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Biometrics & Biostatistics International Journal

Research Article Volume 7 Issue 4

On circular systematic sampling in the presence of linear trend

Subramani J

Department of Statistics, Pondicherry university, India

Correspondence: Subramani J, Department of Statistics, Pondicherry University, RV Nagar, Kalapet, Puducherry?605 014, India

Received: April 13, 2018 | Published: July 20, 2018

Citation: Subramani J. On circular systematic sampling in the presence of linear trend. Biom Biostat Int J. 2018;7(4):286-292. DOI: 10.15406/bbij.2018.07.00220

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Abstract

The present paper deals with the computation of the circular systematic sample mean and its variance in the presence of linear trend among the population values. As a result, explicit expression for the variance of circular systematic sample mean is obtained for a pre-assigned fixed sample size and the population size .The efficiency of circular systematic sampling with that of simple random sampling without replacement is assessed algebraically and also for certain natural populations. It is observed that circular systematic sampling performs better than the simple random sampling without replacement.

Keywords: circular systematic sampling, linear systematic sampling, linear trend, optimal circular systematic sampling, simple random sampling, trend free sampling, yates type end corrections

Introduction

Let U=( U 1 , U 2 ,, U N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8xvaiabg2da9maabmaapaqaa8qacaWFvbWdamaaBaaa juaibaWdbiaaigdaaKqba+aabeaapeGaaiilaiaa=vfapaWaaSbaaK qbGeaapeGaaGOmaaqcfa4daeqaa8qacaGGSaGaeyOjGWRaaiilaiaa =vfapaWaaSbaaKqbGeaapeGaa8Ntaaqcfa4daeqaaaWdbiaawIcaca GLPaaaaaa@45E0@ be a finite population with distinct and identifiable units and Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8xwaaaa@3780@  be a real variable with value Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMfada WgaaqcfasaaiaadMgaaKqbagqaaaaa@3922@ measured on giving a vector of measurements Y=( Y 1 , Y 2 ,, Y N ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8xwaiabg2da9maabmaapaqaa8qacaWFzbWdamaaBaaa juaibaWdbiaaigdaaKqba+aabeaapeGaaiilaiaa=LfapaWaaSbaaK qbGeaapeGaaGOmaaqcfa4daeqaa8qacaGGSaGaeyOjGWRaaiilaiaa =LfapaWaaSbaaKqbGeaapeGaa8Ntaaqcfa4daeqaaaWdbiaawIcaca GLPaaaaaa@45F1@ . If Y i =a+ib, i=1, 2, , N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8xwa8aadaWgaaqcfasaa8qacaWFPbaajuaGpaqabaWd biabg2da9iaa=fgacqGHRaWkcaWFPbGaa8NyaiaacYcacaWFGcGaa8 xAaiabg2da9iaaigdacaGGSaGaa8hOaiaaikdacaGGSaGaa8hOaiab gAci8kaacYcacaWFGcGaa8Ntaaaa@4B24@ then the population is called a labelled population with a perfect linear trend among the population values. The problem is in general, to estimate the population mean Y ¯ = 1 N i=1 N Y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmywayaaraGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaac bmWdbiaa=5eaaaWaaybCaeqajuaipaqaa8qacaWFPbGaeyypa0JaaG ymaaWdaeaapeGaa8Ntaaqcfa4daeaapeGaeyyeIuoaaiaa=LfapaWa aSbaaKqbGeaapeGaa8xAaaqcfa4daeqaaaaa@440D@ on the basis of a sample of size selected from the finite population U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqbacbaaa aaaaaapeGaa8xvaaaa@3778@ . Any ordered sequence S=( u 1 , u 2 ,, u n )=( U i1 , U i2 ,, U in ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaC4uaiabg2da9maabmaapaqaa8qacaWH1bWdamaaBaaajuai baWdbiaaigdaaKqba+aabeaapeGaaiilaiaahwhapaWaaSbaaKqbGe aapeGaaGOmaaqcfa4daeqaa8qacaGGSaGaeyOjGWRaaiilaiaahwha paWaaSbaaKqbGeaapeGaaCOBaaqcfa4daeqaaaWdbiaawIcacaGLPa aacqGH9aqpdaqadaWdaeaapeGaaCyva8aadaWgaaqcfasaa8qacaWH PbGaaGymaaqcfa4daeqaa8qacaGGSaGaaCyva8aadaWgaaqcfasaa8 qacaWHPbGaaGOmaaqcfa4daeqaa8qacaGGSaGaeyOjGWRaaiilaiaa hwfapaWaaSbaaKqbGeaapeGaaCyAaiaah6gaaKqba+aabeaaa8qaca GLOaGaayzkaaaaaa@57F6@ 1 i 1 N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGymaiabgsMiJkaadMgadaWgaaqcfasaaiaaigdaaKqbagqa aiabgsMiJkaad6eaaaa@3E17@ ,  and 1ln MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGymaiabgsMiJkaadYgacqGHKjYOcaWGUbaaaa@3CA2@ is called a sample of size . Several sampling schemes like, simple random sampling without replacement, systematic sampling are available in the literature for selecting a sample of fixed size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaaaa@378C@ from a finite population of size N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaaaa@376C@ . For the case of finite population with a linear trend among the population values and N=kn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaadUgacaWGUbaaaa@3A55@ , the linear systematic sampling (LSS) is normally recommended for selecting a random sample of fixed size . Further it is shown algebraically that the estimator from linear systematic sampling is better than the estimator provided by simple random sampling without replacement in the presence of linear trend. The performance of systematic sample mean can be improved further by introducing some modifications on the selection of the samples which includes the centered systematic sampling,1 balanced systematic sampling,2 modified systematic sampling3 and also by introducing changes in the estimator itself like, Yates type end corrections.4 In recent times several attempts are made to find an alternative to LSS. In this connection it is worth to note the following works which are alternative to LSS. Diagonal systematic sampling,5 Generalized diagonal systematic sampling,6 ,6 Determinant sampling,8 Modified linear systematic sampling,9 -11 Generalized modified linear systematic sampling,12 Star type systematic sampling,13 Remainder linear systematic sampling,14 Generalized systematic sampling,15 Remainder linear systematic sampling,14 Modified balanced circular systematic sampling,16 Modified systematic sampling by Huang,17 Lahiri,18 Leu & Tsui,19 Sampath & Uthayakumaran,20 Singh & Garg,121 Singh & Singh,22 Uthayakumaran,23 . For further discussions on linear systematic sampling the readers are referred to Cochran,24 , Gautschi,25 Khan et al.,26 ,27 Gupta & Kabe,28 Murthy,29 Singh,30 Sukhatme et al.,31 Fountain & Pathak,32 Mukerjee & Sengupta,33 Murthy & Rao,34 Wu,35 Zinger36 and the references cited there in.

If the population size N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaaaa@376C@ is not a multiple of sample size n(Nkn) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaiaacIcacaGGobGaeyiyIKRaai4Aaiaac6gacaGGPaaa aa@3D5F@ then the linear systematic sampling is not applicable for selecting a sample of fixed size .In such situations, circular systematic sampling (CSS) introduced by Lahiri18 cited in Murthy29 provides a constant sample size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaaaa@378C@ and the selected units are distinct if and only if N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaaaa@376C@ and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ are relatively prime numbers. However the circular systematic samples are multiple copies of linear systematic samples when N=kn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaadUgacaWGUbaaaa@3A55@ and provides repetition of sampled units when N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiOtaaaa@376B@ and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ are not relatively prime numbers. As pointed out by Subramani et al.,13 the problems in circular systematic sampling are the following:

  1. The choice for the sampling interval, which ensures the distinct units in the sample and minimum variance
  2. The explicit expressions for the variance of CSS sample mean which is useful to assess the efficiency of the CSS with other sampling schemes, particularly SRSWOR.

Several attempts have been made in the past to get a suitable value for the sampling interval for the given values of population size N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaaaa@376C@ and sample size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaaaa@378C@ . Murthy & Rao34 has given the choice for k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ asIt may be noted that the sample mean is unbiased for the population mean for all values of k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ , through the spread of the sample and hence efficiency is better if k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ is taken as an integer nearest to N/n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiaac+cacaGGUbaaaa@3911@ . However, if repetition of the same unit in a sample is to be avoided, then it is desirable to take the sampling interval as [N/n] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaai4waiaad6eacaGGVaGaaiOBaiaac2faaaa@3AD1@ . It is shown that necessary and sufficient condition for all samples in CSS to have distinct units is that N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaaaa@376C@ and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ are relatively co-prime”.37 Bell house38–40 has suggested that the choice for the sampling interval k=[ ( N/n )+( 1/2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9maadmaabaWaaeWaaeaacaWGobGaai4laiaa d6gaaiaawIcacaGLPaaacqGHRaWkdaqadaqaaiaaigdacaGGVaGaaG OmaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@4318@ when N( n1 )k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabgcMi5oaabmaabaGaamOBaiabgkHiTiaaigdaaiaa wIcacaGLPaaacaWGRbaaaa@3E47@ and k=( N/n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9maabmaabaGaamOtaiaac+cacaWGUbaacaGL OaGaayzkaaaaaa@3C91@ when N=( n1 )k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9maabmaabaGaamOBaiabgkHiTiaaigdaaiaa wIcacaGLPaaacaWGRbaaaa@3D86@ .

