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Research Article Volume 6 Issue 1

Modified Maximum Likelihood Estimation in Poisson Regression

Evrim Oral

Department of Biostatistics Program, LSUHSC School of Public Health, USA

Received: April 25, 2017 | Published: May 25, 2017

citation: Oral E (2017) Modified Maximum Likelihood Estimation in Poisson Regression. Biom Biostat Int J 6(1): 00154. DOI: 10.15406/bbij.2017.06.00154

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Abstract

In Generalized Linear Models, likelihood equations are intractable and do not have explicit solutions; thus, they must be solved by using Newton-type algorithms. Solving these equations by iterations, however, can be problematic: the iterations might converge to wrong values or the iterations might not converge at all. In this study, we derive the modified maximum likelihood estimators for Poisson regression model and study their properties. We also search the robustness of these estimators when there are outliers in the covariates.

Keywords: Count data; Poisson regression; Modified maximum likelihood; Newton-type algorithms; Dixon’s outlier model

Introducton

Poisson regression is widely used for modeling count data, especially when there is no over- or under- dispersion [1]. Since the likelihood equations from this model are intractable, solving these equations requires using iterative methods, such as Newton Raphson or Fisher scoring. However, using iterative methods to find maximum likelihood estimators (MLEs) can generally be problematic and time should be spent to investigate the stability of such solutions [2-4]. Specifically the following difficulties can arise: the iterations might converge to wrong values if the likelihood equations have multiple roots, or the iterations might not converge at all. See [5] and [6] for a discussion about situations where one encounters these difficulties in solving MLEs. The common software, Stata for example, is known to be very sensitive to numerical iterations. Researchers have reported problems in getting Poisson regression estimates with the “poisson” command, which encounters problems in locating the maximum and does not converge [7]. Note that in Poisson regression modeling, additional problems might occur. The most common problem is the over- or under- dispersion in data, in which case using a more flexible model such as negative binomial regression is more appropriate. The second problem is analogous to the complete separation or quasi-complete separation problem in binary regression: the MLEs may not exist for certain data configurations, see [7,8]. In this study we do not consider either of these problems and focus only on the case where it is appropriate to model the data with Poisson regression.

Unlike the maximum likelihood (ML) technique, modified maximum likelihood (MML) methodology produces explicit estimates. MML achieves explicit estimates by linearizing intractable functions within the likelihood equations using the ordered statistics [9]. Asymptotically, MML estimators (MMLEs) are known to be unbiased and have minimum variances, i.e. they are fully efficient. For small sample sizes MMLEs have negligible bias and their variances are only marginally bigger than the minimum variance bounds, i.e. they are highly efficient [10-15].

In the GLM setting, Tiku and Vaughan [3] used the MML methodology to extend the techniques of traditional logistic regression to non-logistic density functions. Oral and Gunay [16] and Oral [17] later extended the work in [3] to the binary regression model with one stochastic covariate. Oral [18] derived the MMLEs in general GLMs which use canonical link when there is only one risk factor. In this study, we derive the explicit MMLEs of Poisson regression model, generalize the derivations to more than one covariate, and study their robustness properties via simulations.

Methods

The Univariate Poisson regression model is given by

E( Y i | X i = x i )= μ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aacIcadaabcaqaaiaadMfadaWgaaqcfasaaiaadMgaaeqaaaqcfaOa ayjcSdGaamiwamaaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0Jaam iEamaaBaaajuaibaGaamyAaaqabaqcfaOaaiykaiabg2da9iabeY7a TnaaBaaajuaibaGaamyAaaqabaaaaa@4755@

=exp( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaciyzaiaacIhacaGGWbGaaiikaiaadQhadaWgaaqcfasaaiaadMga aeqaaKqbakaacMcaaaa@3E88@  (1)

where z i =α+β x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0JaeqySdeMaey4kaSIa eqOSdiMaamiEamaaBaaajuaibaGaamyAaaqabaaaaa@40B0@ , for 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai abgsMiJkaadMgacqGHKjYOcaWGUbaaaa@3C8A@ , and the outcome Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ has the probability distribution

f y ( y i )= exp( μ i ) μ y i y i ! , y i =0,1,2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaamyEaaqabaqcfaOaaiikaiaadMhadaWgaaqcfasa aiaadMgaaeqaaKqbakaacMcacqGH9aqpdaWcaaqaaiGacwgacaGG4b GaaiiCaiaacIcacqGHsislcqaH8oqBdaWgaaqcfasaaiaadMgaaeqa aKqbakaacMcacaaMc8UaeqiVd02aaWbaaKqbGeqabaGaamyEaKqbao aaBaaajuaibaGaamyAaaqabaaaaaqcfayaaiaadMhadaWgaaqcfasa aiaadMgaaeqaaKqbakaacgcaaaGaaiilaiaaykW7caaMc8UaamyEam aaBaaajuaibaGaamyAaaqabaqcfaOaeyypa0JaaGimaiaacYcacaaM c8UaaGymaiaacYcacaaMc8UaaGOmaiaacYcacaaMc8UaaiOlaiaac6 cacaGGUaaaaa@6407@ (2)

Note that in equation (2), Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@ is presumed to increase with X so that b is a priori greater than zero. For the model given in (1)-(2), the log-likelihood function of the random sample ( y i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aadMhadaWgaaqcfasaaiaadMgaaeqaaKqbakaaykW7caGGSaGaaGPa VlaadIhadaWgaaqcfasaaiaadMgaaeqaaKqbakaacMcaaaa@4134@  can be written as

lnL i=1 n y i z i i=1 n g( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyyhIu7aaabCaeaacaWG5bWaaSbaaeaacaWGPbaa beaacaWG6bWaaSbaaeaacaWGPbaabeaaaeaajugWaiaadMgacqGH9a qpcaaIXaaajuaGbaqcLbmacaWGUbaajuaGcqGHris5aiabgkHiTmaa qahabaGaam4zaiaacIcacaWG6bWcdaWgaaqcfayaaKqzadGaamyAaa qcfayabaGaaiykaaqaaKqzadGaamyAaiabg2da9iaaigdaaKqbagaa jugWaiaad6gaaKqbakabggHiLdaaaa@58F1@ , (3)

where g( z i )=exp( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zai aacIcacaWG6bWaaSbaaKqbGeaacaWGPbaabeaajuaGcaGGPaGaeyyp a0JaciyzaiaacIhacaGGWbGaaiikaiaadQhadaWgaaqcfasaaiaadM gaaeqaaKqbakaacMcaaaa@4397@ . The likelihood equations for estimating a and b do not have explicit solutions because of the nonlinear function g( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zai aacIcacaWG6bWaaSbaaKqbGeaacaWGPbaajuaGbeaacaGGPaaaaa@3B93@ , 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGymai abgsMiJkaadMgacqGHKjYOcaWGUbaaaa@3C8A@ . To obtain the MMLEs, we first express the likelihood equations in terms of the ordered variates z (1) z (2) ... z (n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOEam aaBaaajuaibaGaaiikaiaaigdacaGGPaaabeaajuaGcqGHKjYOcaWG 6bWaaSbaaKqbGeaacaGGOaGaaGOmaiaacMcaaeqaaKqbakabgsMiJk aac6cacaGGUaGaaiOlaiabgsMiJkaadQhadaWgaaqcfasaaiaacIca caWGUbGaaiykaaqabaaaaa@4934@ . The likelihood equations can be re-written as

lnL α = i=1 n { y [i] g( z (i) ) } =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITciGGSbGaaiOBaiaadYeaaeaacqGHciITcqaHXoqyaaGa eyypa0ZaaabCaeaadaGadaqaaiaadMhadaWgaaqcfasaaiaacUfaca WGPbGaaiyxaaqabaqcfaOaeyOeI0Iaam4zaiaacIcacaWG6bWaaSba aKqbGeaacaGGOaGaamyAaiaacMcaaKqbagqaaiaacMcaaiaawUhaca GL9baaaKqbGeaacaWGPbGaeyypa0JaaGymaaqcKvaq=haajugWaiaa d6gaaKqbakabggHiLdGaeyypa0JaaGimaaaa@57F0@  (4)

and

lnL β = i=1 n x (i) { y [i] g( z (i) ) } =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITciGGSbGaaiOBaiaadYeaaeaacqGHciITcqaHYoGyaaGa eyypa0ZaaabCaeaacaWG4bWaaSbaaKqbGeaacaGGOaGaamyAaiaacM caaeqaaKqbaoaacmaabaGaamyEamaaBaaajuaibaGaai4waiaadMga caGGDbaabeaajuaGcqGHsislcaWGNbGaaiikaiaadQhadaWgaaqcfa saaiaacIcacaWGPbGaaiykaaqabaqcfaOaaiykaaGaay5Eaiaaw2ha aaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIu oacqGH9aqpcaaIWaaaaa@5934@  (5)

