabaGaamOzamaaBaaajuaibaGaaGymaaqcfayabaGaaiik aiaadEhacaGGPaGaciiBaiaac6gaaKqbGeaacqGHsislcqGHEisPae aacqGHEisPaKqbakabgUIiYdGaamOzamaaBaaajuaibaGaaGymaaqc fayabaGaaiikaiaadEhacaGGPaGaamizaiaadEhacaGGSaaaaa@5F2B@

and

L 2 (w; F 1 , F 2 )=ln f 2 (w) f 1 (w)ln f 2 (w)dw. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aaBaaajuaibaGaaGOmaaqcfayabaGaaiikaiaadEhacaGG7aGaamOr amaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiaadAeadaWgaaqcfa saaiaaikdaaKqbagqaaiaacMcacqGH9aqpciGGSbGaaiOBaiaadAga daWgaaqcfasaaiaaikdaaKqbagqaaiaacIcacaWG3bGaaiykaiabgk HiTmaapehabaGaamOzamaaBaaajuaibaGaaGymaaqcfayabaGaaiik aiaadEhacaGGPaGaciiBaiaac6gaaKqbGeaacqGHsislcqGHEisPae aacqGHEisPaKqbakabgUIiYdGaamOzamaaBaaajuaibaGaaGOmaaqc fayabaGaaiikaiaadEhacaGGPaGaamizaiaadEhacaGGUaaaaa@5F31@

Note that L 1 (w; F 1 (w), F 1 (w)) d F 1 (w)=0 and   L 2 (w; F 1 (w), F 2 (w)) d F 1 (w)=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qaae aacaWGmbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGOaGaam4Daiaa cUdacaWGgbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGOaGaam4Dai aacMcacaGGSaGaamOramaaBaaajuaibaGaaGymaaqcfayabaGaaiik aiaadEhacaGGPaGaaiykaaqabeqacqGHRiI8aiaadsgacaWGgbWaaS baaKqbGeaacaaIXaaajuaGbeaacaGGOaGaam4DaiaacMcacqGH9aqp caaIWaGaaeiiaiaabggacaqGUbGaaeizaiaabccacaqGGaWaa8qaae aacaWGmbWaaSbaaKqbGeaacaaIYaaajuaGbeaacaGGOaGaam4Daiaa cUdacaWGgbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGOaGaam4Dai aacMcacaGGSaGaamOramaaBaaajuaibaGaaGOmaaqcfayabaGaaiik aiaadEhacaGGPaGaaiykaaqabeqacqGHRiI8aiaadsgacaWGgbWaaS baaKqbGeaacaaIXaaajuaGbeaacaGGOaGaam4DaiaacMcacqGH9aqp caaIWaGaaiOlaaaa@6D9F@ Now using this functional representation of D( f 1 , f 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai aacIcacaWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOz amaaBaaajuaibaGaaGOmaaqcfayabaGaaiykaaaa@3E5D@ , then as in Samawi et al. [30] and Serfing [29],

C ( D ^ D( f 1 , f 2 )) L N(0, σ D ^ 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacaWGdbaabeaacaGGOaGabmirayaajaGaeyOeI0IaamiraiaacIca caWGMbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamOzamaaBa aajuaibaGaaGOmaaqcfayabaGaaiykaiaacMcadaGdKaqaaiaadYea aeqacaGLsgcacaWGobGaaiikaiaaicdacaGGSaGaeq4Wdm3aa0baaK qbGeaaceWGebGbaKaaaeaacaaIYaaaaKqbakaacMcacaGGSaaaaa@4D24@

where σ D ^ 2 = L 1 2 (w; F 1 , F 1 )d F 1 + L 2 2 (w; F 1 , F 2 )d F 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aa0baaKqbGeaaceWGebGbaKaaaeaacaaIYaaaaKqbakabg2da9maa peaabaGaamitamaaDaaajuaibaGaaGymaaqaaiaaikdaaaqcfaOaai ikaiaadEhacaGG7aGaamOramaaBaaajuaibaGaaGymaaqcfayabaGa aiilaiaadAeadaWgaaqcfasaaiaaigdaaKqbagqaaiaacMcacaWGKb GaamOramaaBaaajuaibaGaaGymaaqcfayabaaabeqabiabgUIiYdGa ey4kaSYaa8qaaeaacaWGmbWaa0baaKqbGeaacaaIYaaabaGaaGOmaa aajuaGcaGGOaGaam4DaiaacUdacaWGgbWaaSbaaKqbGeaacaaIXaaa juaGbeaacaGGSaGaamOramaaBaaajuaibaGaaGOmaaqcfayabaGaai ykaiaadsgacaWGgbWaaSbaaKqbGeaacaaIXaaajuaGbeaaaeqabeGa ey4kIipaaaa@5E62@