Sengupta & Chattopadhyay33 have proposed the following: “A necessary and sufficient condition for a circular systematic sampling of size, drawn from a population of N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaaaa@376C@ units with sampling interval k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ ,to contain all distinct units is that [ N,k ]/kn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaamWaaeaacaWGobGaaiilaiaacUgaaiaawUfacaGLDbaacaGG VaGaam4AaiabgwMiZkaad6gaaaa@3F59@ or equivalently, N/( n,k )n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiaac+cadaqadaqaaiaac6gacaGGSaGaam4AaaGaayjk aiaawMcaaiabgwMiZkaad6gaaaa@3EF3@ where [ N,k ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaamWaaeaacaWGobGaaiilaiaacUgaaiaawUfacaGLDbaaaaa@3AFD@ and ( n,k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaaeaacaGGUbGaaiilaiaadUgaaiaawIcacaGLPaaaaaa@3AB4@ denote respectively the least common multiple and the greatest common divisor of N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaaaa@376C@ and k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaaaa@3789@ ”. However it seems there is no theoretical result or empirical study available to justify the choice of k which ensures the efficient estimator or the estimator with minimum variance compared to other choices ofk .”

Recently Subramani et al.,41 and Subramani & Singh42 have made attempts to address the above problems and introduced the optimal circular systematic sampling (OCSS) together with the explicit expressions for its variance and the sampling interval in the presence of linear trend. In OCSS, the choice for the sampling interval k is

kn mod N=±1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiaad6gacaGGGcGaamyBaiaad+gacaWGKbGaaiiOaiaa d6eacqGH9aqpcqGHXcqScaaIXaaaaa@4215@ (1.1)

where kn mod N=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiaad6gacaGGGcGaamyBaiaad+gacaWGKbGaaiiOaiaa d6eacqGH9aqpcqGHsislcaaIXaaaaa@4114@ represents kn mod N=N1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiaad6gacaGGGcGaamyBaiaad+gacaWGKbGaaiiOaiaa d6eacqGH9aqpcaWGobGaeyOeI0IaaGymaaaa@41E7@

For the hypothetical population with values Y i =a+ib, i=1, 2, , N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaeywa8aadaWgaaqcfasaa8qacaqGPbaajuaGpaqabaWdbiab g2da9iaabggacqGHRaWkcaqGPbGaaeOyaiaacYcacaqGGcGaaeyAai abg2da9iaaigdacaGGSaGaaeiOaiaaikdacaGGSaGaaeiOaiabgAci 8kaacYcacaqGGcGaaeOtaaaa@4B2D@ ,

The variance of OCSS sample mean is given as

V( y ¯ ocss )= ( N1 )( N+1 ) 12 n 2 b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOvamaabmaapaqaaiqadMhagaqeamaaBaaajuaibaWdbiaa d+gacaWGJbGaam4CaiaadohaaKqba+aabeaaa8qacaGLOaGaayzkaa Gaeyypa0ZaaSaaa8aabaWdbmaabmaapaqaa8qacaWGobGaeyOeI0Ia aGymaaGaayjkaiaawMcaamaabmaapaqaa8qacaWGobGaey4kaSIaaG ymaaGaayjkaiaawMcaaaWdaeaapeGaaGymaiaaikdacaWGUbWdamaa Caaabeqcfasaa8qacaaIYaaaaaaajuaGcaWGIbWdamaaCaaabeqcfa saa8qacaaIYaaaaaaa@4EBE@ (1.2)

The variance of SRSWOR sample mean is given as

V( y ¯ r )  = ( Nn )( N+1 ) 12n b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOvamaabmaapaqaaiqadMhagaqeamaaBaaajuaibaWdbiaa dkhaaKqba+aabeaaa8qacaGLOaGaayzkaaGaaiiOaiaacckacqGH9a qpdaWcaaWdaeaapeWaaeWaa8aabaWdbiaad6eacqGHsislcaWGUbaa caGLOaGaayzkaaWaaeWaa8aabaWdbiaad6eacqGHRaWkcaaIXaaaca GLOaGaayzkaaaapaqaa8qacaaIXaGaaGOmaiaad6gaaaGaamOya8aa daahaaqabKqbGeaapeGaaGOmaaaaaaa@4CB0@  (1.3)

Example 1.1: The procedure of obtaining the optimum value of is explained for the fixed values of sample size and the population size .

If N=12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaigdacaaIYaaaaa@39E9@ and n=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaiabg2da9iaaiwdaaaa@3951@ then k=7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9iaaiEdaaaa@3950@ . That is kn mod N=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiaad6gacaGGGcGaamyBaiaad+gacaWGKbGaaiiOaiaa d6eacqGH9aqpcqGHsislcaaIXaaaaa@4114@
If N=12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOtaiabg2da9iaaigdacaaIYaaaaa@39E9@ and n=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaiabg2da9iaaiwdaaaa@3951@ then k=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9iaaiwdaaaa@394E@ . That is kn mod N=+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiaad6gacaGGGcGaamyBaiaad+gacaWGKbGaaiiOaiaa d6eacqGH9aqpcqGHRaWkcaaIXaaaaa@4109@

The selected OCSS samples, their means, expected value and the variance are given for the sampling interval k=7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9iaaiEdaaaa@3950@ and k=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9iaaiwdaaaa@394E@ in the following Table 1.1 & 1.2:

For both the cases of sampling interval k=7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9iaaiEdaaaa@3950@ and k=5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Aaiabg2da9iaaiwdaaaa@394E@ , it is obtained that  V( y ¯ ocss )= 1 N i=1 N y ¯ i 2 Y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaeiOaiaadAfadaqadaWdaeaaceWG5bGbaebadaWgaaqcfasa a8qacaWGVbGaam4yaiaadohacaWGZbaajuaGpaqabaaapeGaayjkai aawMcaaiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaWGobaa amaawahabeqcfaYdaeaapeGaamyAaiabg2da9iaaigdaa8aabaWdbi aad6eaaKqba+aabaWdbiabggHiLdaapaGabmyEayaaraWaaSbaaKqb GeaapeGaamyAaaqcfa4daeqaamaaCaaabeqcfasaa8qacaaIYaaaaK qbakabgkHiTiqadMfagaqea8aadaahaaqabKqbGeaapeGaaGOmaaaa aaa@5193@

= 512.72 12 6.5 2 =42.7266742.25=0.476667 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeyypa0ZaaSaaa8aabaWdbiaaiwdacaaIXaGaaGOmaiaac6ca caaI3aGaaGOmaaWdaeaapeGaaGymaiaaikdaaaGaeyOeI0IaaGOnai aac6cacaaI1aWdamaaCaaabeqcfasaa8qacaaIYaaaaKqbakabg2da 9iaaisdacaaIYaGaaiOlaiaaiEdacaaIYaGaaGOnaiaaiAdacaaI3a GaeyOeI0IaaGinaiaaikdacaGGUaGaaGOmaiaaiwdacqGH9aqpcaaI WaGaaiOlaiaaisdacaaI3aGaaGOnaiaaiAdacaaI2aGaaG4naaaa@5516@

The value of the variance given above is coincided with the value obtained through the formula given in Table 1.2

Sample Number

Sample Values

OCSS Mean

1

1

6

11

4

9

6.2

2

2

7

12

5

10

7.2

3

3

8

1

6

11

5.8

4

4

9

2

7

12

6.8

5

5

10

3

8

1

5.4

6

6

11

4

9

2

6.4

7

7

12

5

10

3

7.4

8

8

1

6

11

4

6.0

9

9

2

7

12

5

7.0

10

10

3

8

1

6

5.6

11

11

4

9

2

7

6.6

12

12

5

10

3

8

7.6

Table 1.2 OCSS samples and their means for the sampling interval

Further it seems, no attempt is made to derive the explicit expression for the variance of circular systematic sample mean even after 65 years of its introduction for the case of labelled population with a perfect linear trend. As a consequence, the efficiency of circular systematic sampling is not assessed algebraically with that of simple random sampling without replacement.

The points noted above are motivating the present study, which deals with the following:

  1. To derive the explicit expression for the variance of circular systematic sample mean for the population with a perfect linear trend among the population values
  2. To derive the explicit expressions for the Yates type end corrections for further improvements on the circular systematic sampling
  3. To assess the relative performance of circular systematic sampling with that of simple random sampling without replacement and the optimal circular systematic sampling algebraically and also for certain natural populations.
  4. To deduce the optimum values for the sampling fraction and the optimum variance for the circular systematic sampling.

Circular systematic sampling
As stated earlier, the LSS is not applicable when the population size is not a multiple of sample size for selecting a sample of fixed size whereas the CSS introduced by Lahiri18 cited in Murthy,29 provides a constant sample size . The steps involved in CSS for selecting a sample of size with sampling interval are given below:
Step 1: Arrange the population units around a circle
Step 2: Select a random number such that
Step3: For selecting a circular systematic sample of size select every elements from the random start in the circle until elements are accumulated.

