Journal of Statistical Theory and Applications

Volume 19, Issue 1, March 2020, Pages 109 - 117

Sample Design and Estimation of Parameters of Half Logistic Distribution Using Generalized Ranked-Set Sampling

Authors
A. Adatia1, *, A.K.MD. Ehsanes Saleh2
1Department of Computer Science and Mathematics, University of Lethbridge, Lethbridge, Alberta, Canada
2Professor Emeritus and Distinguished Research Professor, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada
*Corresponding author. Email: aminmohamed.adatia@uleth.ca
Corresponding Author
A. Adatia
Received 1 September 2017, Accepted 27 May 2019, Available Online 10 March 2020.
DOI
10.2991/jsta.d.200303.001How to use a DOI?
Keywords
Half logistic distribution; Order statistics; Linear estimation; Generalized ranked-set sampling; Sampling efficiency
Abstract

The ranked-set sampling technique has been generalized so that a more efficient estimator may be obtained. This technique allows more than one unit from each set to be quantified. Consequently, the number of units to be sampled may be reduced significantly and as a result, the corresponding cost would also be reduced. The generalized ranked-set sampling technique is applied in the estimation of parameters of the half logistic distribution. New estimators are proposed which include linear minimum variance unbiased estimators and ranked-set sample estimators. The coefficients, variances and relative efficiencies are tabulated. The estimators are compared to the best linear unbiased estimator of the parameters. Sample design strategy is also considered.

Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

McIntyre [1] advocated the use of ranked-set sampling when experimenters encountered situations where the actual measurements of the sample observations were difficult to make due to constraints like cost, time and other factors. However, ranking of the potential sample data is relatively easy. Since then, this technique has been studied and applied to several areas of applied research. Some of the applications are in forestry [2], medicine [3], environmental monitoring [4,5], population genetics [6], clinical trials [6], agriculture and entomology [7]. A comprehensive review of the subject has been provided by Wolfe [8].

In this paper, the generalized ranked-set sampling (RSS) procedure proposed in Adatia [9] is extended to allow more than one unit from each set to be quantified. Unbiased estimators of the (location, scale) parameters of any location-scale family of distributions can be obtained. The performance of these estimators can be investigated by comparing their efficiency against that of best linear unbiased estimators (BLUEs). Also, the number of units to be sampled is less than that required for balanced RSS.

The generalized ranked-set sampling procedure proposed in this paper also enables us to provide a sampling design strategy to explore the relationship between the efficiency of the estimators and the number of units sampled. Suppose we can afford to take up to N = 15 ranked-set observations. We can select one ranked-set sample of size N = 15, sample design 15×1 (with 225 units selected) or in replicated smaller samples (e.g. five ranked-set samples of size N = 3, sample design 3×5 (with 45 units selected) or three ranked-set samples of size N = 5, sample design 5×3 (with 75 units selected)). For each sample design and number of sampling units selected, the efficiency is computed and can be compared. This is relevant in applications where RSS might be beneficial, but the cost involved in sampling and ranking cannot be completely ignored. Several authors have addressed this concern by introducing cost models [1012]. Also, since ranking large sets is difficult and introduces room for error, an appropriate sample design can be selected with the advantage of known efficiency.

2. APPLICATION

The generalized ranked-set sampling technique is applied in the estimation of parameters of the half logistic distribution. The development of the generalized minimum variance estimator with 2 or 3 observations kept in each set to be quantified, is a continuation of the method presented by Adatia [9]. This paper provides sufficient details for practitioners to construct this type of estimators for their applications. New estimators proposed are generalized ranked-set minimum variance unbiased estimators (GR-MVUEs) and generalized ranked-set sample estimators (GR-RSSs). Coefficients, variances and relative efficiencies are derived. The estimators are compared to the BLUEs of the parameters. In the case of half logistic distribution, GR-MVUE have advantage compared to maximum likelihood estimators (MLEs) as it can be directly calculated from a sample even if both (location, scale) parameters are unknown. Giles [13] gives a good account on the complexity of MLE's for the half logistic distribution.