A consistent estimate for σ D ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aa0baaKqbGeaaceWGebGbaKaaaeaacaaIYaaaaaaa@3A2C@ is given by σ ^ D ^ 2 = 1 C i=1 C L 1 2 (w; F ^ 1 , F ^ 1 ) + 1 C i=1 C L 2 2 (w; F ^ 1 , F ^ 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafq4Wdm NbaKaadaqhaaqcfasaaiqadseagaqcaaqaaiaaikdaaaqcfaOaeyyp a0ZaaSaaaeaacaaIXaaabaGaam4qaaaadaaeWbqaaiaadYeadaqhaa qcfasaaiaaigdaaeaacaaIYaaaaKqbakaacIcacaWG3bGaai4oaiqa dAeagaqcamaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiqadAeaga qcamaaBaaajuaibaGaaGymaaqcfayabaGaaiykaaqcfasaaiaadMga cqGH9aqpcaaIXaaabaGaam4qaaqcfaOaeyyeIuoacqGHRaWkdaWcaa qaaiaaigdaaeaacaWGdbaaamaaqahabaGaamitamaaDaaajuaibaGa aGOmaaqaaiaaikdaaaqcfaOaaiikaiaadEhacaGG7aGabmOrayaaja WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGabmOrayaajaWaaSba aKqbGeaacaaIYaaajuaGbeaacaGGPaaajuaibaGaamyAaiabg2da9i aaigdaaeaacaWGdbaajuaGcqGHris5aiaacYcaaaa@64E4@

Where, L 1 ( w i ; F ^ 1 , F ^ 1 )=ln f ^ 1 ( w i ) D ^ 11 ( f ^ 1 ( w i ), f ^ 1 ( w i )) and  L 2 ( w i ; F ^ 1 , F ^ 2 )=ln f ^ 2 ( w i ) D ^ 12 ( f ^ 1 ( w i ), f ^ 2 ( w i )),i=1,2,...,C, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitam aaDaaajuaibaGaaGymaaqcfayaaaaacaGGOaGaam4DamaaBaaajuai baGaamyAaaqcfayabaGaai4oaiqadAeagaqcamaaBaaajuaibaGaaG ymaaqcfayabaGaaiilaiqadAeagaqcamaaBaaajuaibaGaaGymaaqc fayabaGaaiykaiabg2da9iGacYgacaGGUbGabmOzayaajaWaaSbaaK qbGeaacaaIXaaajuaGbeaacaGGOaGaam4DamaaBaaajuaibaGaamyA aaqcfayabaGaaiykaiabgkHiTiqadseagaqcamaaBaaajuaibaGaaG ymaiaaigdaaKqbagqaaiaacIcaceWGMbGbaKaadaWgaaqcfasaaiaa igdaaKqbagqaaiaacIcacaWG3bWaaSbaaKqbGeaacaWGPbaajuaGbe aacaGGPaGaaiilaiqadAgagaqcamaaBaaajuaibaGaaGymaaqcfaya baGaaiikaiaadEhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMcaca GGPaGaaeiiaiaabggacaqGUbGaaeizaiaabccacaWGmbWaa0baaKqb GeaacaaIYaaajuaGbaaaaiaacIcacaWG3bWaaSbaaKqbGeaacaWGPb aajuaGbeaacaGG7aGabmOrayaajaWaaSbaaKqbGeaacaaIXaaajuaG beaacaGGSaGabmOrayaajaWaaSbaaKqbGeaacaaIYaaajuaGbeaaca GGPaGaeyypa0JaciiBaiaac6gaceWGMbGbaKaadaWgaaqcfasaaiaa ikdaaKqbagqaaiaacIcacaWG3bWaaSbaaKqbGeaacaWGPbaajuaGbe aacaGGPaGaeyOeI0IabmirayaajaWaaSbaaKqbGeaacaaIXaGaaGOm aaqcfayabaGaaiikaiqadAgagaqcamaaBaaajuaibaGaaGymaaqcfa yabaGaaiikaiaadEhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacMca caGGSaGabmOzayaajaWaaSbaaKqbGeaacaaIYaaajuaGbeaacaGGOa Gaam4DamaaBaaajuaibaGaamyAaaqcfayabaGaaiykaiaacMcacaGG SaGaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaai Olaiaac6cacaGGSaGaam4qaiaacYcaaaa@984C@

Where in our case f 1 ( w i )=f( w i ) and  f 2 ( w i )=f( w i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaGaaiikaiaadEhadaWgaaqcfasa aiaadMgaaKqbagqaaiaacMcacqGH9aqpcaWGMbGaaiikaiaadEhada WgaaqcfasaaiaadMgaaKqbagqaaiaacMcacaqGGaGaaeyyaiaab6ga caqGKbGaaeiiaiaadAgadaWgaaqcfasaaiaaikdaaKqbagqaaiaacI cacaWG3bWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGPaGaeyypa0Ja amOzaiaacIcacqGHsislcaWG3bWaaSbaaKqbGeaacaWGPbaajuaGbe aacaGGPaaaaa@54DC@ .

For discussions about different methods addressing the issue of the performance of kernel density estimation at the boundary, see Hall & Park [31].

Simulation Study

As in Samawi et al. [9], to gain some insight of our procedure, a simulation study was conducted to investigate the performance of our new test of symmetry based on D ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiray aajaaaaa@375D@ . We compared our proposed test of symmetry with the test proposed by McWilliams [5], Modarres & Gastwirth [32], Mira [8] Bonferroni’s test, and Samawi et al. [9] tests of symmetry.