The selected units U r , U r+k , U r+2k ,, U r+( n1 )k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8xva8aadaWgaaqcfasaa8qacaWFYbaajuaGpaqabaWd biaacYcacaWFvbWdamaaBaaajuaibaWdbiaa=jhacqGHRaWkcaWFRb aajuaGpaqabaWdbiaacYcacaWFvbWdamaaBaaajuaibaWdbiaa=jha cqGHRaWkcaaIYaGaa83Aaaqcfa4daeqaa8qacaGGSaGaeyOjGWRaai ilaiaa=vfapaWaaSbaaKqbGeaapeGaa8NCaiabgUcaRKqbaoaabmaa juaipaqaa8qacaWFUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaa=T gaaKqba+aabeaaaaa@5194@ be the circular systematic sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaaaa@378C@ for the random start r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOCaaaa@3790@ . If r+jk>N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8NCaiabgUcaRiaa=PgacaWFRbGaeyOpa4Jaa8Ntaaaa @3C28@ then select the item corresponding to { r+jkN } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaiWaa8aabaWdbiaahkhacqGHRaWkcaWHQbGaaC4AaiabgkHi Tiaah6eaaiaawUhacaGL9baaaaa@3E71@

The variance of the circular systematic sample mean is obtained as given below:

V( y ¯ css )= 1 N i=1 N ( y ¯ i Y ¯ ) 2 = 1 N i=1 N y ¯ i 2 Y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8Nvamaabmaapaqaa8qaceWF5bGbaebapaWaaSbaaKqb GeaapeGaa83yaiaa=nhacaWFZbaajuaGpaqabaaapeGaayjkaiaawM caaiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaWFobaaamaa wahabeqcfaYdaeaapeGaa8xAaiabg2da9iaaigdaa8aabaWdbiaa=5 eaaKqba+aabaWdbiabggHiLdaadaqadaWdaeaapeGab8xEayaaraWd amaaBaaajuaibaWdbiaa=LgaaKqba+aabeaapeGaeyOeI0Iab8xway aaraaacaGLOaGaayzkaaWdamaaCaaabeqcfasaa8qacaaIYaaaaKqb akabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaWFobaaamaawa habeqcfaYdaeaapeGaa8xAaiabg2da9iaaigdaa8aabaWdbiaa=5ea aKqba+aabaWdbiabggHiLdaaceWF5bGbaebapaWaaSbaaeaapeGaa8 xAaaWdaeqaamaaCaaabeqcfasaa8qacaaIYaaaaKqbakabgkHiTiqa =Lfagaqea8aadaahaaqabKqbGeaapeGaaGOmaaaaaaa@60B9@  (2.1)

Example 2.1: The procedure of obtaining the value of the sampling interval in the case of circular systematic sampling is explained for the fixed values of sample size and the population size .

If and then . That is k=Int( N n )=Int( 12 5 )=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa83Aaiabg2da9iaa=LeacaWFUbGaa8hDamaabmaapaqa a8qadaWcaaWdaeaapeGaa8NtaaWdaeaapeGaa8NBaaaaaiaawIcaca GLPaaacqGH9aqpcaWFjbGaa8NBaiaa=rhadaqadaWdaeaapeWaaSaa a8aabaWdbiaaigdacaaIYaaapaqaa8qacaaI1aaaaaGaayjkaiaawM caaiabg2da9iaaikdaaaa@489B@

The selected CSS samples, their means, expected value and the variance are given for the sampling interval in the following Table 2.1.

Sample number

Sample values

CSS mean

1

1

3

5

7

9

5

2

2

4

6

8

10

6

3

3

5

7

9

11

7

4

4

6

8

10

12

8

5

5

7

9

11

1

6.6

6

6

8

10

12

2

7.6

7

7

9

11

1

3

6.2

8

8

10

12

2

4

7.2

9

9

11

1

3

5

5.8

10

10

12

2

4

6

6.8

11

11

1

3

5

7

5.4

12

12

2

4

6

8

6.4

Table 2.1 CSS samples and their means for the sampling interval

For the cases of sampling interval and , it is obtained that V( y ¯ css ) = 1 N i=1 N y ¯ i 2 Y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOvamaabmaapaqaaiqadMhagaqeamaaBaaajuaibaWdbiaa dogacaWGZbGaam4Caaqcfa4daeqaaaWdbiaawIcacaGLPaaacaGGGc Gaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6eaaaWaaybC aeqajuaipaqaa8qacaWGPbGaeyypa0JaaGymaaWdaeaapeGaamOtaa qcfa4daeaapeGaeyyeIuoaa8aaceWG5bGbaebadaWgaaqcfasaa8qa caWGPbaajuaGpaqabaWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaey OeI0IabmywayaaraWdamaaCaaabeqcfasaa8qacaaIYaaaaaaa@50A0@

= 515.6 12 6.5 2 =42.9666742.25=0.716667 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeyypa0ZaaSaaa8aabaWdbiaaiwdacaaIXaGaaGynaiaac6ca caaI2aaapaqaa8qacaaIXaGaaGOmaaaacqGHsislcaaI2aGaaiOlai aaiwdapaWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaeyypa0JaaGin aiaaikdacaGGUaGaaGyoaiaaiAdacaaI2aGaaGOnaiaaiEdacqGHsi slcaaI0aGaaGOmaiaac6cacaaIYaGaaGynaiabg2da9iaaicdacaGG UaGaaG4naiaaigdacaaI2aGaaGOnaiaaiAdacaaI3aaaaa@545F@

Computation of circular systematic sample means Consider the labelled population with the population values Y i =a+ib, i=1, 2, , N.  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadKqbacbaaa aaaaaapeGaa8xwa8aadaWgaaqcfasaa8qacaWFPbaajuaGpaqabaWd biabg2da9iaa=fgacqGHRaWkcaWFPbGaa8NyaiaacYcacaWFGcGaa8 xAaiabg2da9iaaigdacaGGSaGaa8hOaiaaikdacaGGSaGaa8hOaiab gAci8kaacYcacaWFGcGaa8Ntaiaac6cacaqGGcaaaa@4CF8@  

The population mean is Y ¯ =a+[ ( N+1 ) 2 ]b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qaceWFzbGbaebacqGH9aqpcaWFHbGaey4kaSYaamWaa8aabaWd bmaalaaapaqaa8qadaqadaWdaeaapeGaa8NtaiabgUcaRiaaigdaai aawIcacaGLPaaaa8aabaWdbiaaikdaaaaacaGLBbGaayzxaaGaa8Ny aaaa@42BE@    (2.2)

After a little algebra the circular systematic sample means are obtained as:

y ¯ i. = a+[ i+ k( n1 ) 2 ]b,  i=1,2,3Nk( n1 )=L( say ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGab8xEay aaraWaaSbaaSqaaabaaaaaaaaapeGaaCyAaiaac6caa8aabeaak8qa cqGH9aqppaqbaeaabeGaaaqaa8qacaWHHbGaey4kaSYaamWaa8aaba WdbiaahMgacqGHRaWkdaWcaaWdaeaapeGaaC4Aamaabmaapaqaa8qa caWHUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGOmaa aaaiaawUfacaGLDbaacaWHIbGaaiilaiaacckaa8aabaWdbiaahMga cqGH9aqpcaaIXaGaaiilaiaaikdacaGGSaGaaG4maiabgAci8kaah6 eacqGHsislcaWHRbWaaeWaa8aabaWdbiaah6gacqGHsislcaaIXaaa caGLOaGaayzkaaGaeyypa0JaaCitamaabmaapaqaa8qacaWHZbGaaC yyaiaahMhaaiaawIcacaGLPaaaaaaaaa@5E84@

y ¯ i. = a+[ i+ k( n1 ) 2 N n ]b,  i=L+1, L+2,,L+k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGab8xEay aaraWaaSbaaSqaaabaaaaaaaaapeGaa8xAaiaac6caa8aabeaak8qa cqGH9aqppaqbaeqabeGaaaqaa8qacaWFHbGaey4kaSYaamWaa8aaba Wdbiaa=LgacqGHRaWkdaWcaaWdaeaapeGaa83Aamaabmaapaqaa8qa caWFUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGOmaa aacqGHsisldaWcaaWdaeaapeGaa8NtaaWdaeaapeGaa8NBaaaaaiaa wUfacaGLDbaacaWFIbGaaiilaiaa=bkaa8aabaWdbiaa=LgacqGH9a qpcaWFmbGaey4kaSIaaGymaiaacYcacaWFGcGaa8htaiabgUcaRiaa ikdacaGGSaGaeyOjGWRaaiilaiaa=XeacqGHRaWkcaWFRbaaaaaa@5AE6@

y ¯ i. = a+[ i+ k( n1 ) 2 2N n ]b,  i=L+k+1, L+k+2,,L+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGab8xEay aaraWaaSbaaSqaaabaaaaaaaaapeGaa8xAaiaac6caa8aabeaak8qa cqGH9aqppaqbaeqabeGaaaqaa8qacaWFHbGaey4kaSYaamWaa8aaba Wdbiaa=LgacqGHRaWkdaWcaaWdaeaapeGaa83Aamaabmaapaqaa8qa caWFUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGOmaa aacqGHsisldaWcaaWdaeaapeGaaGOmaiaa=5eaa8aabaWdbiaa=5ga aaaacaGLBbGaayzxaaGaa8NyaiaacYcacaWFGcaapaqaa8qacaWFPb Gaeyypa0Jaa8htaiabgUcaRiaa=TgacqGHRaWkcaaIXaGaaiilaiaa =bkacaWFmbGaey4kaSIaa83AaiabgUcaRiaaikdacaGGSaGaeyOjGW Raaiilaiaa=XeacqGHRaWkcaaIYaGaa83Aaaaaaaa@5FFA@