Where y [i] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEam aaBaaabaqcLbmacaGGBbGaamyAaiaac2faaKqbagqaaaaa@3C0D@  is the concomitant of x (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaS WaaSbaaKqbagaajugWaiaacIcacaWGPbGaaiykaaqcfayabaaaaa@3C3E@   (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aIXaGaeyizImQaamyAaiabgsMiJkaad6gacaGGPaaaaa@3E36@  and g( z (i) )=exp( z (i) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgaca GGOaGaamOEaKGbaoaaBaaabaGaaiikaiaadMgacaGGPaaabeaajuaG caGGPaGaeyypa0JaciyzaiaacIhacaGGWbGaaiikaiaadQhajyaGda WgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaaiykaaaa@475E@ . Linearizing the intractable function g( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgaca GGOaGaamOEaKGbaoaaBaaabaGaamyAaaqabaqcfaOaaiykaaaa@3C47@  by using the first two terms of its Taylor series expansion around the population quantiles t (i) =E( z (i) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaeyypa0Jaamyr aiaacIcacaWG6bqcga4aaSbaaeaacaGGOaGaamyAaiaacMcaaeqaaK qbakaacMcaaaa@4302@  ( 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaaaa@3CDD@ ), we find

g( z (i) ) a i + b i z (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadEgaca aMc8UaaiikaiaadQhadaWgaaqcgayaaiaacIcacaWGPbGaaiykaaqc fayabaGaaiykaiabgwKiajaadggajyaGdaWgaaqaaiaadMgaaeqaaK qbakabgUcaRiaadkgajyaGdaWgaaqaaiaadMgaaeqaaKqbakaaykW7 caWG6bqcga4aaSbaaeaacaGGOaGaamyAaiaacMcaaeqaaaaa@4CE6@ , (6)

where a i =exp( t (i) ){ 1 t (i) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggajy aGdaWgaaqaaiaadMgaaeqaaKqbakabg2da9iGacwgacaGG4bGaaiiC aiaacIcacaWG0bWaaSbaaKGbagaacaGGOaGaamyAaiaacMcaaKqbag qaaiaacMcadaGadaqaaiaaykW7caaIXaGaeyOeI0IaamiDaKGbaoaa BaaabaGaaiikaiaadMgacaGGPaaabeaaaKqbakaawUhacaGL9baaaa a@4D83@  and b i =exp( t (i) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkgajy aGdaWgaaqaaiaadMgaaeqaaKqbakabg2da9iGacwgacaGG4bGaaiiC aiaacIcacaWG0bqcga4aaSbaaeaacaGGOaGaamyAaiaacMcaaeqaaK qbakaacMcaaaa@43A2@  for 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaaaa@3CDD@ . In order to calculate t (i) =E( z (i) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaeyypa0Jaamyr aiaacIcacaWG6bqcga4aaSbaaeaacaGGOaGaamyAaiaacMcaaeqaaK qbakaacMcaaaa@4302@  values, we define a dummy random variable U with the probability density function

f(u)=exp(u),u<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgaca GGOaGaamyDaiaacMcacqGH9aqpciGGLbGaaiiEaiaacchacaGGOaGa amyDaiaacMcacaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaamyDaiabgYda8iaaicdaaaa@4CF3@  (7)

and re-write the model (1) as

E( Y i | X i = x i )= μ i =F( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweaca GGOaWaaqGaaeaacaWGzbWaaSbaaKGbagaacaWGPbaajuaGbeaaaiaa wIa7aiaadIfajyaGdaWgaaqaaiaadMgaaeqaaKqbakabg2da9iaadI hajyaGdaWgaaqaaiaadMgaaeqaaKqbakaacMcacqGH9aqpcqaH8oqB jyaGdaWgaaqaaiaadMgaaeqaaKqbakabg2da9iaadAeadaqadaqaai aadQhajyaGdaWgaaqaaiaadMgaaeqaaaqcfaOaayjkaiaawMcaaaaa @503F@ , (8)

where F(u)=exp(u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeaca GGOaGaamyDaiaacMcacqGH9aqpciGGLbGaaiiEaiaacchacaGGOaGa amyDaiaacMcaaaa@4029@ , u<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwhacq GH8aapcaaIWaaaaa@398F@ . Thus, the t (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaaaaa@3AC7@  values can be obtained from the equation

t (i) =ln( i/ ( n+1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshada WgaaqcgayaaiaacIcacaWGPbGaaiykaaqcfayabaGaeyypa0JaciiB aiaac6gadaqadaqaamaalyaabaGaamyAaaqaamaabmaabaGaamOBai abgUcaRiaaigdaaiaawIcacaGLPaaaaaaacaGLOaGaayzkaaaaaa@44E5@ , (9)

For 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaaaa@3CDD@ ; asymptotically z (i) t (i) 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQhajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaeyOeI0IaamiD aKGbaoaaBaaabaGaaiikaiaadMgacaGGPaaabeaajuaGcqGHfjcqca aIWaGaaiOlaaaa@4365@ Alternatively, when n is large one can utilize the standard normal distribution

t (i) = Φ 1 ( i/ ( n+1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaeyypa0JaeuOP dy0aaWbaaeqajyaGbaGaeyOeI0IaaGymaaaajuaGdaqadaqaamaaly aabaGaamyAaaqaamaabmaabaGaamOBaiabgUcaRiaaigdaaiaawIca caGLPaaaaaaacaGLOaGaayzkaaaaaa@4762@ . (10)

Incorporating (6) into (4)-(5) and solving the resulting modified likelihood equations yield the explicit MMLEs below:

α ^ = δ m β ^ x ¯ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaGaeyypa0ZaaSaaaeaacqaH0oazaeaacaWGTbaaaiabgkHiTiqb ek7aIzaajaGaaGPaVlqadIhagaqeaKGbaoaaBaaabaGaamyyaaqaba aaaa@4307@  and β ^ = i=1 n δ i ( x (i) x ¯ a ) i=1 n b i ( x (i) x ¯ a ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaGaeyypa0ZaaSaaaeaadaaeWbqaaiabes7aKLGbaoaaBaaabaGa amyAaaqabaqcfaOaaiikaiaadIhajyaGdaWgaaqaaiaacIcacaWGPb GaaiykaaqabaqcfaOaeyOeI0IabmiEayaaraqcga4aaSbaaeaacaWG HbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabgg HiLdGaaiykaaqaamaaqahabaGaamOyaKGbaoaaBaaabaGaamyAaaqa baqcfaOaaiikaiaadIhajyaGdaWgaaqaaiaacIcacaWGPbGaaiykaa qabaqcfaOaeyOeI0IabmiEayaaraqcga4aaSbaaeaacaWGHbaabeaa juaGcaGGPaWaaWbaaeqajyaGbaGaaGOmaaaaaeaacaWGPbGaeyypa0 JaaGymaaqaaiaad6gaaKqbakabggHiLdaaaaaa@6217@ , (10)

where

δ= i=1 n δ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes7aKj abg2da9maaqahabaGaeqiTdqwcga4aaSbaaeaacaWGPbaabeaaaeaa caWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaaaa@431F@ , δ i = y [i] a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes7aKL GbaoaaBaaabaGaamyAaaqabaqcfaOaeyypa0JaamyEaKGbaoaaBaaa baGaai4waiaadMgacaGGDbaabeaajuaGcqGHsislcaWGHbqcga4aaS baaeaacaWGPbaabeaaaaa@4409@ , m= i=1 n b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2gacq GH9aqpdaaeWbqaaiaadkgajyaGdaWgaaqaaiaadMgaaeqaaaqaaiaa dMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoaaaa@41AE@ , (11)

and

x ¯ a = ( i=1 n b x i (i) )/m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qeamaaBaaajyaGbaGaamyyaaqcfayabaGaeyypa0ZaaSGbaeaadaqa daqaamaaqahabaGaamOyaKGbaoaaBeaabaGaamyAaaqabaqcfaOaam iEaKGbaoaaBaaabaGaaiikaiaadMgacaGGPaaabeaaaeaacaWGPbGa eyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaacaGLOaGaayzkaa aabaGaamyBaaaaaaa@4B09@ . (12)