As in McWilliams [5], the runs test is described as follows: For any random sample of size n, let Y (1) , Y (2),... , Y (n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGzb WaaSbaaKqbGeaacaGGOaGaaGymaiaacMcaaKqbagqaaiaacYcacaWG zbWaaSbaaeaajuaicaGGOaGaaGOmaiaacMcajuaGcaGGSaGaaiOlai aac6cacaGGUaGaaeiiaiaacYcaaeqaaiaadMfadaWgaaqcfasaaiaa cIcacaWGUbGaaiykaaqabaaaaa@4682@ denote the sample values ordered from the smallest to largest according to their absolute value (signs are retained), and S 1 , S 2,..., S n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGtb WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaam4uamaaBaaabaqc faIaaGOmaKqbakaacYcacaGGUaGaaiOlaiaac6cacaGGSaaabeaaca WGtbWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@4250@ denote indicator variables designating the sign of the Y (j) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGzb WaaSbaaKqbGeaacaGGOaGaamOAaiaacMcaaeqaaaaa@3A16@ values [ S j =1 if  Y (j)  is nonnegative, 0 otherwise MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGtb WaaSbaaKqbGeaacaWGQbaajuaGbeaacqGH9aqpcaaIXaGaaeiiaiaa bMgacaqGMbGaaeiiaiaadMfadaWgaaqcfasaaiaacIcacaWGQbGaai ykaaqabaqcfaOaaeiiaiaabMgacaqGZbGaaeiiaiaab6gacaqGVbGa aeOBaiaab6gacaqGLbGaae4zaiaabggacaqG0bGaaeyAaiaabAhaca qGLbGaaeilaiaabccacaqGWaGaaeiiaiaab+gacaqG0bGaaeiAaiaa bwgacaqGYbGaae4DaiaabMgacaqGZbGaaeyzaaaa@5AA8@ ]. Thus, the test statistic used for testing symmetry is R * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGsb Waa0baaeaaaKqbGeaacaGGQaaaaaaa@3876@ = the number of runs in S 1 , S 2,..., S n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGtb WaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaam4uamaaBaaabaqc faIaaGOmaKqbakaacYcacaGGUaGaaiOlaiaac6cacaGGSaaabeaaca WGtbWaaSbaaKqbGeaacaWGUbaajuaGbeaaaaa@4250@ sequence= 1+ j=2 n I j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaaIXa Gaey4kaSYaaabCaeaacaWGjbWaaSbaaKqbGeaacaWGQbaajuaGbeaa aKqbGeaacaWGQbGaeyypa0JaaGOmaaqaaiaad6gaaKqbakabggHiLd aaaa@4162@ , where

I j ={ 0   if  S j = S j1 1   if  S j S j1   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGjb WaaSbaaKqbGeaacaWGQbaajuaGbeaacqGH9aqpdaGabaqaauaabeqa ceaaaeaacaaIWaGaaeiiaiaabccacaqGGaGaaeyAaiaabAgacaqGGa Gaam4uamaaBaaajuaibaGaamOAaaqcfayabaGaeyypa0Jaam4uamaa BaaajuaibaGaamOAaiabgkHiTiaaigdaaKqbagqaaaqaaiaaigdaca qGGaGaaeiiaiaabccacaqGPbGaaeOzaiaabccacaWGtbWaaSbaaKqb GeaacaWGQbaajuaGbeaacqGHGjsUcaWGtbWaaSbaaKqbGeaacaWGQb GaeyOeI0IaaGymaaqcfayabaGaaeiiaaaaaiaawUhaaaaa@56EF@

We reject the null hypothesis if R * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGsb Waa0baaeaaaKqbGeaacaGGQaaaaaaa@3876@ is smaller than a critical value ( c α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGJb WaaSbaaKqbGeaacqaHXoqyaKqbagqaaaaa@3A05@ ) at the pre-specified value of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXo qyaaa@3840@ . Moreover, Mira [8] Bonferroni’s test is γ 1 ( F n )=2( X ¯ n X s:n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZo WzdaWgaaqcfasaaiaaigdaaKqbagqaaiaacIcacaWGgbWaaSbaaKqb GeaacaWGUbaajuaGbeaacaGGPaGaeyypa0JaaGOmaiaacIcaceWGyb GbaebadaWgaaqcfasaaiaad6gaaKqbagqaaiabgkHiTiaadIfadaWg aaqcfasaaiaadohacaGG6aGaamOBaaqcfayabaGaaiykaaaa@4904@ , where X s:n =Median( X 1 , X 2 ,..., X n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGyb WaaSbaaKqbGeaacaWGZbGaaiOoaiaad6gaaKqbagqaaiabg2da9iaa d2eacaWGLbGaamizaiaadMgacaWGHbGaamOBaiaacIcacaWGybWaaS baaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiwamaaBaaajuaibaGa aGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGyb WaaSbaaKqbGeaacaWGUbaajuaGbeaacaGGPaaaaa@4E8D@ . The process is to reject the null hypothesis if | γ 1 ( F n )| a n n S c ( γ 1 , F n ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaGG8b Gaeq4SdC2aaSbaaKqbGeaacaaIXaaajuaGbeaacaGGOaGaamOramaa BaaajuaibaGaamOBaaqcfayabaGaaiykaiaacYhacqGHLjYSdaWcaa qaaiaadggadaWgaaqcfasaaiaad6gaaKqbagqaaaqaamaakaaabaGa amOBaaqabaaaaiaadofadaWgaaqcfasaaiaadogaaKqbagqaaiaacI cacqaHZoWzdaWgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWGgbWa aSbaaKqbGeaacaWGUbaajuaGbeaacaGGPaGaaiilaaaa@5093@ where