y ¯ i. = a+[ i+ k( n1 ) 2 3N n ]b, i=L+2k+1, L+2k+2,,L+3k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGab8xEay aaraWaaSbaaSqaaabaaaaaaaaapeGaa8xAaiaac6caa8aabeaak8qa cqGH9aqppaqbaeqabeGaaaqaa8qacaWFHbGaey4kaSYaamWaa8aaba Wdbiaa=LgacqGHRaWkdaWcaaWdaeaapeGaa83Aamaabmaapaqaa8qa caWFUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGOmaa aacqGHsisldaWcaaWdaeaapeGaaG4maiaa=5eaa8aabaWdbiaa=5ga aaaacaGLBbGaayzxaaGaa8NyaiaacYcaa8aabaWdbiaa=LgacqGH9a qpcaWFmbGaey4kaSIaaGOmaiaa=TgacqGHRaWkcaaIXaGaaiilaiaa =bkacaWFmbGaey4kaSIaaGOmaiaa=TgacqGHRaWkcaaIYaGaaiilai abgAci8kaacYcacaWFmbGaey4kaSIaaG4maiaa=Tgaaaaaaa@6053@

y ¯ i. = a+[ i+ k( n1 ) 2 ( n1 )N n ]b, i=L+( n2 )k+1, L+( n2 )k+2,,L+( n1 )k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGab8xEay aaraWaaSbaaSqaaabaaaaaaaaapeGaa8xAaiaac6caa8aabeaak8qa cqGH9aqppaqbaeqabeGaaaqaa8qacaWFHbGaey4kaSYaamWaa8aaba Wdbiaa=LgacqGHRaWkdaWcaaWdaeaapeGaa83Aamaabmaapaqaa8qa caWFUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGOmaa aacqGHsisldaWcaaWdaeaapeWaaeWaa8aabaWdbiaa=5gacqGHsisl caaIXaaacaGLOaGaayzkaaGaa8NtaaWdaeaapeGaa8NBaaaaaiaawU facaGLDbaacaWFIbGaaiilaaWdaeaapeGaa8xAaiabg2da9iaa=Xea cqGHRaWkdaqadaWdaeaapeGaa8NBaiabgkHiTiaaikdaaiaawIcaca GLPaaacaWFRbGaey4kaSIaaGymaiaacYcacaWFGcGaa8htaiabgUca Rmaabmaapaqaa8qacaWFUbGaeyOeI0IaaGOmaaGaayjkaiaawMcaai aa=TgacqGHRaWkcaaIYaGaaiilaiabgAci8kaacYcacaWFmbGaey4k aSYaaeWaa8aabaWdbiaa=5gacqGHsislcaaIXaaacaGLOaGaayzkaa Gaa83Aaaaaaaa@6E5F@ (2.3)

Remark 2.1: Since L=Nk( n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFmbGaeyypa0Jaa8NtaiabgkHiTiaa=TgadaqadaWdaeaa peGaa8NBaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaa@3FAA@  then L+( n1 )k=N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFmbGaey4kaSYaaeWaa8aabaWdbiaa=5gacqGHsislcaaI XaaacaGLOaGaayzkaaGaa83Aaiabg2da9iaa=5eaaaa@3F9F@

From the above expressions the sum of the CSS sample means is obtained as

i=1 N y ¯ i = i=1 N [ a+ [ i+ k( n1 ) 2 ]b ] Nb n j=1 n1 i=1 k j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGfWbqabSWdaeaaieWapeGaa8xAaiabg2da9iaaigdaa8aabaWd biaa=5eaa0WdaeaapeGaeyyeIuoaaOWdaiqa=LhagaqeamaaBaaale aapeGaa8xAaaWdaeqaaOWdbiabg2da9maawahabeWcpaqaa8qacaWF PbGaeyypa0JaaGymaaWdaeaapeGaa8Ntaaqdpaqaa8qacqGHris5aa GcdaWadaWdaeaapeGaa8xyaiabgUcaRiaa=bkadaWadaWdaeaapeGa a8xAaiabgUcaRmaalaaapaqaa8qacaWFRbWaaeWaa8aabaWdbiaa=5 gacqGHsislcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaIYaaaaaGa ay5waiaaw2faaiaa=jgaaiaawUfacaGLDbaacqGHsisldaWcaaWdae aapeGaa8Ntaiaa=jgaa8aabaWdbiaa=5gaaaWaaybCaeqal8aabaWd biaa=PgacqGH9aqpcaaIXaaapaqaa8qacaWFUbGaeyOeI0IaaGymaa qdpaqaa8qacqGHris5aaGcdaGfWbqabSWdaeaapeGaa8xAaiabg2da 9iaaigdaa8aabaWdbiaa=Tgaa0WdaeaapeGaeyyeIuoaaOGaa8NAaa aa@69AA@

i=1 N y ¯ i =Na+( N( N+1 ) 2 + Nk( n1 ) 2 )b kNbn( n1 ) 2n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGfWbqabSWdaeaaieWapeGaa8xAaiabg2da9iaaigdaa8aabaWd biaa=5eaa0WdaeaapeGaeyyeIuoaaOWdaiqa=LhagaqeamaaBaaale aapeGaa8xAaaWdaeqaaOWdbiabg2da9iaa=5eacaWFHbGaey4kaSYa aeWaa8aabaWdbmaalaaapaqaa8qacaWFobWaaeWaa8aabaWdbiaa=5 eacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaIYaaaaiab gUcaRmaalaaapaqaa8qacaWFobGaa83Aamaabmaapaqaa8qacaWFUb GaeyOeI0IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGOmaaaaaiaa wIcacaGLPaaacaWFIbGaeyOeI0YaaSaaa8aabaWdbiaa=TgacaWFob Gaa8Nyaiaa=5gadaqadaWdaeaapeGaa8NBaiabgkHiTiaaigdaaiaa wIcacaGLPaaaa8aabaWdbiaaikdacaWFUbaaaaaa@5E34@

1 N i=1 N y ¯ i =a+( N+1 2 )b= Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaaGymaaWdaeaaieWapeGaa8NtaaaadaGfWbqa bSWdaeaapeGaa8xAaiabg2da9iaaigdaa8aabaWdbiaa=5eaa0Wdae aapeGaeyyeIuoaaOWdaiqa=LhagaqeamaaBaaaleaapeGaa8xAaaWd aeqaaOWdbiabg2da9iaa=fgacqGHRaWkdaqadaWdaeaapeWaaSaaa8 aabaWdbiaa=5eacqGHRaWkcaaIXaaapaqaa8qacaaIYaaaaaGaayjk aiaawMcaaiaa=jgacqGH9aqpceWFzbGbaebaaaa@4C26@ (2.4)

 

That is, the CSS sample mean is an unbiased estimator for its population mean.

Computation of variance of circular systematic sample mean
For the labelled population and the corresponding CSS sample means defined in Section 2.1, the derivation of the variance of circular systematic sample mean is given below.

Consider 1 N i=1 N y ¯ i =a+( N+1 2 )b= Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaaGymaaWdaeaaieWapeGaa8NtaaaadaGfWbqa bSWdaeaapeGaa8xAaiabg2da9iaaigdaa8aabaWdbiaa=5eaa0Wdae aapeGaeyyeIuoaaOWdaiqa=LhagaqeamaaBaaaleaapeGaa8xAaaWd aeqaaOWdbiabg2da9iaa=fgacqGHRaWkdaqadaWdaeaapeWaaSaaa8 aabaWdbiaa=5eacqGHRaWkcaaIXaaapaqaa8qacaaIYaaaaaGaayjk aiaawMcaaiaa=jgacqGH9aqpceWFzbGbaebaaaa@4C26@

= 1 N i=1 N y ¯ i 2 Y ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaaieWapeGaa8Ntaaaa daGfWbqabSWdaeaapeGaa8xAaiabg2da9iaaigdaa8aabaWdbiaa=5 eaa0WdaeaapeGaeyyeIuoaaOWdaiqa=LhagaqeamaaBaaaleaapeGa a8xAaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaakiabgkHiTiqa=L fagaqea8aadaahaaWcbeqaa8qacaaIYaaaaaaa@4643@ (2.5)

By substituting the CSS sample means and the population means in the above expression, the variance of CSS sample mean for the labelled population is obtained as