The MMLEs derived above are asymptotically equivalent to their corresponding MLEs, giving them the same attractive asymptotic properties; however, one can refine the estimates by re-calculating a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggajy aGdaWgaaqaaiaadMgaaeqaaaaa@395B@  and b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkgajy aGdaWgaaqaaiaadMgaaeqaaaaa@395C@  values by replacing the theoretical population quantiles with their estimated values t (i) = α ^ + β ^ x (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaeyypa0JafqyS deMbaKaacqGHRaWkcuaHYoGygaqcaiaaykW7caWG4bqcga4aaSbaae aacaGGOaGaamyAaiaacMcaaeqaaaaa@461C@ (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aMc8UaaGymaiabgsMiJkaadMgacqGHKjYOcaWGUbGaaiykaaaa@3FC1@ . This process might be repeated until a desired convergence is met. The stabilization generally is reached within a few iterations.

Asymptotic variances and Co-variances

Vaughan and Tiku [15] proved rigorously that the MMLEs are asymptotically unbiased and their variances and co-variances are exactly the same as those of the MLEs. In the present situation, therefore, the asymptotic variances and the covariance of α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@  and β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@  are given by I 1 ( α,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbakaa=L eajyaGdaahaaqabeaacqGHsislcaaIXaaaaKqbaoaabmaabaGaeqyS deMaaiilaiabek7aIbGaayjkaiaawMcaaaaa@400D@ , where I is the Fisher Information matrix consisting of the elements E( 2 ln L * / α 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalyaabaGaeyOaIy7aaWbaaeqajyaGbaGaaGOm aaaajuaGciGGSbGaaiOBaiaadYeadaahaaqabeaacaGGQaaaaaqaai abgkGi2kabeg7aHLGbaoaaCaaabeqaaiaaikdaaaaaaaqcfaOaayjk aiaawMcaaiaaykW7caGGSaaaaa@484E@ E( 2 ln L * / αβ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalyaabaGaeyOaIy7aaWbaaeqajyaGbaGaaGOm aaaajuaGciGGSbGaaiOBaiaadYeadaahaaqabeaacaGGQaaaaaqaai abgkGi2kabeg7aHjaaykW7cqGHciITcqaHYoGyaaaacaGLOaGaayzk aaaaaa@48AA@ , and E( 2 ln L * / β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabgkHiTi aadweadaqadaqaamaalyaabaGaeyOaIy7aaWbaaeqajyaGbaGaaGOm aaaajuaGciGGSbGaaiOBaiaadYeadaahaaqabeaacaGGQaaaaaqaai abgkGi2kabek7aInaaCaaabeqcgayaaiaaikdaaaaaaaqcfaOaayjk aiaawMcaaaaa@4615@ . From the modified likelihood equations

lnL α ln L * α = i=1 n { y i ( a i + b i z i ) } =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaciiBaiaac6gacaWGmbaabaGaeyOaIyRaeqySdegaaiab g2Hi1oaalaaabaGaeyOaIyRaciiBaiaac6gacaWGmbWaaWbaaeqaba GaaiOkaaaaaeaacqGHciITcqaHXoqyaaGaeyypa0ZaaabCaeaadaGa daqaaiaadMhajyaGdaWgaaqaaiaadMgaaeqaaKqbakabgkHiTiaacI cacaWGHbqcga4aaSbaaeaacaWGPbaabeaajuaGcqGHRaWkcaWGIbqc ga4aaSbaaeaacaWGPbaabeaajuaGcaWG6bWaaSbaaKGbagaacaWGPb aajuaGbeaacaGGPaaacaGL7bGaayzFaaaajyaGbaGaamyAaiabg2da 9iaaigdaaeaacaWGUbaajuaGcqGHris5aiabg2da9iaaicdaaaa@6309@ ,

lnL β ln L * β = i=1 n x i { y i ( a i + b i z i ) } =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaeyOaIyRaciiBaiaac6gacaWGmbaabaGaeyOaIyRaeqOSdigaaiab g2Hi1oaalaaabaGaeyOaIyRaciiBaiaac6gacaWGmbWaaWbaaeqaba GaaiOkaaaaaeaacqGHciITcqaHYoGyaaGaeyypa0ZaaabCaeaacaWG 4bWaaSbaaKGbagaacaWGPbaajuaGbeaadaGadaqaaiaadMhadaWgaa qcgayaaiaadMgaaKqbagqaaiabgkHiTiaacIcacaWGHbqcga4aaSba aeaacaWGPbaabeaajuaGcqGHRaWkcaWGIbqcga4aaSbaaeaacaWGPb aabeaajuaGcaWG6bqcga4aaSbaaeaacaWGPbaabeaajuaGcaGGPaaa caGL7bGaayzFaaaajyaGbaGaamyAaiabg2da9iaaigdaaeaacaWGUb aajuaGcqGHris5aiabg2da9iaaicdaaaa@6636@ ,

the Fisher Information matrix can be easily obtained as

V= I 1 ( γ 0 , γ 1 )= [ i=1 n Q i i=1 n Q i x i i=1 n Q i x i i=1 n Q i x i 2 ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbakaa=z facaWF9aGaa8xsaKGbaoaaCaaabeqaaKqzadGaeyOeI0IaaGymaaaa juaGcaGGOaGaeq4SdCwcga4aaSbaaeaacaaIWaaabeaajuaGcaGGSa Gaeq4SdCwcga4aaSbaaeaacaaIXaaabeaajuaGcaGGPaGaeyypa0Za amWaaeaafaqabeGacaaabaWaaabCaeaacaWGrbqcga4aaSbaaeaaca WGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakab ggHiLdaabaWaaabCaeaacaWGrbWaaSbaaKGbagaacaWGPbaajuaGbe aacaWG4bWaaSbaaKGbagaacaWGPbaajuaGbeaaaKGbagaacaWGPbGa eyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaabaWaaabCaeaaca WGrbqcga4aaSbaaeaacaWGPbaabeaajuaGcaWG4bqcga4aaSbaaeaa caWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbak abggHiLdaabaWaaabCaeaacaWGrbqcga4aaSbaaeaacaWGPbaabeaa juaGcaWG4bqcga4aa0baaeaacaWGPbaabaGaaGOmaaaaaeaacaWGPb Gaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaaaaGaay5waiaa w2faaKGbaoaaCaaabeqaaiabgkHiTiaaigdaaaaaaa@7A8D@ , (13)

where Q i =exp( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgfada WgaaqcgayaaiaadMgaaKqbagqaaiabg2da9iGacwgacaGG4bGaaiiC aiaacIcacaWG6bqcga4aaSbaaeaacaWGPbaabeaajuaGcaGGPaaaaa@423E@ . V is estimated by replacing Q i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgfajy aGdaWgaaqaaiaadMgaaeqaaaaa@394B@  with its estimate Q ^ i =exp( z ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadgfaga qcaKGbaoaaBaaabaGaamyAaaqabaqcfaOaeyypa0JaciyzaiaacIha caGGWbGaaiikaiqadQhagaqcaKGbaoaaBaaabaGaamyAaaqabaqcfa Oaaiykaaaa@425E@ , z ^ i = α ^ + β ^ x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadQhaga qcaKGbaoaaBaaabaGaamyAaaqabaqcfaOaeyypa0JafqySdeMbaKaa cqGHRaWkcuaHYoGygaqcaiaaykW7caWG4bqcga4aaSbaaeaacaWGPb aabeaaaaa@4380@  ( 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaaaa@3CDD@ ). Since z i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQhada WgaaqcgayaaiaadMgaaKqbagqaaaaa@3A02@  values converge to t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaadMgaaeqaaaaa@396E@  values as n tends to infinity, a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadggajy aGdaWgaaqaaiaadMgaaeqaaaaa@395B@  and b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkgada WgaaqcgayaaiaadMgaaKqbagqaaaaa@39EA@  values are treated as constant coefficients for large n, see also [3]. Hence, the asymptotic variances can be estimated by