In generalized ranked-set sampling, first a set of N elements is randomly selected from a given population. The sample is ordered without making actual measurements. The unit identified with the N11 rank is accurately measured. Next, a second set of N elements is randomly selected from the population. Again the units are ordered and the unit with the N12 rank is accurately measured. The process is continued until N set of N elements is selected. The units are again ordered and the unit with N1N rank is accurately measured. The ordered sample of the N sets can be represented as follows:

Set1X(11)X(12)X(1N)Set2X(21)X(22)X(2N)SetNX(N1)X(N2)X(NN)

In generalized ranked-set sampling where two units are selected from each set, first a set of N elements is randomly selected from a given population. The sample is ordered without making actual measurements. The units identified with the N11 and N21 ranks are accurately measured. Next, a second set of N elements is randomly selected from the population. Again the units are ordered and the units with the N12 and N22 ranks are accurately measured. The process is continued until M = N/2 set of N elements is selected. The units are again ordered and the units with N1M and N2M ranks are accurately measured. The ordered sample of the M = N/2 sets can be represented as follows: XN11,XN21,,XN1M,XN2M where 1N1i<N2iN and 1 ≤ i ≤ M.

Similarly, the generalized ranked-set sample of size N where three units are selected from each set consists of units which are accurately measured, i.e. XN11,XN21,XN31,,XN1M,XN2M,XN3M where 1N1(i)<N2(i)<N3(i)N,1iM and M = N/3.

We note that from the generalized ranked-set sampling procedure, the balanced ranked-set sample is obtained by selecting the ordered unit N1(i)=i(i) from set i(1iN).

2.1. Estimators Based on Generalized Ranked-Set Sampling

Let the random variable X have a half logistic distribution with probability density function (f(x))

f(x)=2exp[(xμ)/σ]σ[1+exp[(xμ)/σ]]2,  xμ,σ>0
where µ and σ are the location and the scale parameters respectively.

Let

  • Z(Nj(i))=(X(Nj(i))μ)/σαNj(i)=E(Z(Nj(i)))

  • ωNjiNki=CovZNj(i),ZNki

  • X_(i)=XN1i,XN2iT be the two order statistics of ranks N1i and N2i in the ith set.

Then the generalized ranked-set sample is given by X_S2=X_(1),X_(2),,X_(M), the expected value of the standardized X_S2 is given by α_S2T=α_1T,,α_MT and the variance and covariance matrix is given by ΩS2=DiagonalΩ_1,,Ω_M where

S2=N1(1),N2(1),,N1(M),N2(M)Ω_i=ωN1(i)N1(i)ωN1(i)N2(i)ωN1(i)N2(i)ωN1(i)N1(i),N1(i)<N2(i)andi=1,,Mα_iT=αN1(i),αN2(i)T

Least squares estimator of the parameter based on generalized ranked-set sample when two order statistics are selected from each sample:

S2=N11,N21,,N1M,N2Mμ^S2=i=1MaN1(i)(i)XN1(i)+aN2(i)(i)XN2(i)σ^S2=i=1MbN1(i)(i)XN1(i)+bN2(i)(i)XN2(i)
where
aN1(i)(i)=T1T1T2T32ωN2(i)N2(i)ωN1(i)N2(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2T3T1T2T32ωN2(i)N2(i)αN1(i)ωN1(i)N2(i)αN2(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2
aN2(i)(i)=T1T1T2T32ωN1(i)N1(i)ωN1(i)N2(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2T3T1T2T32ωN1(i)N1(i)αN2(i)ωN1(i)N2(i)αN1(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2
bN1(i)(i)=T2T1T2T32ωN2(i)N2(i)αN1(i)ωN1(i)N2(i)αN2(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2T3T1T2T32ωN2(i)N2(i)ωN1(i)N2(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2
bN2(i)(i)=T2T1T2T32ωN1(i)N1(i)αN2(i)ωN1(i)N2(i)αN1(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2T3T1T2T32ωN1(i)N1(i)ωN1(i)N2(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2
T1S2=αS2TΩS21αS2
T2S2=1TΩS211
T3S2=1TΩS21αS2

The variances and covariances of these estimators are given by

Varμ^S2=T1S2σ2/T1S2T2S2T3S22
Varσ^S2=T2S2σ2/T1S2T2S2T3S22
Covμ^S2,σ^S2=T3S2σ2/T1S2T2S2T3S22

The generalized variance is

Gvarμ^S2,σ^S2=Vμ^S2Vσ^S2Covμ^S2,σ^S22
and when three order statistics are selected from each set, the generalized ranked-set sample is S3=(N1(1),N2(1),N3(1),,N1(M),N2(M),N3(M)).