V( y ¯ css )= b 2 N { i=1 N [  i+ k( n1 ) 2 ] 2 + N 2 n 2 j=1 n1 i=1 k j 2 2N n j=1 n1 i=L+( j1 )k+1 L+jk j( i+ k( n1 ) 2 ) N ( N+1 ) 2 4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF JbGaa83Caiaa=nhaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpda WcaaWdaeaapeGaa8Nya8aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqa a8qacaWFobaaamaacmaapaqaa8qadaGfWbqabSWdaeaapeGaa8xAai abg2da9iaaigdaa8aabaWdbiaa=5eaa0WdaeaapeGaeyyeIuoaaOWa amWaa8aabaWdbiaa=bkacaWFPbGaey4kaSYaaSaaa8aabaWdbiaa=T gadaqadaWdaeaapeGaa8NBaiabgkHiTiaaigdaaiaawIcacaGLPaaa a8aabaWdbiaaikdaaaaacaGLBbGaayzxaaWdamaaCaaaleqabaWdbi aaikdaaaGccqGHRaWkdaWcaaWdaeaapeGaa8Nta8aadaahaaWcbeqa a8qacaaIYaaaaaGcpaqaa8qacaWFUbWdamaaCaaaleqabaWdbiaaik daaaaaaOWaaybCaeqal8aabaWdbiaa=PgacqGH9aqpcaaIXaaapaqa a8qacaWFUbGaeyOeI0IaaGymaaqdpaqaa8qacqGHris5aaGcdaGfWb qabSWdaeaapeGaa8xAaiabg2da9iaaigdaa8aabaWdbiaa=Tgaa0Wd aeaapeGaeyyeIuoaaOGaa8NAa8aadaahaaWcbeqaa8qacaaIYaaaaO GaeyOeI0YaaSaaa8aabaWdbiaaikdacaWFobaapaqaa8qacaWFUbaa amaawahabeWcpaqaa8qacaWFQbGaeyypa0JaaGymaaWdaeaapeGaa8 NBaiabgkHiTiaaigdaa0WdaeaapeGaeyyeIuoaaOWaaybCaeqal8aa baWdbiaa=LgacqGH9aqpcaWFmbGaey4kaSYaaeWaa8aabaWdbiaa=P gacqGHsislcaaIXaaacaGLOaGaayzkaaGaa83AaiabgUcaRiaaigda a8aabaWdbiaa=XeacqGHRaWkcaWFQbGaa83Aaaqdpaqaa8qacqGHri s5aaGccaWFQbWaaeWaa8aabaWdbiaa=LgacqGHRaWkdaWcaaWdaeaa peGaa83Aamaabmaapaqaa8qacaWFUbGaeyOeI0IaaGymaaGaayjkai aawMcaaaWdaeaapeGaaGOmaaaaaiaawIcacaGLPaaacqGHsisldaWc aaWdaeaapeGaa8Ntamaabmaapaqaa8qacaWFobGaey4kaSIaaGymaa GaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qa caaI0aaaaaGaay5Eaiaaw2haaaaa@9A8A@

After a little algebra, the variance of CSS sample mean is obtained as

V( y ¯ css )= b 2 N { N( N+1 )( 2N+1 ) 6 + N k 2 ( n1 ) 2 4 + 2k( n1 )N( N+1 ) 4 + k N 2 ( n1 )n( 2n1 ) 6 n 2 kN( n1 )[ 6N+k( n2 )+3 ] 6 N ( N+1 ) 2 4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF JbGaa83Caiaa=nhaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpda WcaaWdaeaapeGaa8Nya8aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqa a8qacaWFobaaamaacmaapaqaa8qadaWcaaWdaeaapeGaa8Ntamaabm aapaqaa8qacaWFobGaey4kaSIaaGymaaGaayjkaiaawMcaamaabmaa paqaa8qacaaIYaGaa8NtaiabgUcaRiaaigdaaiaawIcacaGLPaaaa8 aabaWdbiaaiAdaaaGaey4kaSYaaSaaa8aabaWdbiaa=5eacaWFRbWd amaaCaaaleqabaWdbiaaikdaaaGcdaqadaWdaeaapeGaa8NBaiabgk HiTiaaigdaaiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaa aOWdaeaapeGaaGinaaaacqGHRaWkdaWcaaWdaeaapeGaaGOmaiaa=T gadaqadaWdaeaapeGaa8NBaiabgkHiTiaaigdaaiaawIcacaGLPaaa caWFobWaaeWaa8aabaWdbiaa=5eacqGHRaWkcaaIXaaacaGLOaGaay zkaaaapaqaa8qacaaI0aaaaiabgUcaRmaalaaapaqaa8qacaWFRbGa a8Nta8aadaahaaWcbeqaa8qacaaIYaaaaOWaaeWaa8aabaWdbiaa=5 gacqGHsislcaaIXaaacaGLOaGaayzkaaGaa8NBamaabmaapaqaa8qa caaIYaGaa8NBaiabgkHiTiaaigdaaiaawIcacaGLPaaaa8aabaWdbi aaiAdacaWFUbWdamaaCaaaleqabaWdbiaaikdaaaaaaOGaeyOeI0Ya aSaaa8aabaWdbiaa=TgacaWFobWaaeWaa8aabaWdbiaa=5gacqGHsi slcaaIXaaacaGLOaGaayzkaaWaamWaa8aabaWdbiaaiAdacaWFobGa ey4kaSIaa83Aamaabmaapaqaa8qacaWFUbGaeyOeI0IaaGOmaaGaay jkaiaawMcaaiabgUcaRiaaiodaaiaawUfacaGLDbaaa8aabaWdbiaa iAdaaaGaeyOeI0YaaSaaa8aabaWdbiaa=5eadaqadaWdaeaapeGaa8 NtaiabgUcaRiaaigdaaiaawIcacaGLPaaapaWaaWbaaSqabeaapeGa aGOmaaaaaOWdaeaapeGaaGinaaaaaiaawUhacaGL9baaaaa@91A6@

By simplifying the above expression one may get

V( y ¯ css )={ ( N1 )( N+1 ) 12 k( n 2 1 )( 2Nkn ) 12n } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF JbGaa83Caiaa=nhaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpda GadaWdaeaapeWaaSaaa8aabaWdbmaabmaapaqaa8qacaWFobGaeyOe I0IaaGymaaGaayjkaiaawMcaamaabmaapaqaa8qacaWFobGaey4kaS IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGymaiaaikdaaaGaeyOe I0YaaSaaa8aabaWdbiaa=TgadaqadaWdaeaapeGaa8NBa8aadaahaa Wcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa bmaapaqaa8qacaaIYaGaa8NtaiabgkHiTiaa=TgacaWFUbaacaGLOa Gaayzkaaaapaqaa8qacaaIXaGaaGOmaiaa=5gaaaaacaGL7bGaayzF aaGaa8Nya8aadaahaaWcbeqaa8qacaaIYaaaaaaa@5CB4@  (2.6)

Computation of optimum values for the sampling fraction  and the variance of circular systematic sample mean We know that the sampling fraction  is obtained as  or the positive integer closest to . Without loss of generality, let us assume that . That is  is the difference between  and  and . By replacing the values of k in the variance expression, one may get

V( y ¯ css )={ ( N1 )( N+1 ) 12 k( n 2 1 )( 2Nkn ) 12n } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF JbGaa83Caiaa=nhaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpda GadaWdaeaapeWaaSaaa8aabaWdbmaabmaapaqaa8qacaWFobGaeyOe I0IaaGymaaGaayjkaiaawMcaamaabmaapaqaa8qacaWFobGaey4kaS IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGymaiaaikdaaaGaeyOe I0YaaSaaa8aabaWdbiaa=TgadaqadaWdaeaapeGaa8NBa8aadaahaa Wcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa bmaapaqaa8qacaaIYaGaa8NtaiabgkHiTiaa=TgacaWFUbaacaGLOa Gaayzkaaaapaqaa8qacaaIXaGaaGOmaiaa=5gaaaaacaGL7bGaayzF aaGaa8Nya8aadaahaaWcbeqaa8qacaaIYaaaaaaa@5CB4@

={ ( N1 )( N+1 ) 12 ( Nd )( n 2 1 )( 2N( Nd ) ) 12 n 2 } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaGadaWdaeaapeWaaSaaa8aabaWdbmaabmaapaqaaGqa d8qacaWFobGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaapaqaa8 qacaWFobGaey4kaSIaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGym aiaaikdaaaGaeyOeI0YaaSaaa8aabaWdbmaabmaapaqaa8qacaWFob GaeS4eI0Maa8hzaaGaayjkaiaawMcaamaabmaapaqaa8qacaWFUbWd amaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIXaaacaGLOaGaay zkaaWaaeWaa8aabaWdbiaaikdacaWFobGaeyOeI0YaaeWaa8aabaWd biaa=5eacqWItisBcaWFKbaacaGLOaGaayzkaaaacaGLOaGaayzkaa aapaqaa8qacaaIXaGaaGOmaiaa=5gapaWaaWbaaSqabeaapeGaaGOm aaaaaaaakiaawUhacaGL9baacaWFIbWdamaaCaaaleqabaWdbiaaik daaaaaaa@5D6B@

      ={ ( N1 )( N+1 ) 12 ( Nd )( n 2 1 )( N±d ) 12 n 2 } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFGcGaeyypa0ZaaiWaa8aabaWdbmaalaaapaqaa8qadaqa daWdaeaapeGaa8NtaiabgkHiTiaaigdaaiaawIcacaGLPaaadaqada WdaeaapeGaa8NtaiabgUcaRiaaigdaaiaawIcacaGLPaaaa8aabaWd biaaigdacaaIYaaaaiabgkHiTmaalaaapaqaa8qadaqadaWdaeaape Gaa8NtaiabloHiTjaa=rgaaiaawIcacaGLPaaadaqadaWdaeaapeGa a8NBa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGymaaGaay jkaiaawMcaamaabmaapaqaa8qacaWFobGaeyySaeRaa8hzaaGaayjk aiaawMcaaaWdaeaapeGaaGymaiaaikdacaWFUbWdamaaCaaaleqaba WdbiaaikdaaaaaaaGccaGL7bGaayzFaaGaa8Nya8aadaahaaWcbeqa a8qacaaIYaaaaaaa@5B27@           