Var( α ^ )= i=1 n Q ^ i x i 2 / { i=1 n Q ^ i i=1 n Q ^ i x i 2 ( i=1 n Q ^ i x i ) 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabAfaca qGHbGaaeOCaiaacIcacuaHXoqygaqcaiaacMcacqGH9aqpdaWcgaqa amaaqahabaGabmyuayaajaqcga4aaSbaaeaacaWGPbaabeaajuaGca WG4bqcga4aa0baaeaacaWGPbaabaGaaGOmaaaaaeaacaWGPbGaeyyp a0JaaGymaaqaaiaad6gaaKqbakabggHiLdaabaWaaiWaaeaadaaeWb qaaiqadgfagaqcamaaBaaabaGaamyAaaqabaaajyaGbaGaamyAaiab g2da9iaaigdaaeaacaWGUbaajuaGcqGHris5amaaqahabaGabmyuay aajaqcga4aaSbaaeaacaWGPbaabeaajuaGcaWG4bqcga4aa0baaeaa caWGPbaabaGaaGOmaaaajuaGcqGHsisldaqadaqaamaaqahabaGabm yuayaajaqcga4aaSbaaeaacaWGPbaabeaajuaGcaWG4bqcga4aaSba aeaacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaK qbakabggHiLdaacaGLOaGaayzkaaqcga4aaWbaaeqabaGaaGOmaaaa aeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaaca GL7bGaayzFaaaaaaaa@72BC@ , (14)

Var( β ^ )= i=1 n Q ^ i / { i=1 n Q ^ i i=1 n Q ^ i x i 2 ( i=1 n Q ^ i x i ) 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabAfaca qGHbGaaeOCaiaacIcacuaHYoGygaqcaiaacMcacqGH9aqpdaWcgaqa amaaqahabaGabmyuayaajaqcga4aaSbaaeaacaWGPbaabeaaaeaaca WGPbGaeyypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdaabaWaaiWa aeaadaaeWbqaaiqadgfagaqcaKGbaoaaBaaabaGaamyAaaqabaaaba GaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5amaaqaha baGabmyuayaajaqcga4aaSbaaeaacaWGPbaabeaajuaGcaWG4bqcga 4aa0baaeaacaWGPbaabaGaaGOmaaaajuaGcqGHsisldaqadaqaamaa qahabaGabmyuayaajaqcga4aaSbaaeaacaWGPbaabeaajuaGcaWG4b WaaSbaaKGbagaacaWGPbaajuaGbeaaaKGbagaacaWGPbGaeyypa0Ja aGymaaqaaiaad6gaaKqbakabggHiLdaacaGLOaGaayzkaaWaaWbaae qajyaGbaGaaGOmaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6ga aKqbakabggHiLdaacaGL7bGaayzFaaaaaaaa@6FF5@ . (15)

Hypothesis Testing

Testing the null hypothesis H 0 :β=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeajy aGdaWgaaqaaiaaicdaaeqaaKqbakaacQdacqaHYoGycqGH9aqpcaaI Waaaaa@3DBB@  is of great practical importance in Poisson regression modelling. The likelihood ratio statistic for testing H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeajy aGdaWgaaqaaiaaicdaaeqaaaaa@390E@  is LR=2( L 0 L 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeaca WGsbGaeyypa0JaeyOeI0IaaGOmaiaacIcacaWGmbWaaSbaaKGbagaa caaIWaaajuaGbeaacqGHsislcaWGmbqcga4aaSbaaeaacaaIXaaabe aajuaGcaGGPaaaaa@4307@ , where L 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeada WgaaqcgayaaiaaicdaaKqbagqaaaaa@39A0@  and L 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadYeajy aGdaWgaaqaaiaaigdaaeqaaaaa@3913@  denote the maximized log-modified likelihood functions under the null and alternative hypotheses, respectively. The null distribution of LR is asymptotically a chi-square with 1 degree of freedom. Large values of LR lead the rejection of H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeajy aGdaWgaaqaaiaaicdaaeqaaaaa@390E@ . Alternatively, the Wald statistic W (the ratio of β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@  to its standard error) might be used. Since β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@  is asymptotically equivalent to the MLE, the null distribution of W is asymptotically normal N(0,1). Large values of W lead to the rejection of H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIeada WgaaqcgayaaiaaicdaaKqbagqaaaaa@399C@ .

Numerical Example

To compare ML and MML estimates numerically, we analyzed the data given on page 82 of Agresti [19]. The data is from a study of nesting horseshoe crabs where the response Y is the number of satellites that each female crab has, and the corresponding values of the covariate X is the carapace width of 173 crabs. The study investigates the relationship between Y and X. We calculated the MMLEs from equations (10)-(12) and their approximate standard errors from (14)-(15). The FORTRAN code written to carry out the calculations can be obtained from the author. Our results are completely consistent with those given in [19], which is expected; see Table 1.

Coefficient

Estimate

SE

W

LR

ML

a

-3.3048

0.5422

b

0.164

0.02

8.2

64.9

Coefficient

Estimate

SE

W

LR

MML

a

-3.3047

0.5423

b

0.164

0.0199

8.241

64.91

Table 1: MLEs and MMLEs along with their standard errors for horseshoe crab data.

Remark: In solving (10)-(12), Oral [18] proposed to calculate the initial t (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshada WgaaqcgayaaiaacIcacaWGPbGaaiykaaqcfayabaaaaa@3B55@  values from the least squares estimators (LSEs), which is a different approach than using equation (9) (Approach 1) or equation (10) (Approach 2). Since t (i) =E(α+β x (i) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaeyypa0Jaamyr aiaacIcacqaHXoqycqGHRaWkcqaHYoGycaaMc8UaamiEaKGbaoaaBa aabaGaaiikaiaadMgacaGGPaaabeaajuaGcaGGPaaaaa@48AD@ , t (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaGGOaGaamyAaiaacMcaaeqaaaaa@3962@  values can be approximated by t ˜ (i) = α ˜ + β ˜ x (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadshaga acamaaBaaajyaGbaGaaiikaiaadMgacaGGPaaajuaGbeaacqGH9aqp cuaHXoqygaacaiabgUcaRiqbek7aIzaaiaGaaGPaVlaadIhajyaGda WgaaqaaiaacIcacaWGPbGaaiykaaqabaaaaa@4629@ , where