a n z 1 α 2  as n, S c 2 ( γ 1 , F n )=4 σ ^ 2 + ( D n,c ) 2 4 D n,c S μ ¯ F , σ ^ 2 = 1 n1 i=1 n ( X i X ¯ n ) 2 , S μ ¯ F = X ¯ n 2 n i=1 n X i I( X i X s:n ),  D n,c = n 1/5 2c ( X [(n/2)+c n 4/5 ]:n X [(n/2)+c n 4/5 +1]:n ),and c=0.5. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqbak aadggadaWgaaqcfasaaiaad6gaaKqbagqaaiabgkziUkaadQhadaWg aaqcfasaaiaaigdacqGHsisljuaGdaWcbaqcfasaaiabeg7aHbqaai aaikdaaaaajuaGbeaacaqGGaGaaeyyaiaabohacaqGGaGaamOBaiab gkziUkabg6HiLkaacYcacaWGtbWaa0baaKqbGeaacaWGJbaabaGaaG OmaaaajuaGcaGGOaGaeq4SdC2aaSbaaKqbGeaacaaIXaaajuaGbeaa caGGSaGaamOramaaBaaajuaibaGaamOBaaqcfayabaGaaiykaiabg2 da9iaaisdacuaHdpWCgaqcamaaCaaabeqcfasaaiaaikdaaaqcfaOa ey4kaSIaaiikaiaadseadaWgaaqcfasaaiaad6gacaGGSaGaam4yaa qcfayabaGaaiykamaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0Ia aGinaiaadseadaWgaaqcfasaaiaad6gacaGGSaGaam4yaaqcfayaba Gaam4uamaaBaaajuaibaGafqiVd0MbaebajuaGdaWgaaqcfasaaiaa dAeaaeqaaaqcfayabaGaaiilaiqbeo8aZzaajaWaaWbaaeqajuaiba GaaGOmaaaajuaGcqGH9aqpdaWcaaqaaiaaigdaaeaacaWGUbGaeyOe I0IaaGymaaaadaaeWbqaaiaacIcacaWGybWaaSbaaKqbGeaacaWGPb aajuaGbeaacqGHsislceWGybGbaebadaWgaaqcfasaaiaad6gaaKqb agqaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaey yeIuoacaGGPaWaaWbaaeqajuaibaGaaGOmaaaajuaGcaGGSaGaam4u amaaBaaajuaibaGafqiVd0MbaebajuaGdaWgaaqcfasaaiaadAeaae qaaaqcfayabaGaeyypa0JabmiwayaaraWaaSbaaKqbGeaacaWGUbaa juaGbeaacqGHsisldaWcaaqaaiaaikdaaeaacaWGUbaaamaaqahaba GaamiwamaaBaaajuaibaGaamyAaaqcfayabaaajuaibaGaamyAaiab g2da9iaaigdaaeaacaWGUbaajuaGcqGHris5aiaadMeacaGGOaGaam iwamaaBaaajuaibaGaamyAaaqcfayabaGaeyizImQaamiwamaaBaaa juaibaGaam4CaiaacQdacaWGUbaajuaGbeaacaGGPaGaaiilaiaabc cacaWGebWaaSbaaKqbGeaacaWGUbGaaiilaiaadogaaKqbagqaaiab g2da9aGcbaqcfa4aaSaaaeaacaWGUbWaaWbaaeqajuaibaGaaGymai aac+cacaaI1aaaaaqcfayaaiaaikdacaWGJbaaaiaacIcacaWGybWa aSbaaKqbGeaacaGGBbGaaiikaiaad6gacaGGVaGaaGOmaiaacMcacq GHRaWkcaWGJbGaamOBaKqbaoaaCaaajuaibeqaaiaaisdacaGGVaGa aGynaaaacaGGDbGaaiOoaiaad6gaaKqbagqaaiabgkHiTiaadIfada WgaaqcfasaaiaacUfacaGGOaGaamOBaiaac+cacaaIYaGaaiykaiab gUcaRiaadogacaWGUbqcfa4aaWbaaKqbGeqabaGaaGinaiaac+caca aI1aaaaiabgUcaRiaaigdacaGGDbGaaiOoaiaad6gaaKqbagqaaiaa cMcacaGGSaGaaeyyaiaab6gacaqGKbGaaeiiaiaadogacqGH9aqpca aIWaGaaiOlaiaaiwdacaGGUaaaaaa@DD4D@

The Modarres & Gastwirth [32] test is the hybrid test of sign test in the first stage and a percentile-modified two-sample Wilcoxon see Gastwirth [33] test in the second stage. Finally, Samawi et al. [9] test of symmetry is based on kernel estimate of the overlap measure.