The variance and covariance matrix is given by

ΩS3=DiagonalΩ_1,,Ω_M
where
Ω_i=ωN1(i)N1(i)ωN1(i)N2(i)ωN1(i)N3(i)ωN2(i)N1(i)ωN2(i)N2(i)ωN2(i)N3(i)ωN3(i)N1(i)ωN3(i)N2(i)ωN3(i)N3(i),N1(i)<N2(i)<N3(i)andi=1,,M
μ^S3=i=1MaN1(i)(i)XN1(i)+aN2(i)(i)XN2(i)+aN3(i)(i)XN3(i)
σ^S3=i=1MbN1(i)(i)XN1(i)+bN2(i)(i)XN2(i)+bN3(i)(i)XN3(i)
aN1(i)(i)=T2T1T2T32|ΩS3|ωN2(i)N3(i)ωN1(i)N2(i)ωN2(i)N3(i)+ωN1(i)N3(i)ωN2(i)N3(i)ωN2(i)N2(i)+ωN3(i)N3(i)ωN2(i)N2(i)ωN1(i)N2(i)T3T1T2T32|ΩS3|αN1(i)ωN2(i)N2(i)ωN3(i)N3(i)ωN2(i)N3(i)2+αN2(i)ωN1(i)N3(i)ωN2(i)N3(i)ωN1(i)N2(i)ωN3(i)N3(i)+αN3(i)ωN1(i)N2(i)ωN2(i)N3(i)ωN1(i)N3(i)ωN2(i)N2(i)
aN2(i)(i)=T2T1T2T32|ΩS3|ωN1(i)N3(i)ωN1(i)N2(i)ωN1(i)N3(i)+ωN2(i)N3(i)ωN1(i)N3(i)ωN2(i)N3(i)+ωN3(i)N3(i)ωN1(i)N1(i)ωN1(i)N2(i)T3T1T2T32|ΩS3|αN1(i)ωN1(i)N3(i)ωN2(i)N3(i)ωN1(i)N2(i)ωN3(i)N3(i)+αN2(i)ωN1(i)N1(i)ωN3(i)N3(i)ωN1(i)N3(i)2+αN3(i)ωN1(i)N2(i)ωN1(i)N3(i)ωN1(i)N1(i)ωN2(i)N3(i)
aN3(i)(i)=T2T1T2T32|ΩS3|ωN2(i)N2(i)ωN1(i)N1(i)ωN1(i)N3(i)+ωN2(i)N2(i)ωN1(i)N3(i)ωN1(i)N2(i)+ωN2(i)N3(i)ωN1(i)N2(i)ωN1(i)N1(i)T3T1T2T32|ΩS3|αN1(i)ωN1(i)N2(i)ωN2(i)N3(i)ωN1(i)N3(i)ωN2(i)N2(i)+αN2(i)ωN1(i)N2(i)ωN1(i)N3(i)ωN1(i)N1(i)ωN2(i)N3(i)+αN3(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2
bN1(i)(i)=T2T1T2T32|ΩS3|αN1(i)ωN2(i)N2(i)ωN3(i)N3(i)ωN2(i)N3(i)2+αN2(i)ωN1(i)N3(i)ωN2(i)N3(i)ωN1(i)N2(i)ωN3(i)N3(i)+αN3(i)ωN1(i)N2(i)ωN2(i)N3(i)ωN1(i)N3(i)ωN2(i)N2(i)T3T1T2T32|ΩS3|ωN2(i)N3(i)ωN1(i)N2(i)ωN2(i)N3(i)+ωN1(i)N3(i)ωN2(i)N3(i)ωN2(i)N2(i)+ωN3(i)N3(i)ωN2(i)N2(i)ωN1(i)N2(i)
bN2(i)(i)=T2T1T2T32|ΩS3|αN1(i)ωN1(i)N3(i)ωN2(i)N3(i)ωN1(i)N2(i)ωN3(i)N3(i)+αN2(i)ωN1(i)N1(i)ωN3(i)N3(i)ωN1(i)N3(i)2+αN3(i)ωN1(i)N2(i)ωN1(i)N3(i)ωN1(i)N1(i)ωN2(i)N3(i)T3T1T2T32|ΩS3|ωN1(i)N3(i)ωN1(i)N2(i)ωN1(i)N3(i)+ωN2(i)N3(i)ωN1(i)N3(i)ωN2(i)N3(i)+ωN3(i)N3(i)ωN1(i)N1(i)ωN1(i)N2(i)
bN3(i)(i)=T2T1T2T32|ΩS3|αN1(i)ωN1(i)N2(i)ωN2(i)N3(i)ωN1(i)N3(i)ωN2(i)N2(i)+αN2(i)ωN1(i)N2(i)ωN1(i)N3(i)ωN1(i)N1(i)ωN2(i)N3(i)+αN3(i)ωN1(i)N1(i)ωN2(i)N2(i)ωN1(i)N2(i)2T3T1T2T32|ΩS3|ωN2(i)N2(i)ωN1(i)N1(i)ωN1(i)N3(i)+ωN2(i)N2(i)ωN1(i)N3(i)ωN1(i)N2(i)+ωN2(i)N3(i)ωN1(i)N2(i)ωN1(i)N1(i)
where
T1S3=αS3TΩS31αS3
T2S3=1TΩS311
T3S3=1TΩS31αS3