α ˜ = y ¯ β ˜ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aaiaGaeyypa0JabmyEayaaraGaeyOeI0IafqOSdiMbaGaacaaMc8Ua bmiEayaaraaaaa@3FDE@  and β ˜ = i=1 n ( x i x ¯ ) y i / i=1 n ( x i x ¯ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aaiaGaeyypa0ZaaSGbaeaadaaeWbqaamaabmaabaGaamiEaKGbaoaa BaaabaGaamyAaaqabaqcfaOaeyOeI0IabmiEayaaraaacaGLOaGaay zkaaGaamyEaKGbaoaaBaaabaGaamyAaaqabaaabaGaamyAaiabg2da 9iaaigdaaeaacaWGUbaajuaGcqGHris5aaqaamaaqahabaWaaeWaae aacaWG4bqcga4aaSbaaeaacaWGPbaabeaajuaGcqGHsislceWG4bGb aebaaiaawIcacaGLPaaajyaGdaahaaqabeaacaaIYaaaaaqaaiaadM gacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoaaaaaaa@57C8@

are the LSEs; see also [3] and [20]. When using this approach (say, Approach 3), the t (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaaaaa@3AC7@  values need to be revised after the first iteration with their estimated values t (i) = α ^ + β ^ x (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaqcfaOaeyypa0JafqyS deMbaKaacqGHRaWkcuaHYoGygaqcaiaaykW7caWG4bqcga4aaSbaae aacaGGOaGaamyAaiaacMcaaeqaaaaa@461C@ (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aMc8UaaGymaiabgsMiJkaadMgacqGHKjYOcaWGUbGaaiykaaaa@3FC1@  as described above. Estimating population quantiles from the LSEs changes neither the derivations/solutions (10)-(12) nor the results. However, the total revision number needed for stabilization under different approaches is not the same, see also [3] page 889. To compare the performance of these three approaches, we conducted a simulation study where we calculated the bias values and variances of the resulting MLEs as well as the coverage probabilities. We also provided average revision numbers needed for stabilization from each approach. We set α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHb aa@3876@  to zero and considered various values for β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@3878@  and sample size n. Our results from 10,000 Monte Carlo runs are given in Table 2. 

As can be seen from Table 2, all approaches provide same biases and variances after stabilization, which is expected. The resulting coverage probabilities from all approaches are close to 0.95, and as sample size increases, both bias values and variances decrease, also as expected. For a given (α,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcacq aHXoqycaGGSaGaeqOSdiMaaiykaaaa@3C20@  value, it can be seen that the fastest stabilization is achieved by Approach 2 (i.e. equation (10)). Thus, although all three approaches yield the same results, we suggest to calculate initial t (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaaaaa@3AC7@  values (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aMc8UaaGymaiabgsMiJkaadMgacqGHKjYOcaWGUbGaaiykaaaa@3FC1@  from equation (10) because the stabilization from this approach is the fastest one.

 

 

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@3878@

0.1

 

0.5

 

1.0

 

n

 

 

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@

β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@

β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@

β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@

30

Approach 1

Bias

0.0355

0.0006

0.0341

0.0067

0.0278

0.0078

 

Variance

0.0373

0.0389

0.0403

0.0369

0.0451

0.0328

 

Coverage prob.

0.9521

0.9537

0.9522

0.9541

0.9525

0.9488

 

No of revisions

4.67

 

3.45

 

6.10

 

Approach 2

Bias

0.0355

0.0006

0.0341

0.0067

0.0279

0.0079

 

Variance

0.0373

0.0388

0.0403

0.0369

0.0451

0.0328

 

Coverage prob.

0.9521

0.9537

0.9522

0.9541

0.9525

0.9488

 

No of revisions

2.65

 

2.00

 

1.81

 

Approach 3

Bias

0.0355

0.0006

0.0341

0.0067

0.0278

0.0079

 

Variance

0.0372

0.0389

0.0402

0.0368

0.0451

0.0328

 

Coverage prob.

0.9521

0.9537

0.9523

0.9540

0.9525

0.9488

 

No of revisions

2.98

 

3.04

 

4.63

 

100

Approach 1

Bias

0.0098

0.0007

0.0079

0.0002

0.0081

0.0017

 

Variance

0.0104

0.0104

0.0113

0.0094

0.0126

0.0074

 

Coverage prob.

0.9504

0.9518

0.9470

0.9494

0.9538

0.9495

 

No of revisions

4.78

 

3.24

 

6.59

 

Approach 2

Bias

0.0098

0.0007

0.0079

0.0001

0.0081

0.0016

 

Variance

0.0103

0.0104

0.0114

0.0095

0.0126

0.0073

 

Coverage prob.

0.9504

0.9518

0.9470

0.9494

0.9538

0.9496

 

No of revisions

2.91

 

2.01

 

1.32

 

Approach 3

Bias

0.0098

0.0007

0.0078

0.0002

0.0082

0.0017

 

Variance

0.0104

0.0104

0.0113

0.0094

0.0126

0.0073

 

Coverage prob.

0.9505

0.9518

0.9470

0.9494

0.9539

0.9495

 

No of revisions

2.99

 

3.00

 

4.69

 

250

Approach 1

Bias

0.0041

0.0002

0.0041

0.0000

0.0013

0.0007

 

Variance

0.0041

0.0040

0.0044

0.0036

0.0049

0.0026

 

Coverage prob.

0.9506

0.9452

0.9500

0.9479

0.9507

0.9513

 

No of revisions

4.78

 

3.11

 

6.87

 

Approach 2

Bias

0.0041

0.0002

0.0040

0.0000

0.0012

0.0007

 

Variance

0.0040

0.0041

0.0044

0.0036

0.0049

0.0026

 

Coverage prob.

0.9506

0.9452

0.9500

0.9479

0.9507

0.9513

 

No of revisions

2.90

 

2.00

 

1.12

 

Approach 3

Bias

0.0041

0.0002

0.0041

0.0001

0.0013

0.0007

 

Variance

0.0041

0.0041

0.0044

0.0036

0.0049

0.0026

 

Coverage prob.

0.9506

0.9452

0.9500

0.9479

0.9507

0.9513

 

No of revisions

2.99

 

3.00

 

4.76

 

Table 2: Bias values, variances, convergence probabilities and average revision numbers using three different approaches to calculate t (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaaaaa@3AC7@  values (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aMc8UaaGymaiabgsMiJkaadMgacqGHKjYOcaWGUbGaaiykaaaa@3FC1@ .

Generalization to Multivariable Case

Now consider k (k2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca WGRbGaeyyzImRaaGOmaiaacMcaaaa@3BA2@  covariates and assume all of them take positive values without loss of generality. The Poisson regression model with k covariates can be written as

E( Y i | x i1 , x i2 ,..., x ik )=F( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweaca GGOaWaaqGaaeaacaWGzbqcga4aaSbaaeaacaWGPbaabeaaaKqbakaa wIa7aiaadIhadaWgaaqcgayaaiaadMgacaaIXaaajuaGbeaacaGGSa GaamiEaKGbaoaaBaaabaGaamyAaiaaikdaaeqaaKqbakaacYcacaGG UaGaaGPaVlaac6cacaaMc8UaaiOlaiaacYcacaWG4bWaaSbaaKGbag aacaWGPbGaam4AaaqcfayabaGaaiykaiabg2da9iaadAeadaqadaqa aiaadQhajyaGdaWgaaqaaiaadMgaaeqaaaqcfaOaayjkaiaawMcaaa aa@573D@  (16)

where F( z i )=exp( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAeada qadaqaaiaadQhadaWgaaqcgayaaiaadMgaaKqbagqaaaGaayjkaiaa wMcaaiabg2da9iGacwgacaGG4bGaaiiCaiaacIcacaWG6bqcga4aaS baaeaacaWGPbaabeaajuaGcaGGPaaaaa@44BB@  and

z i = β 0 + j=1 k β j x ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQhada WgaaqcgayaaiaadMgaaKqbagqaaiabg2da9iabek7aILGbaoaaBaaa baGaaGimaaqabaqcfaOaey4kaSYaaabCaeaacqaHYoGydaWgaaqcga yaaiaadQgaaKqbagqaaiaadIhajyaGdaWgaaqaaiaadMgacaWGQbaa beaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaadUgaaKqbakabggHiLd aaaa@4D33@ , (17)