In the following simulation, SAS version 9.3 {proc kde; method=srot} is used. As in McWilliams [5], the generalized lambda distribution see, Ramberg & Schmeiser [34] is used in our simulation with following set of parameters:

To generate the observations we used x i = λ 1 + 1 λ 2 ( u i λ 3 (1 u i ) λ 4 , i=1,...,m, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWG4b WaaSbaaKqbGeaacaWGPbaajuaGbeaacqGH9aqpcqaH7oaBdaWgaaqc fasaaiaaigdaaKqbagqaaiabgUcaRmaalaaabaGaaGymaaqaaiabeU 7aSnaaBaaajuaibaGaaGOmaaqcfayabaaaaiaacIcacaWG1bWaa0ba aKqbGeaacaWGPbaabaGaeq4UdWwcfa4aaSbaaKqbGeaacaaIZaaabe aaaaqcfaOaeyOeI0IaaiikaiaaigdacqGHsislcaWG1bWaaSbaaKqb GeaacaWGPbaajuaGbeaacaGGPaWaaWbaaeqabaGaeq4UdW2aaSbaaK qbGeaacaaI0aaajuaGbeaaaaGaaiilaiaabccacaWGPbGaeyypa0Ja aGymaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaamyBaiaacYcaaa a@5CB3@ where u i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG1bWaaS baaSqaaiaadMgaaeqaaaaa@3827@ a uniform random number. The significance level used in the simulation is α=0.05, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXo qycqGH9aqpcaaIWaGaaiOlaiaaicdacaaI1aGaaiilaaaa@3CDB@ with sample sizes n=30, 50, and 100. To investigate the Type I error, the symmetric distributions used in the simulation are the first case of the generalized lambda and the normal. Our simulation is based on 5000 simulated samples. The 95% confidence intervals of the true probability of type I error under the null hypothesis with α=0.05 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXo qycqGH9aqpcaaIWaGaaiOlaiaaicdacaaI1aaaaa@3C2B@  are (0.04396, 0.05504).

Table 1.1 shows the estimated probability of type I error. Our test is an asymptotic test with a slight bias in D(., .) and in the variance estimation for small sample size. For sample sizes more than 30, the test seems to have an estimated probability of type I error close to the nominal value 0.05. However, Bonferroni’s test seems to be conservative test procedure, while Modarres, Gastwirth test is slightly conservative for small sample size. Table 1.2 and Table 1.3 show that using D(., .) based test is more powerful than McWilliams [5], Bonferroni’s, Modarres & Gastwirth [32] and Samawi et al. [9] tests in all of the presented cases. The efficiency increases as the sample size increases.

Distribution

n

Run Tests

Test Based on the Overlap

Bonferroni’s γ 1 ( F n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZo WzdaWgaaqcfasaaiaaigdaaKqbagqaaiaacIcacaWGgbWaaSbaaKqb GeaacaWGUbaajuaGbeaacaGGPaaaaa@3DD4@

Modarres and Gastwirth (1998) Test W 0.80 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGxb WaaSbaaKqbGeaacaaIWaGaaiOlaiaaiIdacaaIWaaajuaGbeaaaaa@3B42@

Test Based on Kullback-Leibler Information

Case #1 generalized lambda λ 1 =0, λ 2 =0.197454, λ 3 =0.134915, λ 4 =0.134915,  α 3 =0, α 4 =3.0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqbak abeU7aSnaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaaGimaiaa cYcacqaH7oaBdaWgaaqcfasaaiaaikdaaKqbagqaaiabg2da9iaaic dacaGGUaGaaGymaiaaiMdacaaI3aGaaGinaiaaiwdacaaI0aGaaiil aiabeU7aSnaaBaaajuaibaGaaG4maaqcfayabaGaeyypa0JaaGimai aac6cacaaIXaGaaG4maiaaisdacaaI5aGaaGymaiaaiwdacaGGSaaa keaajuaGcqaH7oaBdaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9i aaicdacaGGUaGaaGymaiaaiodacaaI0aGaaGyoaiaaigdacaaI1aGa aiilaiaabccacqaHXoqydaWgaaqcfasaaiaaiodaaKqbagqaaiabg2 da9iaaicdacaGGSaGaeqySde2aaSbaaKqbGeaacaaI0aaajuaGbeaa cqGH9aqpcaaIZaGaaiOlaiaaicdaaaaa@6A61@

30

0.046

0.056

0.03

0.027

0.051

50

0.052

0.051

0.032

0.044

0.047

100

0.058

0.052

0.027

0.046

0.051

Normal (0, 1)

30

0.052

0.057

0.03

0.03

0.052

50

0.048

0.055

0.03

0.043

0.051

100

0.051

0.052

0.032

0.048

0.052

Table 1.1: Probability of Type I Error under the Null Hypothesis. (α =0.05).