   ={ ( N1 )( N+1 ) 12 ( n 2 1 )( N 2 d 2 ) 12 n 2 } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaGadaWdaeaapeWaaSaaa8aabaWdbmaabmaapaqaaGqa d8qacaWFobGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaapaqaa8 qacaWFobGaey4kaSIaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGym aiaaikdaaaGaeyOeI0YaaSaaa8aabaWdbmaabmaapaqaa8qacaWFUb WdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIXaaacaGLOaGa ayzkaaWaaeWaa8aabaWdbiaa=5eapaWaaWbaaSqabeaapeGaaGOmaa aakiabgkHiTiaa=rgapaWaaWbaaSqabeaapeGaaGOmaaaaaOGaayjk aiaawMcaaaWdaeaapeGaaGymaiaaikdacaWFUbWdamaaCaaaleqaba WdbiaaikdaaaaaaaGccaGL7bGaayzFaaGaa8Nya8aadaahaaWcbeqa a8qacaaIYaaaaaaa@569A@

By simplifying the above expression one may get

V( y ¯ css )={ ( N1 )( N+1 ) 12 n 2 + ( n 2 1 )( d 2 1 ) 12 n 2 } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF JbGaa83Caiaa=nhaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpda GadaWdaeaapeWaaSaaa8aabaWdbmaabmaapaqaa8qacaWFobGaeyOe I0IaaGymaaGaayjkaiaawMcaamaabmaapaqaa8qacaWFobGaey4kaS IaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGymaiaaikdacaWFUbWd amaaCaaaleqabaWdbiaaikdaaaaaaOGaey4kaSYaaSaaa8aabaWdbm aabmaapaqaa8qacaWFUbWdamaaCaaaleqabaWdbiaaikdaaaGccqGH sislcaaIXaaacaGLOaGaayzkaaWaaeWaa8aabaWdbiaa=rgapaWaaW baaSqabeaapeGaaGOmaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaa a8aabaWdbiaaigdacaaIYaGaa8NBa8aadaahaaWcbeqaa8qacaaIYa aaaaaaaOGaay5Eaiaaw2haaiaa=jgapaWaaWbaaSqabeaapeGaaGOm aaaaaaa@5E1C@ (2.7)

The above expression attains minimum at , which implies or .

That is, the optimum variance of CSS sample mean is exactly the same as given in (1.2).

V( y ¯ ocss )={ ( N1 )( N+1 ) 12 n 2 } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF VbGaa83yaiaa=nhacaWFZbaapaqabaaak8qacaGLOaGaayzkaaGaey ypa0ZaaiWaa8aabaWdbmaalaaapaqaa8qadaqadaWdaeaapeGaa8Nt aiabgkHiTiaaigdaaiaawIcacaGLPaaadaqadaWdaeaapeGaa8Ntai abgUcaRiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaaigdacaaIYaGa a8NBa8aadaahaaWcbeqaa8qacaaIYaaaaaaaaOGaay5Eaiaaw2haai aa=jgapaWaaWbaaSqabeaapeGaaGOmaaaaaaa@4FCC@

Hence we conclude that the optimum value of the sampling fraction  is obtained as  stated by Subramani et al.,41 and Subramani & Singh42 as given in (Table 1.1)

Sample number

Sample values

OCSS mean

1

1

8

3

10

5

5.4

2

2

9

4

11

6

6.4

3

3

10

5

12

7

7.4

4

4

11

6

1

8

6

5

5

12

7

2

9

7

6

6

1

8

3

10

5.6

7

7

2

9

4

11

6.6

8

8

3

10

5

12

7.6

9

9

4

11

6

1

6.2

10

10

5

12

7

2

7.2

11

11

6

1

8

3

5.8

12

12

7

2

9

4

6.8

Table 1 OCSS samples and their means for the sampling interval

Comparison of the efficiency of CSS and SRSWOR sample means

By comparing the variance expressions for a SRSWOR sample mean (1.3) and a CSS sample mean (2.7) one can easily show that

V( y ¯ r )V( y ¯ css )= ( Nn )( N+1 ) b 2 12n { ( N1 )( N+1 ) 12 n 2 ( n 2 1 )( d 2 1 ) 12 n 2 } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF Ybaapaqabaaak8qacaGLOaGaayzkaaGaeyOeI0Iaa8Nvamaabmaapa qaaiqa=LhagaqeamaaBaaaleaapeGaa83yaiaa=nhacaWFZbaapaqa baaak8qacaGLOaGaayzkaaGaeyypa0ZaaSaaa8aabaWdbmaabmaapa qaa8qacaWFobGaeyOeI0Iaa8NBaaGaayjkaiaawMcaamaabmaapaqa a8qacaWFobGaey4kaSIaaGymaaGaayjkaiaawMcaaiaa=jgapaWaaW baaSqabeaapeGaaGOmaaaaaOWdaeaapeGaaGymaiaaikdacaWFUbaa aiabgkHiTmaacmaapaqaa8qadaWcaaWdaeaapeWaaeWaa8aabaWdbi aa=5eacqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaa8aabaWdbiaa =5eacqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaIXaGaaG Omaiaa=5gapaWaaWbaaSqabeaapeGaaGOmaaaaaaGccqGHsisldaWc aaWdaeaapeWaaeWaa8aabaWdbiaa=5gapaWaaWbaaSqabeaapeGaaG OmaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaadaqadaWdaeaapeGa a8hza8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGymaaGaay jkaiaawMcaaaWdaeaapeGaaGymaiaaikdacaWFUbWdamaaCaaaleqa baWdbiaaikdaaaaaaaGccaGL7bGaayzFaaGaa8Nya8aadaahaaWcbe qaa8qacaaIYaaaaaaa@71EA@

={ ( Nn )( N+1 ) 12n ( N1 )( N+1 ) 12 n 2 + ( n 2 1 )( d 2 1 ) 12 n 2 } b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaGadaWdaeaapeWaaSaaa8aabaWdbmaabmaapaqaaGqa d8qacaWFobGaeyOeI0Iaa8NBaaGaayjkaiaawMcaamaabmaapaqaa8 qacaWFobGaey4kaSIaaGymaaGaayjkaiaawMcaaaWdaeaapeGaaGym aiaaikdacaWFUbaaaiabgkHiTmaalaaapaqaa8qadaqadaWdaeaape Gaa8NtaiabgkHiTiaaigdaaiaawIcacaGLPaaadaqadaWdaeaapeGa a8NtaiabgUcaRiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaaigdaca aIYaGaa8NBa8aadaahaaWcbeqaa8qacaaIYaaaaaaakiabgUcaRmaa laaapaqaa8qadaqadaWdaeaapeGaa8NBa8aadaahaaWcbeqaa8qaca aIYaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaapaqaa8qa caWFKbWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIXaaaca GLOaGaayzkaaaapaqaa8qacaaIXaGaaGOmaiaa=5gapaWaaWbaaSqa beaapeGaaGOmaaaaaaaakiaawUhacaGL9baacaWFIbWdamaaCaaale qabaWdbiaaikdaaaaaaa@6372@

={ ( N+1 )( n1 )( Nn+1 ) 12 n 2 + ( n 2 1 )( d 2 1 ) 12 n 2 } b 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH9aqpdaGadaWdaeaapeWaaSaaa8aabaWdbmaabmaapaqaaGqa d8qacaWFobGaey4kaSIaaGymaaGaayjkaiaawMcaamaabmaapaqaa8 qacaWFUbGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaapaqaa8qa caWFobGaeyOeI0Iaa8NBaiabgUcaRiaaigdaaiaawIcacaGLPaaaa8 aabaWdbiaaigdacaaIYaGaa8NBa8aadaahaaWcbeqaa8qacaaIYaaa aaaakiabgUcaRmaalaaapaqaa8qadaqadaWdaeaapeGaa8NBa8aada ahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMca amaabmaapaqaa8qacaWFKbWdamaaCaaaleqabaWdbiaaikdaaaGccq GHsislcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaIXaGaaGOmaiaa =5gapaWaaWbaaSqabeaapeGaaGOmaaaaaaaakiaawUhacaGL9baaca WFIbWdamaaCaaaleqabaWdbiaaikdaaaGccqGHLjYScaaIWaaaaa@6004@

That is, the circular systematic sampling is more efficient than the simple random sampling without replacement. i.e. V( y ¯ css )V( y ¯ r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF JbGaa83Caiaa=nhaa8aabeaaaOWdbiaawIcacaGLPaaacqGHKjYOca WFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWFYbaapaqa baaak8qacaGLOaGaayzkaaaaaa@4430@

Some modifications on circular systematic sample mean

It has been shown in Section 3 that the circular systematic sampling performs better than the simple random sampling without replacement. However it is not a trend free sampling33 which can be achieved by introducing Yates type end corrections4 as given below:
The modification involves the usual circular systematic sampling but the modified sample mean is defined as

y ¯ css * = y ¯ css +a( y 1 y n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmGab8xEay aaraWaa0baaSqaaabaaaaaaaaapeGaa83yaiaa=nhacaWFZbaapaqa a8qacaWFQaaaaOGaeyypa0Zdaiqa=LhagaqeamaaBaaaleaapeGaa8 3yaiaa=nhacaWFZbaapaqabaGcpeGaey4kaSIaa8xyamaabmaapaqa a8qacaWF5bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgkHiTi aa=LhapaWaaSbaaSqaa8qacaWFUbaapaqabaaak8qacaGLOaGaayzk aaaaaa@4A06@ (4.1)

That is, the units selected first and last are given the weights and respectively whereas the remaining units get the weight .By equating for the population with a perfect linear trend, we get the values for from (4.1) as: Here one may have the following two situations: (i).The random start  is less than or equal to and (ii). The random start is greater than .
Case (i). When the random start  is less than or equal to
By setting (4.1) is equal to we get