for 1in. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaiaac6caaaa@3D8F@ , 1jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGQbGaeyizImQaam4Aaaaa@3CDB@ . Following the same lines of [3], in order to rank the z-values we can assume that all covariates are equally effective in increasing the response Y, i.e. we initially take β j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIL GbaoaaBaaabaGaamOAaaqabaaaaa@3A17@ ’s all equal, and order the z-values that would correspond to the ordered x-values, where x i = x i1 + x i2 +...+ x ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhajy aGdaWgaaqaaiaadMgaaeqaaKqbakabg2da9iaadIhadaWgaaqcgaya aiaadMgacaaIXaaajuaGbeaacqGHRaWkcaWG4bqcga4aaSbaaeaaca WGPbGaaGOmaaqabaqcfaOaey4kaSIaaiOlaiaac6cacaGGUaGaey4k aSIaamiEamaaBaaajyaGbaGaamyAaiaadUgaaKqbagqaaaaa@4BA4@ (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aIXaGaeyizImQaamyAaiabgsMiJkaad6gacaGGPaaaaa@3E36@ . In other words, the ordered z-values become

z (i) = β 0 + β 1 x i1 * +...+ β k x ik * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQhada WgaaqcgayaaiaacIcacaWGPbGaaiykaaqcfayabaGaeyypa0JaeqOS diwcga4aaSbaaeaacaaIWaaabeaajuaGcqGHRaWkcqaHYoGyjyaGda WgaaqaaiaaigdaaeqaaKqbakaadIhajyaGdaqhaaqaaiaadMgacaaI XaaabaGaaiOkaaaajuaGcqGHRaWkcaGGUaGaaGPaVlaac6cacaaMc8 UaaiOlaiabgUcaRiabek7aILGbaoaaBaaabaGaam4AaaqabaqcfaOa amiEaKGbaoaaDaaabaGaamyAaiaadUgaaeaacaGGQaaaaaaa@5802@ , (18)

where the vector [ 1 x i1 * x i2 * . . x ik * ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaaba qbaeqabeGbaaaabaGaaGymaaqaaiaadIhajyaGdaqhaaqaaiaadMga caaIXaaabaGaaiOkaaaaaKqbagaacaWG4bqcga4aa0baaeaacaWGPb GaaGOmaaqaaiaacQcaaaaajuaGbaGaaiOlaaqaaiaac6caaeaacaWG 4bqcga4aa0baaeaacaWGPbGaam4AaaqaaiaacQcaaaaaaaqcfaOaay 5waiaaw2faamaaCaaabeqaaaaaaaa@490E@  is the ith row of the matrix

X * =[ 1 x 11 * x 12 * ... x 1k * 1 x 21 * x 22 * ... x 2k * . . . ... . . . . ... . 1 x n1 * x n2 * ... x nk * ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada ahaaqabKGbagaacaGGQaaaaKqbakabg2da9maadmaabaqbaeqGbwqb aaaaaeaacaaIXaaabaGaamiEaKGbaoaaDaaabaGaaGymaiaaigdaae aacaGGQaaaaaqcfayaaiaadIhajyaGdaqhaaqaaiaaigdacaaIYaaa baGaaiOkaaaaaKqbagaacaGGUaGaaiOlaiaac6caaeaacaWG4bqcga 4aa0baaeaacaaIXaGaam4AaaqaaiaacQcaaaaajuaGbaGaaGymaaqa aiaadIhajyaGdaqhaaqaaiaaikdacaaIXaaabaGaaiOkaaaaaKqbag aacaWG4bqcga4aa0baaeaacaaIYaGaaGOmaaqaaiaacQcaaaaajuaG baGaaiOlaiaac6cacaGGUaaabaGaamiEaKGbaoaaDaaabaGaaGOmai aadUgaaeaacaGGQaaaaaqcfayaaiaac6caaeaacaGGUaaabaGaaiOl aaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6caaeaacaGGUaaabaGaai Olaaqaaiaac6caaeaacaGGUaGaaiOlaiaac6caaeaacaGGUaaabaGa aGymaaqaaiaadIhajyaGdaqhaaqaaiaad6gacaaIXaaabaGaaiOkaa aaaKqbagaacaWG4bqcga4aa0baaeaacaWGUbGaaGOmaaqaaiaacQca aaaajuaGbaGaaiOlaiaac6cacaGGUaaabaGaamiEaKGbaoaaDaaaba GaamOBaiaadUgaaeaacaGGQaaaaaaaaKqbakaawUfacaGLDbaaaaa@78A0@ , (19)

which is constructed by arranging the rows of the X matrix

X=[ 1 x 11 x 12 ... x 1k 1 x 21 x 22 ... x 2k . . . ... . . . . ... . 1 x n1 x n2 ... x nk ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfacq GH9aqpdaWadaqaauaabeyGfuaaaaaabaGaaGymaaqaaiaadIhajyaG daWgaaqaaiaaigdacaaIXaaabeaaaKqbagaacaWG4bWaaSbaaKGbag aacaaIXaGaaGOmaaqcfayabaaabaGaaiOlaiaac6cacaGGUaaabaGa amiEaKGbaoaaBaaabaGaaGymaiaadUgaaeqaaaqcfayaaiaaigdaae aacaWG4bWaaSbaaKGbagaacaaIYaGaaGymaaqcfayabaaabaGaamiE amaaBaaajyaGbaGaaGOmaiaaikdaaKqbagqaaaqaaiaac6cacaGGUa GaaiOlaaqaaiaadIhadaWgaaqcgayaaiaaikdacaWGRbaajuaGbeaa aeaacaGGUaaabaGaaiOlaaqaaiaac6caaeaacaGGUaGaaiOlaiaac6 caaeaacaGGUaaabaGaaiOlaaqaaiaac6caaeaacaGGUaaabaGaaiOl aiaac6cacaGGUaaabaGaaiOlaaqaaiaaigdaaeaacaWG4bqcga4aaS baaeaacaWGUbGaaGymaaqabaaajuaGbaGaamiEamaaBaaajyaGbaGa amOBaiaaikdaaKqbagqaaaqaaiaac6cacaGGUaGaaiOlaaqaaiaadI hajyaGdaWgaaqaaiaad6gacaWGRbaabeaaaaaajuaGcaGLBbGaayzx aaaaaa@708C@ ,

so as to correspond to the ordered x (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIhajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaaaaa@3ACB@  value (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aIXaGaeyizImQaamyAaiabgsMiJkaad6gacaGGPaaaaa@3E36@ . The MMLEs can be obtained along the same lines as in the Univariate case:

Γ ^ = ( X * T M X * ) 1 X * T Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbfo5ahz aajaGaeyypa0ZaaeWaaeaacaWGybqcga4aaWbaaeqabaGaaiOkaaaa daahaaqabeaacaWGubaaaKqbakaad2eacaaMc8UaaGPaVlaadIfada ahaaqabKGbagaacaGGQaaaaaqcfaOaayjkaiaawMcaaKGbaoaaCaaa beqaaiabgkHiTiaaigdaaaqcfaOaamiwaKGbaoaaCaaabeqaaiaacQ caaaWaaWbaaeqabaGaamivaaaajuaGcqqHuoaraaa@4D67@  (20)

where Δ= [ δ 1 δ 2 . . δ n ] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGabKqbakab=r 5aejabg2da9maadmaabaqbaeqabeqbaaaabaGaeqiTdqwcga4aaSba aeaacaaIXaaabeaaaKqbagaacqaH0oazjyaGdaWgaaqaaiaaikdaae qaaaqcfayaaiaac6caaeaacaGGUaaabaGaeqiTdq2aaSbaaKGbagaa caWGUbaajuaGbeaaaaaacaGLBbGaayzxaaqcga4aaWbaaeqabaGaam ivaaaaaaa@494E@  , δ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes7aKn aaBaaajyaGbaGaamyAaaqcfayabaaaaa@3AA8@  is given by (11) and M is the nxn diagonal matrix

M=[ b 1 0 ... 0 0 b 2 ... 0 ... .. ... .. 0 0 ... b n ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eacq GH9aqpdaWadaqaauaabeqaeqaaaaaabaGaamOyaKGbaoaaBaaabaGa aGymaaqabaaajuaGbaGaaGimaaqaaiaac6cacaGGUaGaaiOlaaqaai aaicdaaeaacaaIWaaabaGaamOyamaaBaaajyaGbaGaaGOmaaqcfaya baaabaGaaiOlaiaac6cacaGGUaaabaGaaGimaaqaaiaac6cacaGGUa GaaiOlaaqaaiaac6cacaGGUaaabaGaaiOlaiaac6cacaGGUaaabaGa aiOlaiaac6caaeaacaaIWaaabaGaaGimaaqaaiaac6cacaGGUaGaai OlaaqaaiaadkgajyaGdaWgaaqaaiaad6gaaeqaaaaaaKqbakaawUfa caGLDbaaaaa@552C@ .