Case #

n

Run Test

Test Based on the Overlap

Bonferroni’s γ 1 ( F n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZo WzdaWgaaqcfasaaiaaigdaaKqbagqaaiaacIcacaWGgbWaaSbaaKqb GeaacaWGUbaajuaGbeaacaGGPaaaaa@3DD4@

Modarres and Gastwirth (1998) Test W 0.80 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGxb WaaSbaaKqbGeaacaaIWaGaaiOlaiaaiIdacaaIWaaajuaGbeaaaaa@3B42@

Test based on Kullback-Leibler Information

-2
λ 1 =0, λ 2 =1, λ 3 =1.4, λ 4 =0.25  α 3 =0.5, α 4 =2.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaaicdacaGGSaGa eq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaaIXaGaai ilaiabeU7aSnaaBaaajuaibaGaaG4maaqcfayabaGaeyypa0JaaGym aiaac6cacaaI0aGaaiilaiabeU7aSnaaBaaajuaibaGaaGinaaqcfa yabaGaeyypa0JaaGimaiaac6cacaaIYaGaaGynaiaabccacqaHXoqy daWgaaqcfasaaiaabodaaKqbagqaaiaab2dacaqGWaGaaeOlaiaabw dacaqGSaGaeqySde2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqp caaIYaGaaiOlaiaaikdaaaa@5E50@

30

0.282

0.501

0.253

0.495

0.948

50

0.456

0.839

0.352

0.941

0.992

100

0.781

0.999

0.5

1

1

-3
λ 1 =0, λ 2 =1, λ 3 =0.00007, λ 4 =0.1, α 3 =1.5, α 4 =5.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaaicdacaGGSaGa eq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaaIXaGaai ilaiabeU7aSnaaBaaajuaibaGaaG4maaqcfayabaGaeyypa0JaaGim aiaac6cacaaIWaGaaGimaiaaicdacaaIWaGaaG4naiaacYcacqaH7o aBdaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9iaaicdacaGGUaGa aGymaiaacYcacqaHXoqydaWgaaqcfasaaiaaiodaaKqbagqaaiabg2 da9iaaigdacaGGUaGaaGynaiaacYcacqaHXoqydaWgaaqcfasaaiaa isdaaKqbagqaaiabg2da9iaaiwdacaGGUaGaaGioaaaa@60EE@

30

0.444

0.846

0.508

0.61

0.98

50

0.678

0.953

0.756

0.99

0.999

100

0.913

1

0.966

1

1

-4
λ 1 =3.586508, λ 2 =0.04306, λ 3 =0.025213, λ 4 =0.094029 α 3 =0.9, α 4 =4.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqbak abeU7aSnaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaaG4maiaa c6cacaaI1aGaaGioaiaaiAdacaaI1aGaaGimaiaaiIdacaGGSaGaeq 4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaaIWaGaaiOl aiaaicdacaaI0aGaaG4maiaaicdacaaI2aGaaiilaiabeU7aSnaaBa aajuaibaGaaG4maaqcfayabaGaeyypa0JaaGimaiaac6cacaaIWaGa aGOmaiaaiwdacaaIYaGaaGymaiaaiodacaGGSaGaeq4UdW2aaSbaaK qbGeaacaaI0aaajuaGbeaacqGH9aqpcaaIWaGaaiOlaiaaicdacaaI 5aGaaGinaiaaicdacaaIYaGaaGyoaaqaaiabeg7aHnaaBaaajuaiba GaaG4maaqcfayabaGaeyypa0JaaGimaiaac6cacaaI5aGaaiilaiab eg7aHnaaBaaajuaibaGaaGinaaqcfayabaGaeyypa0JaaGinaiaac6 cacaaIYaaaaaa@6E4B@

30

0.12

0.38

0.154

0.179

0.684

50

0.134

0.541

0.26

0.474

0.854

100

0.245

0.761

0.488

0.845

0.946

-5
λ 1 =0, λ 2 =1, λ 3 =0.0075, λ 4 =0.03, α 3 =1.5, α 4 =7.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaaicdacaGGSaGa eq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqGHsislca aIXaGaaiilaiabeU7aSnaaBaaajuaibaGaaG4maaqcfayabaGaeyyp a0JaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaaiEdacaaI1aGaai ilaiabeU7aSnaaBaaajuaibaGaaGinaaqcfayabaGaeyypa0JaeyOe I0IaaGimaiaac6cacaaIWaGaaG4maiaacYcacqaHXoqydaWgaaqcfa saaiaaiodaaKqbagqaaiabg2da9iaaigdacaGGUaGaaGynaiaacYca cqaHXoqydaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9iaaiEdaca GGUaGaaGynaaaa@63BB@

30

0.141

0.451

0.231

0.247

0.81

50

0.201

0.601

0.41

0.652

0.92

100

0.336

0.839

0.741

0.954

0.98

Table 1.2: Power of Kullback-Leibler Information based test, with comparison with other tests Under Alternative Hypotheses (α =0.05).

Case #

n

Runs Test

Test Based on the Overlap

Bonferroni’s γ 1 ( F n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZo WzdaWgaaqcfasaaiaaigdaaKqbagqaaiaacIcacaWGgbWaaSbaaKqb GeaacaWGUbaajuaGbeaacaGGPaaaaa@3DD4@

Modarres and Gastwirth (1998) Test W 0.80 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGxb WaaSbaaKqbGeaacaaIWaGaaiOlaiaaiIdacaaIWaaajuaGbeaaaaa@3B42@