[ i+ k( n1 ) 2 ]+a( y 1 y n )= ( N+1 ) 2 , i=1, 2, , Nk( n1 )=L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWadaWdaeaaieWapeGaa8xAaiabgUcaRmaalaaapaqaa8qacaWF RbWaaeWaa8aabaWdbiaa=5gacqGHsislcaaIXaaacaGLOaGaayzkaa aapaqaa8qacaaIYaaaaaGaay5waiaaw2faaiabgUcaRiaa=fgadaqa daWdaeaapeGaa8xEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacq GHsislcaWF5bWdamaaBaaaleaapeGaa8NBaaWdaeqaaaGcpeGaayjk aiaawMcaaiabg2da9maalaaapaqaa8qadaqadaWdaeaapeGaa8Ntai abgUcaRiaaigdaaiaawIcacaGLPaaaa8aabaWdbiaaikdaaaGaaiil aiaa=bkacaWFPbGaeyypa0JaaGymaiaacYcacaWFGcGaaGOmaiaacY cacaWFGcGaeyOjGWRaaiilaiaa=bkacaWFobGaeyOeI0Iaa83Aamaa bmaapaqaa8qacaWFUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabg2 da9iaa=Xeaaaa@64F8@

By putting ( y 1 y n )=i( i+( n1 )k )=k( n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeGaaCyEa8aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacqGHsislcaWH5bWdamaaBaaaleaapeGaaCOBaaWdaeqaaaGcpe GaayjkaiaawMcaaiabg2da9iaahMgacqGHsisldaqadaWdaeaapeGa aCyAaiabgUcaRmaabmaapaqaa8qacaWHUbGaeyOeI0IaaGymaaGaay jkaiaawMcaaiaahUgaaiaawIcacaGLPaaacqGH9aqpcqGHsislcaWH RbWaaeWaa8aabaWdbiaah6gacqGHsislcaaIXaaacaGLOaGaayzkaa aaaa@50E1@ ,

we get a= 2i+k( n1 )( N+1 ) 2k( n1 ) , i=1,2,3,,Nk( n1 )=L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFHbGaeyypa0ZaaSaaa8aabaWdbiaaikdacaWFPbGaey4k aSIaa83Aamaabmaapaqaa8qacaWFUbGaeyOeI0IaaGymaaGaayjkai aawMcaaiabgkHiTmaabmaapaqaa8qacaWFobGaey4kaSIaaGymaaGa ayjkaiaawMcaaaWdaeaapeGaaGOmaiaa=TgadaqadaWdaeaapeGaa8 NBaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGaaiilaiaa=bkacaWF PbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaey OjGWRaaiilaiaa=5eacqGHsislcaWGRbWdamaabmaabaWdbiaad6ga cqGHsislcaaIXaaapaGaayjkaiaawMcaa8qacqGH9aqpcaWGmbaaaa@5EE9@ (4.2)

Case (ii). When the random start is greater than
Let the random start lies between  and
By setting (4.1) is equal to we get

[ i+ k( n1 ) 2 jN n ]+a( y 1 y n )= ( N+1 ) 2 , i=L+( j1 )k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWadaWdaeaaieWapeGaa8xAaiabgUcaRmaalaaapaqaa8qacaWF RbWaaeWaa8aabaWdbiaa=5gacqGHsislcaaIXaaacaGLOaGaayzkaa aapaqaa8qacaaIYaaaaiabgkHiTmaalaaapaqaa8qacaWFQbGaa8Nt aaWdaeaapeGaa8NBaaaaaiaawUfacaGLDbaacqGHRaWkcaWFHbWaae Waa8aabaWdbiaa=LhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGa eyOeI0Iaa8xEa8aadaWgaaWcbaWdbiaa=5gaa8aabeaaaOWdbiaawI cacaGLPaaacqGH9aqpdaWcaaWdaeaapeWaaeWaa8aabaWdbiaa=5ea cqGHRaWkcaaIXaaacaGLOaGaayzkaaaapaqaa8qacaaIYaaaaiaacY cacaWFGcGaa8xAaiabg2da9iaa=XeacqGHRaWkdaqadaWdaeaapeGa a8NAaiabgkHiTiaaigdaaiaawIcacaGLPaaacaWFRbGaey4kaSIaaG ymaaaa@601D@ to L+jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFmbGaey4kaSIaa8NAaiaa=Tgaaaa@3A76@

By putting ( y 1 y n )=i( i+( n1 )kjN )=jNk( n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeGaaCyEa8aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacqGHsislcaWH5bWdamaaBaaaleaapeGaaCOBaaWdaeqaaaGcpe GaayjkaiaawMcaaiabg2da9iaahMgacqGHsisldaqadaWdaeaapeGa aCyAaiabgUcaRmaabmaapaqaa8qacaWHUbGaeyOeI0IaaGymaaGaay jkaiaawMcaaiaahUgacqGHsislcaWHQbGaaCOtaaGaayjkaiaawMca aiabg2da9iaahQgacaWHobGaeyOeI0IaaC4Aamaabmaapaqaa8qaca WHUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaa@5562@ we get

a= n[ ( N+1 )2ik( n1 ) ]+2jN 2n[ jNk( n1 ) ] ,i=L+( j1 )k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWHHbGaeyypa0ZaaSaaa8aabaacbmWdbiaa=5gadaWadaWdaeaa peWaaeWaa8aabaWdbiaa=5eacqGHRaWkcaaIXaaacaGLOaGaayzkaa GaeyOeI0IaaGOmaiaa=LgacqGHsislcaWFRbWaaeWaa8aabaWdbiaa =5gacqGHsislcaaIXaaacaGLOaGaayzkaaaacaGLBbGaayzxaaGaey 4kaSIaaGOmaiaa=PgacaWFobaapaqaa8qacaaIYaGaa8NBamaadmaa paqaa8qacaWFQbGaa8NtaiabgkHiTiaa=TgadaqadaWdaeaapeGaa8 NBaiabgkHiTiaaigdaaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaGa aiilaiaa=LgacqGH9aqpcaWFmbGaey4kaSYaaeWaa8aabaWdbiaa=P gacqGHsislcaaIXaaacaGLOaGaayzkaaGaa83AaiabgUcaRiaaigda aaa@62E6@ to L+jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGmbGaey4kaSIaamOAaiaadUgaaaa@3A76@    (4.3)

Remark 4.1: In the presence of a perfect linear trend the modified circular systematic sample mean becomes the population mean and hence the . In this case the circular systematic sampling becomes a completely trend free sampling (See Mukerjee and Sengupta, 1990).

Numerical comparisons of circular systematic sampling for certain natural populations

It has been shown in Section 3 that the circular systematic sampling performs well compared to simple random sampling without replacement whenever there exists a perfect linear trend among the population values. However this is an unrealistic assumption in real life situations. Consequently an attempt has been made to study the efficiency of the circular systematic sampling for a population considered by Subramani6 and Murthy.29 The first data were collected for assessing the process capability of a manufacturing process from an auto ancillary manufacturing unit located in Tamilnadu. The data pertain to the measurements taken continuously during the Turning operation performed on the component Torsion bar in Frontier CNC Lathe Machine. The data were collected for estimating the mean value of the outer diameter of the Torsion bar, one of the key components in integrated power steering system. The measurements were taken continuously for the first 50 components produced in a shift. The 50 measurements based on the order of the production are given in Table 3.1. However the first 37 measurements after arranging the data in ascending order are taken to get a linear trend among the population values as given in Table 3.2. The Second data are about the number of workers for 80 factories in a region. However the first 37 measurements of the data are taken to get a linear trend among the population values as given in Table 3.3.

9050

9052

9050

9052

9052

9056

9056

9054

9056

9058

9054

9054

9060

9058

9060

9058

9056

9058

9058

9060

9062

9064

9062

9064

9066

9070

9068

9072

9072

9070

9072

9070

9070

9072

9074

9076

9078

9076

9076

9078

9078

9078

9082

9080

9082

9080

9082

9086

9086

9084

Table 3.1 Data of outer Diameter of Torsion Bar

9050

9050

9052

9052

9052

9054

9054

9054

9056

9056

9056

9056

9058

9058

9058

9058

9058

9060

9060

9060

9062

9062

9064

9064

9066

9068

9070

9070

9070

9070

9072

9072

9072

9072

9074

9076

9078

 

 

 

Table 3.2 The data arranged in ascending order

51

51

52

52

53

54

57

60

65

67

68

70

71

73

74

76

78

80

81

85

87

88

92

93

97

100

107

110

113

116

119

121

125

127

127

131

134

 

Table 3.3 Number of workers in first 37 factories (Murthy, 1967, p.228)

The variances of simple random sample mean, circular systematic sample mean and optimal circular systematic sample mean together with the percentage relative efficiencies are obtained and are presented in Table 3.4. The PREs of the proposed estimator ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaaieWapeGaa8hCaaGaayjkaiaawMcaaaaa@3989@  with respect to an existing estimator ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaaieWapeGaa8xzaaGaayjkaiaawMcaaaaa@397E@ is computed as

PRE( p )= V( e ) V( p ) x100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWHqbGaaCOuaiaahweadaqadaWdaeaaieWapeGaa8hCaaGaayjk aiaawMcaaiabg2da9maalaaapaqaa8qacaWFwbWaaeWaa8aabaWdbi aa=vgaaiaawIcacaGLPaaaa8aabaWdbiaa=zfadaqadaWdaeaapeGa a8hCaaGaayjkaiaawMcaaaaacaWH4bGaaCymaiaahcdacaWHWaaaaa@4761@