The variances and covariance of these estimators are given by

Varμ^S3=T1S3σ2/T1S3T2S3T3S32
Varσ^S3=T2S3σ2/T1S3T2S3T3S32
Covμ^S3,σ^S3=T3S3σ2/T1S3T2S3T3S32

The generalized variance is

Gvarμ^S3,σ^S3=Vμ^S3Vσ^S3Covμ^S3,σ^S32

2.2. Generalized Ranked-Set Minimum Variance Unbiased Estimator

GR-MVUEs are obtained from the generalized ranked-set estimators (RSSs) when all possible choices of S are considered. The best choice of S is the one which gives the minimum generalized variance of the estimators. This S is denoted by SGR-MVUE, The estimators are denoted by μ^GRMVUE and σ^GRMVUE. Table 1 provides ranks SGR-MVUE, variances, covariances and generalized variances of the estimators for N = 2(2)10 when two order statistics are selected from each sample. Table 2 provides ranks SGR-MVUE, variances, covariances and generalized variances of the estimators for N = 3(3)15 when three order statistics are selected from each sample.

N SGR-MVUE Var(μ^GRMVUE)σ2 Var(σ^GRMVUE)σ2 Covμ^GRMVUE,σ^GRMVUEσ2 GVarμ^GRMVUE,σ^GRMVUEσ4 aN1(i)(i) aN2(i)(i) bN1(i)(i) bN2(i)(i)
2 {1,2} 1.00166 0.81616 −0.68028 0.35473 1.62945 −0.62945 −0.81472 0.81472
4 {1,4} 0.10577 0.15425 −0.07402 0.01084 0.59445 −0.09445 −0.22292 −0.22292
6 {1,5} 0.03231 0.06900 −0.02440 0.00163 0.39148 −0.05815 −0.19728 0.19728
8 {1,7} 0.01361 0.03625 −0.00999 0.00040 0.27745 −0.02745 −0.12114 0.12114
10 {1,9} 0.00704 0.02296 −0.00516 0.00014 0.21575 −0.01575 −0.08545 0.08545
Table 1

Variances, covariances, generalized variances and coefficients for μ^GRMVUE and σ^GRMVUE when two order statistics are selected from each set.