Test Based on Kullback-Leibler Information

-6
λ 1 =0.116734, λ 2 =0.351663, λ 3 =0.13, λ 4 =0.16, α 3 =0.8, α 4 =11.4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaKqbak abeU7aSnaaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaeyOeI0Ia aGimaiaac6cacaaIXaGaaGymaiaaiAdacaaI3aGaaG4maiaaisdaca GGSaGaeq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqGH sislcaaIWaGaaiOlaiaaiodacaaI1aGaaGymaiaaiAdacaaI2aGaaG 4maiaacYcacqaH7oaBdaWgaaqcfasaaiaaiodaaKqbagqaaiabg2da 9iabgkHiTiaaicdacaGGUaGaaGymaiaaiodacaGGSaGaeq4UdW2aaS baaKqbGeaacaaI0aaajuaGbeaacqGH9aqpcqGHsislcaaIWaGaaiOl aiaaigdacaaI2aGaaiilaaqaaiabeg7aHnaaBaaajuaibaGaaG4maa qcfayabaGaeyypa0JaaGimaiaac6cacaaI4aGaaiilaiabeg7aHnaa BaaajuaibaGaaGinaaqcfayabaGaeyypa0JaaGymaiaaigdacaGGUa GaaGinaaaaaa@6E36@

30

0.051

0.161

0.034

0.033

0.191

50

0.055

0.174

0.04

0.055

0.225

100

0.053

0.21

0.059

0.12

0.331

-7
λ 1 =0, λ 2 =1, λ 3 =0.1, λ 4 =0.18, α 3 =2.0, α 4 =21.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaaicdacaGGSaGa eq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqGHsislca aIXaGaaiilaiabeU7aSnaaBaaajuaibaGaaG4maaqcfayabaGaeyyp a0JaeyOeI0IaaGimaiaac6cacaaIXaGaaiilaiabeU7aSnaaBaaaju aibaGaaGinaaqcfayabaGaeyypa0JaeyOeI0IaaGimaiaac6cacaaI XaGaaGioaiaacYcacqaHXoqydaWgaaqcfasaaiaaiodaaKqbagqaai abg2da9iaaikdacaGGUaGaaGimaiaacYcacqaHXoqydaWgaaqcfasa aiaaisdaaKqbagqaaiabg2da9iaaikdacaaIXaGaaiOlaiaaikdaaa a@6237@

30

0.101

0.189

0.091

0.092

0.452

50

0.111

0.241

0.155

0.21

0.611

100

0.122

0.361

0.336

0.478

0.737

-8
λ 1 =0, λ 2 =1, λ 3 =0.001, λ 4 =0.13, α 3 =3.16, α 4 =23.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaaicdacaGGSaGa eq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqGHsislca aIXaGaaiilaiabeU7aSnaaBaaajuaibaGaaG4maaqcfayabaGaeyyp a0JaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaaigdacaGGSaGaeq 4UdW2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpcqGHsislcaaI WaGaaiOlaiaaigdacaaIZaGaaiilaiabeg7aHnaaBaaajuaibaGaaG 4maaqcfayabaGaeyypa0JaaG4maiaac6cacaaIXaGaaGOnaiaacYca cqaHXoqydaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9iaaikdaca aIZaGaaiOlaiaaiIdaaaa@6470@

30

0.544

0.98

0.643

0.655

0.993

50

0.752

0.999

0.888

0.992

1

100

0.961

1

0.996

1

1

-9
λ 1 =0, λ 2 =1, λ 3 =0.0001, λ 4 =0.17 α 3 =3.88, α 4 =40.7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaaicdacaGGSaGa eq4UdW2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcqGHsislca aIXaGaaiilaiabeU7aSnaaBaaajuaibaGaaG4maaqcfayabaGaeyyp a0JaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaaicdacaaIXaGaai ilaiabeU7aSnaaBaaajuaibaGaaGinaaqcfayabaGaeyypa0JaeyOe I0IaaGimaiaac6cacaaIXaGaaG4naiabeg7aHnaaBaaajuaibaGaaG 4maaqcfayabaGaeyypa0JaaG4maiaac6cacaaI4aGaaGioaiaacYca cqaHXoqydaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9iaaisdaca aIWaGaaiOlaiaaiEdaaaa@6485@

30

0.571

1

0.685

0.676

0.993

50

0.81

1

0.916

0.995

0.999

100

0.963

1

0.999

1

1

Table 1.3: Power of Overlap based test and Run Tests under Alternative Hypotheses (α =0.05).

Note: The values of skewness ( α 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaGGOa GaeqySde2aaSbaaKqbGeaacaqGZaaajuaGbeaacaqGPaaaaa@3B2B@ and kurtosis ( α 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaGGOa GaeqySde2aaSbaaKqbGeaacaaI0aaajuaGbeaacaGGPaaaaa@3B34@ are from McWilliams [5].

Note: The values of skewness ( α 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaGGOa GaeqySde2aaSbaaKqbGeaacaqGZaaajuaGbeaacaqGPaaaaa@3B2B@ and kurtosis ( α 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVG0dg9vqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaGGOa GaeqySde2aaSbaaKqbGeaacaaI0aaajuaGbeaacaGGPaaaaa@3B34@ are from McWilliams [5].

Illustration Using Base Deficit Data

We applied our new test procedure of symmetry to the base deficit (bd) data as in Samawi et al. [9]. The base deficit score refers to a deficit of "base" present in the blood. Base deficit scores were first established by Davis et al. [35]. The base deficit score has been found correlated to many variables in the trauma population, such as, mechanism of injury, the presence of intra-abdominal injury, transfusion requirements, mortality, the risk of complications, and the number of days spent in the intensive care unit as indicated by Tremblay et al. [36] and Davis et al. [37].   