N

n

Population 1

Population 2

PRE-

PRE-

PRE-

PRE-

37

2

30.59

15.95

15.95

191.81

100

320.89

176.87

176.87

181.43

100

37

3

19.81

6.03

6.03

328.49

100

207.81

68.09

68.09

305.2

100

37

4

14.42

4.96

4.96

290.76

100

151.28

35.99

35.99

420.34

100

37

5

11.19

3.15

2.93

381.51

107.37

117.35

25.19

22.69

517.19

111.02

37

6

9.03

1.81

1.81

498.07

100

94.74

15.68

15.68

604.21

100

37

7

7.49

1.7

1.66

451.2

102.11

78.59

13.74

12.01

654.37

114.4

37

8

6.34

1.51

1.51

420.44

100

66.47

13.17

10.12

656.82

130.14

37

9

5.44

0.81

0.81

673.73

100

57.05

7.57

7.57

753.63

100

37

10

4.72

1.01

0.76

620.11

132.72

49.51

10.02

5.44

910.11

184.19

37

11

4.13

4.13

0.64

641.46

641.46

43.34

13.13

5.67

764.37

231.57

37

12

3.64

3.64

3.64

100

100

38.2

4.73

4.73

807.61

100

37

13

3.23

3.23

0.37

865.15

865.15

33.85

5.55

3.85

879.22

144.16

37

14

2.87

2.87

0.39

743.78

743.78

30.12

11.05

2.86

1053.15

386.36

37

15

2.56

0.23

0.23

1095.3

100

26.89

27.12

2.68

1003.36

1011.94

37

16

2.29

1.37

0.3

754.61

449.67

24.07

14.57

2.5

962.8

582.8

37

17

2.06

0.62

0.21

993.24

300

21.57

6.14

1.72

1254.07

356.98

37

18

1.85

0.24

0.24

781.78

100

19.36

1.88

1.88

1029.79

100

37

19

1.66

0.21

0.21

781.13

100

17.37

1.69

1.69

1027.81

100

37

20

1.49

0.45

0.15

990.67

298.67

15.59

4.43

1.24

1257.26

357.26

Table 3.4 Comparison of optimal circular systematic sampling, circular systematic sampling and simple random sampling without replacement

It is seen from the table values that the optimal circular systematic sampling performs better than the circular systematic sampling and the circular systematic sampling performs better than simple random sampling in all the cases. In general, it is observed that for the population with a linear trend the following inequality is true. That is, V( y ¯ ocss )V( y ¯ css )V( y ¯ r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmaeaaaaaa aaa8qacaWFwbWaaeWaa8aabaGab8xEayaaraWaaSbaaSqaa8qacaWF VbGaa83yaiaa=nhacaWFZbaapaqabaaak8qacaGLOaGaayzkaaGaey izImQaa8Nvamaabmaapaqaaiqa=LhagaqeamaaBaaaleaapeGaa83y aiaa=nhacaWFZbaapaqabaaak8qacaGLOaGaayzkaaGaeyizImQaa8 Nvamaabmaapaqaaiqa=LhagaqeamaaBaaaleaapeGaa8NCaaWdaeqa aaGcpeGaayjkaiaawMcaaaaa@4D87@

Acknowledgements

The Author wishes to record his heartiest thanks for the Editor and the Reviewer for their constructive comments, which have improved the presentation of the paper.

Conflict of interest

Author declares that there is no conflict of interest.

References

  1. Madow WG. On the theory of systematic sampling III. Comparison of centered and random start systematic sampling. Ann Math Statist. 1953;24(1):101–106.
  2. Sethi VK. On optimum pairing of units. Sankhya B. 1965;27(3–4):315–320.
  3. Singh D, Jindal KK, Garg JN On modified systematic sampling. Biometrika. 1968;55(3):541–546.
  4. Yates F Systematic Sampling. Phil Trans Royal Soc A. 1948;241(834):345–377.
  5. Subramani J. Diagonal Systematic Sampling Scheme for Finite Populations. Journal of the Indian Society of Agricultural Statistics. 2000;53(2):187–195.
  6. Subramani J. Further results on diagonal systematic sampling for finite populations. Journal of the Indian Society of Agricultural Statistics. 2009;63(3):277–282.
  7. Subramani J. Generalization of Diagonal Systematic Sampling Scheme for Finite Populations. Model Assisted Statistics and Applications. 2010;5(2):17–128.
  8. Subramani J, Tracy DS. Determinant Sampling Scheme for Finite Populations. Internl J Math & Statist Sci. 1999;8(1):27–41.
  9. Subramani J. A modification on linear systematic sampling for odd sample size. Bonfring International Journal of Data Mining. 2012;2(2):32–36.
  10. Subramani J. A Modification on Systematic Sampling. Model Assisted Statistics and Applications. 2013;8(3):215–227.
  11. Subramani J. A Further Modification on Systematic Sampling for Finite Populations. Journal of Statistical Theory and Practice. 2013;7(3):471–479.
  12. Subramani J, Gupta SN. Generalized modified linear systematic sampling scheme for finite populations. Hacettepe Journal of Mathematics and Statistics (HJMS). 2014;43(3):529–542.
  13. Subramani J. Star Type Systematic Sampling Schemes for Finite Populations. Communications in Statistics–Theory and Methods. 2014;43(1):175–190.
  14. Chang HJ, Huang KC. Remainder linear systematic sampling. Sankhya B. 2000;62(2):249–256.
  15. Khan Z, Shabbir J, Gupta S. A New Sampling Design for Systematic Sampling. Communications in Statistics–Theory and Methods. 2013;42(18): 3359-3370.
  16. Ching Ho Leu, Fei–Fei Kao. Modified balanced circular systematic sampling. Statistics & Probability Letters. 2006;76:373–383.
  17. Huang KC. Mixed random systematic sampling designs. Metrika. 2004;59(1):1–11.
  18. Lahiri DB. A method of sample selection providing unbiased ratio estimates. Ball Internat Statist Inst. 1951;33:33–140.
  19. Leu CH, Tsui KW. New partially systematic sampling. Statist Sinica. 1996;6:617–630.
  20. Sampath S, Uthayakumaran N. Markov systematic sampling. Biometrical J. 1998;40(7):883–895.
  21. Singh P, Garg JN. On balanced random sampling. Sankhya C. 1979;41:60–68.
  22. Singh D, Singh P. New systematic sampling. JSPI. 1977;1:163–177.
  23. Uthayakumaran N. Additional circular systematic sampling methods. Biometrical J. 1998;40(4):467–474.
  24. Cochran WG. Sampling Techniques, 3rd ed. New York: John Wiley and Sons; 1977.
  25. Gautschi W. Some remarks on systematic sampling. AMS. 1957;28:385–394.
  26. Khan Z, Gupta S, Shabbir J. Diagonal Circular Systematic Sampling, to appear in Journal of Statistical Theory and Practice. 2013.
  27. Khan Z, Gupta S, Shabbir J. Generalized Systematic Sampling. Journal of Communications in Statistics–Simulation and Computation. 2013;44(9): 2240-2250.
  28. Gupta AK, Kabe DG. Theory of Sample Surveys. World Scientific Publishing Company. 2011:236.
  29. Murthy MN. Sampling Theory and Methods. India: Statistical Publishing House; 1967.
  30. Sarjinder Singh. Advanced Sampling Theory with Applications. Springer; 2003.
  31. Sukhatme PV, Sukhatme BV, Sukhatme S, et al. Sampling Theory of Surveys with Applications. USA: Iowa State Univ Press; 1984. p. 1–519.
  32. Fountain RL, Pathak P. Systematic and Non–random Sampling in the Presence of Linear Trends. Communications in statistics–Theory and Methods. 1989;18(7):2511–2526.
  33. Mukerjee R, Sengupta S. Optimal Estimation of a Finite Population Means in the Presence of Linear Trend. Biometrika. 1990; 77(3):625–630.
  34. Murthy MN, Rao TJ. Systematic sampling with illustrative examples. In: Krishnaiah PR, Rao CR, editors. Handbook of Statistics. 1988;6:147–185.
  35. WU CFJ. Estimation in systematic sampling with supplementary observations. Sankhya B. 1984;46:306–315.
  36. Zinger A. Variance estimation in partially systematic sampling. JASA. 1980;75(369):206–211.
  37. Sudakar K. A note on circular systematic sampling. Sankhya C. 1978;40:72–73.
  38. Bell House DR. On the choice of sampling interval in circular systematic sampling. Shankya B. 1984;247–248.
  39. Bell House DR. Systematic sampling. In: Krishnaiah PR, Rao CR, editors. Handbook of Statistics. 1988;6:125–145.
  40. Bell House DR, Rao JNK. Systematic sampling in the presence of linear trends. Biometrika. 1975;62(3):694–697.
  41.  Subramani J, Gupta SN, Prabavathy G.): Circular systematic sampling in the presence of linear trend. American Journal of Mathematical and Management Sciences (AJMMS). 2014;33(1):1–19.
  42. Subramani J, Sarjinder Singh. Estimation of Population Mean in the Presence of Linear Trend. Communications in Statistics–Theory and Methods. 2014;43(15):3095–3116.
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