The asymptotic variance-covariance matrix V of the estimators can be derived from the Fisher information matrix V= I 1 ( β 0 , β 1 ,..., β k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaerbdfwBIjxAHb stHrhAaGqbdKqbakaa=zfacaWF9aGaa8xsamaaCaaabeqcgayaaiab gkHiTiaaigdaaaqcfa4aaeWaaeaacqaHYoGydaWgaaqcgayaaiaaic daaKqbagqaaiaacYcacqaHYoGyjyaGdaWgaaqaaiaaigdaaeqaaKqb akaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaeqOSdiwcga4aaSbaae aacaWGRbaabeaaaKqbakaawIcacaGLPaaaaaa@514D@  as given below

V= [ Q i Q i x 1i ... Q i x ki Q i x 1i Q i x 2 1i ... Q i x ki x 1i ... ... ... ... Q i x ki Q i x ki x 1i ... Q i x 2 ki ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAfacq GH9aqpdaWadaqaauaabaqaeqaaaaaabaWaaabqaeaacaWGrbqcga4a aSbaaeaacaWGPbaabeaaaKqbagqabeGaeyyeIuoaaeaadaaeabqaai aadgfadaWgaaqcgayaaiaadMgaaKqbagqaaaqabeqacqGHris5aiaa dIhajyaGdaWgaaqaaiaaigdacaWGPbaabeaaaKqbagaacaGGUaGaai Olaiaac6caaeaadaaeabqaaiaadgfajyaGdaWgaaqaaiaadMgaaeqa aaqcfayabeqacqGHris5aiaadIhadaWgaaqcgayaaiaadUgacaWGPb aajuaGbeaaaeaadaaeabqaaiaadgfajyaGdaWgaaqaaiaadMgaaeqa aaqcfayabeqacqGHris5aiaadIhadaWgaaqcgayaaiaaigdacaWGPb aajuaGbeaaaeaadaaeabqaaiaadgfajyaGdaWgaaqaaiaadMgaaeqa aaqcfayabeqacqGHris5aiaadIhajyaGdaahaaqabeaacaaIYaaaam aaBaaabaGaaGymaiaadMgaaeqaaaqcfayaaiaac6cacaGGUaGaaiOl aaqaamaaqaeabaGaamyuaKGbaoaaBaaabaGaamyAaaqabaaajuaGbe qabiabggHiLdGaamiEaKGbaoaaBaaabaGaam4AaiaadMgaaeqaaKqb akaadIhajyaGdaWgaaqaaiaaigdacaWGPbaabeaaaKqbagaacaGGUa GaaiOlaiaac6caaeaacaGGUaGaaiOlaiaac6caaeaacaGGUaGaaiOl aiaac6caaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caGGUaGaaiOlaiaac6caaeaadaaeabqaaiaa dgfajyaGdaWgaaqaaiaadMgaaeqaaaqcfayabeqacqGHris5aiaadI hadaWgaaqcgayaaiaadUgacaWGPbaajuaGbeaaaeaadaaeabqaaiaa dgfajyaGdaWgaaqaaiaadMgaaeqaaaqcfayabeqacqGHris5aiaadI hadaWgaaqcgayaaiaadUgacaWGPbaajuaGbeaacaWG4bWaaSbaaKGb agaacaaIXaGaamyAaaqcfayabaaabaGaaiOlaiaac6cacaGGUaaaba WaaabqaeaacaWGrbqcga4aaSbaaeaacaWGPbaabeaaaKqbagqabeGa eyyeIuoacaWG4bqcga4aaWbaaeqabaGaaGOmaaaadaWgaaqaaiaadU gacaWGPbaabeaaaaaajuaGcaGLBbGaayzxaaqcga4aaWbaaeqabaGa eyOeI0IaaGymaaaaaaa@B03A@  (21)

where Q i =exp( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgfajy aGdaWgaaqaaiaadMgaaeqaaKqbakabg2da9iGacwgacaGG4bGaaiiC aiaacIcacaWG6bqcga4aaSbaaeaacaWGPbaabeaajuaGcaGGPaaaaa@423E@ . V is estimated by replacing Q i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgfada WgaaqcgayaaiaadMgaaKqbagqaaaaa@39D9@  by

Q ^ i =exp( z ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadgfaga qcamaaBaaajyaGbaGaamyAaaqcfayabaGaeyypa0JaciyzaiaacIha caGGWbGaaiikaiqadQhagaqcamaaBaaajyaGbaGaamyAaaqcfayaba Gaaiykaaaa@425E@ , z ^ (i) = β ^ 0 + β ^ 1 x i1 * +...+ β ^ k x ik * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadQhaga qcaKGbaoaaBaaabaGaaiikaiaadMgacaGGPaaabeaajuaGcqGH9aqp cuaHYoGygaqcaKGbaoaaBaaabaGaaGimaaqabaqcfaOaey4kaSIafq OSdiMbaKaajyaGdaWgaaqaaiaaigdaaeqaaKqbakaadIhajyaGdaqh aaqaaiaadMgacaaIXaaabaGaaiOkaaaajuaGcqGHRaWkcaGGUaGaaG PaVlaac6cacaaMc8UaaiOlaiabgUcaRiqbek7aIzaajaqcga4aaSba aeaacaWGRbaabeaajuaGcaWG4bqcga4aa0baaeaacaWGPbGaam4Aaa qaaiaacQcaaaaaaa@5842@   (1in) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca aIXaGaeyizImQaamyAaiabgsMiJkaad6gacaGGPaaaaa@3E36@ .

Robustness

Measures of influence considered in linear regression models, such as high leverage values, are analogous in the GLM framework. Large leverage values typically mean that there are outliers in covariates. When outliers present in the data, inferences based on MLEs becomes unreliable. In fact, it has been showed that MLEs are not robust in GLMs [21]. In Poisson regression setting, if there are outliers in the continuous covariates, the estimates can be influenced. Thus, we also studied the robustness properties of the derived MMLEs under several outlier models. We considered the Univariate model given by (1) for simplicity. We assumed that α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHj abg2da9iaaicdaaaa@3A36@  and performed a Monte-Carlo study for different values of β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@3878@ , where (n-r) of the observations X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadIfada WgaaqcgayaaiaaigdaaKqbagqaaiaacYcacaaMc8UaamiwaKGbaoaa BaaabaGaaGOmaaqabaqcfaOaaiilaiaaykW7caGGUaGaaGPaVlaac6 cacaaMc8UaaiOlaiaacYcacaWGybWaaSbaaKGbagaacaWGUbaajuaG beaaaaa@49E4@  (we don’t know which) come from the Standard Normal Distribution with σ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZj abg2da9iaaigdaaaa@3A5B@  and the remaining r observations come from the Normal distribution with a scale cσ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadogacq aHdpWCaaa@3982@  where c is a positive constant. We calculated the value of r from the equation r=[ 0.1n+0.5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkhacq GH9aqpdaWadaqaaiaaicdacaGGUaGaaGymaiaaykW7caWGUbGaey4k aSIaaGimaiaac6cacaaI1aaacaGLBbGaayzxaaaaaa@4278@  (Dixon’s outlier model). The outlier models considered are:

(a) (n-r) come from N(0,1) and r come from N(0,1) (No outlier situation),

(b) (n-r) come from N(0,1) and r come from N(0,1.5),

(c) (n-r) come from N(0,1) and r come from N(0,2),

(d) (n-r) come from N(0,1) and r come from N(0,4).