The samples used in this illustration are part from the data collected based on a retrospective study of the trauma registry at a level 1 trauma center between January, 1998 and May, 2000. The primary concern was to determine at what point we can differentiate between life and death based on a base deficit score. A first step in this analysis is to determine if there is a difference in location for the base deficit score of those who survive and those who fail to survive. As is frequently the case in such studies, the underlying distribution is assumed “normal” or at least symmetric and a t-test or a nonparametric test would be performed without checking the assumptions. In either case a test of symmetry is almost never considered as a means of determining how one may proceed in the analysis. Based on the conclusions of a test of symmetry, the analyst can chose the most powerful test for location. The goal is to test the hypothesis that, on average, the base deficit score is the same for those who survive and those who fail to survive their injuries. The injuries of interest in this group of patients are either penetrating injury or blunt injury. However, before deciding on the test procedure, we need to check the assumptions of underlying distribution of the base deficit for both penetrating injury and blunt injury groups of patients. In particular, the assumption of symmetry of the underlying distribution needs to be verified. The data will be centered about the estimated measure of location to perform the tests of symmetry.

Figure 1.1 and Figure 1.2 show the box plot for penetrating injury and blunt injury groups for dead and alive patients respectively. Clearly there is some asymmetry on all four distributions. Also, Table 2.1 and Table 2.2 show summery statistics for penetrating injury and blunt injury groups for dead and alive patients respectively. Table 2.3 shows the overlap based test, the runs test and the proposed test of symmetry based on the Kullback-Leibler information of symmetry for the underlying distribution for patients discharged alive and dead patients of blunt trauma and penetrating trauma. We reject the assumption of symmetry for underlying distribution of these groups.

Descriptives

BD

Type of Wound

Statistic

Std. Error

Penetrating

Mean

-10.81

0.846

95% Confidence Interval for Mean

Lower Bound

-12.49

Upper Bound

-9.12

5% Trimmed Mean

-10.68

Median

-10

Variance

52.904

Std. Deviation

7.274

Minimum

-29

Maximum

9

Range

38

Interquartile Range

10

Skewness

-0.21

0.279

Kurtosis

0.102

0.552

Blunt

Mean

-7.59

0.444

95% Confidence Interval for Mean

Lower Bound

-8.46

Upper Bound

-6.71

5% Trimmed Mean

-7.3

Median

-6

Variance

60.65

Std. Deviation

7.788

Minimum

-37

Maximum

23

Range

60

Interquartile Range

10

Skewness

-0.518

0.139

Kurtosis

1.368

0.277

Table 2.1: Summery statistics for base deficit for dead patients.

Descriptives

Base Deficit

Type of Wound

Statistic

Std. Error

penetrating

Mean

-3.52

0.202

95% Confidence Interval for Mean

Lower Bound

-3.91

Upper Bound

-3.12

5% Trimmed Mean

-3.06

Median

-2.7

Variance

24.683

Std. Deviation

4.968

Minimum

-28

Maximum

12

Range

40

Interquartile Range

5

Skewness

-1.75

0.099

Kurtosis

5.079

0.199

Blunt

Mean

-1.8

0.059

95% Confidence Interval for Mean

Lower Bound

-1.92

Upper Bound

-1.69

5% Trimmed Mean

-1.61

Median

-1.3

Variance

11.601

Std. Deviation

3.406

Minimum

-27

Maximum

13

Range

40

Interquartile Range

3

Skewness

-1.22

0.043

Kurtosis

4.39

0.085

Table 2.2: Summery statistics for base deficit for alive patients.

Injury Type

N

Test

Significance

Kullback-Leibler Information

Penetrating - Dead

74

3.989

<0.0001

Penetrating - alive

603

13.057

<0.0000

Overlap test*

Penetrating - Dead

74

-2.09

0.0183

Penetrating - alive

603

-16.928

<0.0001

Run test*

Penetrating - Dead

74

-2.065

0.0195

Penetrating - alive

603

-16.41

<0.0001

Kullback-Leibler Information

Blunt - Dead

306

13.92

<0.0001

Blunt - alive

3275

8.053

<0.0001

Overlap test*

Blunt - Dead

306

-13.264

<0.0001

Blunt - alive

3275

-79.074

<0.0001

Run test*

Blunt - Dead

306

-10.29

<0.0001

Blunt - alive

3275

-52.405

<0.0001

Table 2.3: Test of symmetry with summary statistics.

Figure 1.1: Box plot to base deficit for dead patients.
Figure 1.2: Box plot to base deficit for alive patients.

The proposed test of symmetry based on the Kullback-Leibler information, appears to outperform the other tests of symmetry in the literature in terms of power. Our test is more sensitive to detect a slight asymmetry in the underlying distribution than other tests proposed in the literature. Moreover, the kernel density estimation literature is very rich and many of the proposed methods and the improved methods are available on statistical software, such as SAS™, S-plus, Stata and R. Since based on the Kullback-Leibler information can be used in multivariate cases as well as in univariate cases, our proposed test of symmetry can be extended to multivariate cases for diagonal symmetry, conditional symmetry and other types of symmetry.

References

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