Note that the model (a) above does not involve outliers and is given for the sake of comparisons. In order to be able to make direct comparisons, after generating the X values we divided them by the standard deviation of the distribution for each model. After generating the X values, we calculated z i =α+β x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadQhada WgaaqcgayaaiaadMgaaKqbagqaaiabg2da9iabeg7aHjabgUcaRiab ek7aIjaadIhajyaGdaWgaaqaaiaadMgaaeqaaiaaykW7aaa@4350@ and μ i =exp( z i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeY7aTL GbaoaaBaaabaGaamyAaaqabaqcfaOaeyypa0JaciyzaiaacIhacaGG WbGaaiikaiaadQhadaWgaaqcgayaaiaadMgaaKqbagqaaiaacMcaaa a@431E@  for 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaaaa@3CDD@  to generate Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMfada WgaaqcgayaaiaadMgaaKqbagqaaaaa@39E1@  values from Poisson distribution with mean μ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeY7aTL GbaoaaBaaabaGaamyAaaqabaaaaa@3A2B@  ( 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaaaa@3CDD@ ). The values obtained from 5000 runs are given in (Table 3).

 

 

Model (a): No outlier

Model (b)

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@3878@

n

Bias( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Bias( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Var( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Var( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Bias( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Bias( β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@ )

Var( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Var( β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@ )

0.1

30

0.0326

0.0075

0.0371

0.0385

0.0330

0.0021

0.0377

0.0388

 

50

0.0195

0.0004

0.0214

0.0219

0.0211

0.0033

0.0214

0.0218

 

100

0.0082

0.0039

0.0103

0.0102

0.0094

0.0008

0.0103

0.0104

0.2

30

0.0316

0.0020

0.0377

0.0388

0.0302

0.0034

0.0388

0.0386

 

50

0.0202

0.0029

0.0217

0.0215

0.0200

0.0007

0.0219

0.0212

 

100

0.0108

0.0000

0.0104

0.0102

0.0115

0.0002

0.0108

0.0103

0.4

30

0.0323

0.0033

0.0391

0.0376

0.0346

0.0012

0.0407

0.0385

 

50

0.0207

0.0005

0.0224

0.0209

0.0165

0.0004

0.0230

0.0209

 

100

0.0086

0.0004

0.0109

0.0097

0.0089

0.0009

0.0113

0.0099

 

 

Model (c)

Model (d)

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabek7aIb aa@3878@

n

Bias( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Bias( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Var( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Var( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Bias( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Bias( β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@ )

Var( α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbeg7aHz aajaaaaa@3886@ )

Var( β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqbek7aIz aajaaaaa@3888@ )

0.1

30

0.02827

0.0003

0.0381

0.0413

0.0385

0.0026

0.0372

0.0477

 

50

0.0192

0.0013

0.0220

0.0220

0.0211

0.0004

0.0213

0.0246

 

100

0.0095

0.0001

0.0105

0.0104

0.0099

0.0002

0.0108

0.0107

0.2

30

0.0362

0.0036

0.0396

0.0419

0.0350

0.0080

0.0375

0.0467

 

50

0.0163

0.0014

0.0208

0.0211

0.0189

0.0057

0.0221

0.0242

 

100

0.0111

0.0010

0.0106

0.0105

0.0109

0.0013

0.0099

0.0103

0.4

30

0.0313

0.0015

0.0396

0.0370

0.0314

0.0017

0.0379

0.0461

 

50

0.0178

0.0030

0.0226

0.0207

0.0166

0.0001

0.0218

0.0223

 

100

0.0079

0.0003

0.0110

0.0093

0.0074

0.0039

0.0111

0.0088

Table 3: Empirical biases and variances from Dixon’s outlier models.

As can be seen from the table, the biases in the estimates are negligible for all models. The variances Var( β ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaabAfaca qGHbGaaeOCaiaacIcacuaHYoGygaqcaiaacMcaaaa@3C93@  (hence the Wald statistics W) are almost the same for a given n for the models (a), (b), (c) and (d), which means that the MMLEs are robust to outliers in the covariate. Note that the MML methodology achieves robustness through the t (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshajy aGdaWgaaqaaiaacIcacaWGPbGaaiykaaqabaaaaa@3AC7@  ( 1in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepKI8Vfc8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaigdacq GHKjYOcaWGPbGaeyizImQaamOBaaaa@3CDD@ ) values.

Conclusion

Poisson regression serves as a useful technique to model count data. The MLEs in Poisson regression are obtained via Newton-type algorithms; however these algorithms might not converge or converge to inaccurate values. In this study we derived the explicit MMLEs for Poisson regression. We also considered the case where there are outliers in the (continuous) covariate, which generally is the case in real life applications, and searched the properties of the derived MMLEs under several data violations. Although the scope of the simulations reported here is limited, we can conclude that MML methodology provides robust estimation in Poisson regression.

Acknowledgement

We are grateful to the reviewer for the helpful comment which helped to improve this manuscript.

References

  1.  Cameron AC, Trivedi PK (1998) Regression Analysis of Count Data. Cambridge Univ. Press, NY, USA.
  2.  Casella G, Berger R (2002) Statistical Inference. (2nd edn), Thomson Learning, Pacific Grove, CA, USA.
  3. Tiku ML, Vaughan DC (1997) Logistic and nonlogistic density functions in binary regression with nonstochastic covariates. Biometrical Journal 39(8): 883-898.
  4.  Vaughan DC (2002) The generalized secant hyperbolic distribution and its properties. Communications in Statistics - Theory and Methods 31(2): 219-238.
  5. Barnett VD (1966) Evaluation of the maximum likelihood estimator when the likelihood equation has multiple roots. Biometrika 53(1): 151-165.
  6. Lee KR, Kapadia CH, Dwight BB (1980) On estimating the scale parameter of Rayleigh distribution from censored samples. Statistische Hefte 21(1): 14-20.
  7. Silva JMC, Tenreyro S (2011) Poisson: Some convergence Issues. The Stata Journal 11(2): 207-212.
  8. Silva JMC, Tenreyro S (2010) On the Existence of the Maximum Likelihood Estimates in Poisson Regression. Economics Letters 107: 310-312.
  9. Tiku ML (1967) Estimating the mean and standard deviation from a censored normal sample. Biometrika 54(1): 155-165.
  10. Bhattacharyya GK (1985). The asymptotics of maximum likelihood and related estimators based on type II censored data. Journal of the American Statistical Association 80: 398-404.
  11. Tan WY (1985) On Tiku’s robust procedure-a Bayesian insight. Journal of Statistical Planning and Inference 11(3): 329-340.
  12. Tiku ML, Tan WY, Balakrishnan N (1986) Robust Inference. Marcel Dekker, New York, USA.
  13. Tiku ML, Suresh RP (1992) A new method of estimation for location and scale parameters. Journal of Statitical Planning and Inference 30: 281-292.
  14. Vaughan DC (1992) On the Tiku-Suresh method of estimation. Communications in Statistics-Theory and Methods 21(2): 451-469.
  15. Vaughan DC, Tiku ML (2000) Estimation and hypothesis testing for nonnormal bivariate distribution with applications. Mathematical and Computer Modelling 32: 53-67.
  16. Oral E, Gunay S (2004) Stochastic Covariates in Binary Regression. Hacettepe Journal of Mathematics and Statistics 33: 97-109.
  17. Oral E (2006) Binary Regression with Stochastic Covariates. Communications in Statistics Theory and Methods 35: 1429-1447.
  18. Oral E (2011) Parameter Estimation in Generalized Linear Models through Modified Maximum Likelihood. International Statistical Institute. Proceedings of the 58th World Statistical Congress.
  19. Agresti A (1996) Categorical Data Analysis. John Wiley and Sons, New York, USA.
  20. Kwan R. Lee, Kapadia C H , Dwight B. Brock (1980) On Estimating the Scale Parameter of the Rayleigh distribution from Doubly Censored Samples. Statistische Hefte 21(1): 14-29.
  21. Cantoni E, Ronchetti E (2001) Robust Inference for Generalized Linear Models. Journal of the American Statistical Association 96(455):1022-1030.
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