N SGR-MVUE Var(μ^GRMVUE)σ2 Var(σ^GRMVUE)σ2 Covμ^GRMVUE,σ^GRMVUEσ2 GVarμ^GRMVUE,σ^GRMVUEσ4 aN1(i)(i) aN2(i)(i) aN3(i)(i) bN1(i)(i) bN2(i)(i) bN3(i)(i)
3 {1,2,3} 0.37765 0.39677 −0.24682 0.08892 1.39350 −0.15472 −0.23878 −0.75455 0.33586 0.41869
6 {1,4,6} 0.04616 0.08090 −0.02948 0.00287 0.57992 0.04437 0.03555 −0.29143 0.18421 0.10722
9 {1,7,9} 0.01404 0.03410 −0.00882 0.00040 0.36374 −0.01957 −0.01084 −0.15754 0.11233 0.04521
12 {1,9,12} 0.00608 0.01881 −0.00382 0.00010 0.26736 0.01206 0.00530 −0.11888 0.09072 0.02817
15 {1,10,14} 0.00318 0.01198 −0.00204 0.00003 0.21227 −0.00661 −0.00566 −0.10531 0.06791 0.03740
Table 2

Variances, covariances, generalized variances and coefficients for μ^GRMVUE and σ^GRMVUE when three order statistics are selected from each set.

2.3. Generalized Ranked-Set Sample Estimator

The RSSs for μ and σ are obtained from the generalized RSSs when S = {1, 2, ‥, N}. These estimators are denoted by μ^GRRSS and σ^GRRSS respectively. Table 3 provides coefficients for estimating the parameters and Table 4 provides variances, covariances and generalized variances of the estimators for N = 2(2)10 when two order statistics are selected from each sample. Table 5 provides coefficients for estimating the parameters and Table 6 provides variances, covariances and generalized variances of the estimators for N = 3(3)15 when three order statistics are selected from each sample.

N = 2
N = 4
N = 6
N = 8
N = 10
i j Nj(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i)
1 1 1 1.62945 −0.81472 1.39598 −0.95936 1.18171 −0.87926 1.04868 −0.81221 0.95884 −0.76147
2 2 −0.62945 0.81472 −0.33002 0.57960 −0.19158 0.40237 −0.12737 0.30237 −0.09238 0.24027
2 1 3 0.10286 0.11900 0.27059 −0.12321 0.29356 −0.18825 0.28928 −0.20735
2 4 −0.16883 0.26076 −0.17828 0.33185 −0.13637 0.28852 −0.10409 0.24253
3 1 5 −0.00826 0.14160 0.09360 0.00737 0.12928 −0.05456
2 6 −0.07418 0.12665 −0.10732 0.20950 −0.09736 0.20999
4 1 7 −0.02403 0.11826 0.03642 0.04345
2 8 −0.04073 0.07444 −0.07029 0.14261
5 1 9 −0.02430 0.09570
2 10 −0.02540 0.04884
Table 3

Coefficients aNj(i)(i) and bNj(i)(i) for computing μ^GRRSS and σ^GRRSS when two order statistics are selected from each set.

N Varμ^GRRSSσ2 Varσ^GRRSSσ2 Covμ^GRRSS,σ^GRRSSσ2 GVarμ^GRRSS,σ^GRRSSσ4 K Varσ^GRRSS,NNσ2
2 1.00166 0.81616 −0.68028 0.35473 1.00000 0.81616
4 0.22636 0.24452 −0.16630 0.02770 0.99312 0.22745
6 0.09334 0.11768 −0.07148 0.00588 0.99979 0.11721
8 0.04959 0.06927 −0.03901 0.00191 0.99988 0.06902
10 0.03031 0.04562 −0.02430 0.00079 1.00015 0.04593
Table 4

Variances, covariances and generalized variances for ranked-set sampling estimators when two order statistics are selected from each set.

N = 3
N = 6
N = 9
N = 12
N = 15
i j Nj(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i) aNj(i)(i) bNj(i)(i)
1 1 1 1.39350 −0.15472 1.20517 −0.83158 1.07394 −0.79372 0.98981 −0.75881 0.93084 −0.73083
2 2 −0.75455 0.33586 −0.00961 0.05816 0.00164 0.01809 0.00274 0.00730 0.00248 0.00336
3 3 1.39350 −0.15472 −0.18179 0.43148 −0.11408 0.32205 −0.07893 0.25062 −0.05879 0.20359
2 1 4 0.11952 0.06356 0.20536 −0.09207 0.21827 −0.13768 0.21668 −0.15328
2 5 −0.06299 0.13816 −0.01614 0.05087 −0.00487 0.02294 −0.00142 0.01197
3 6 −0.07030 0.14022 −0.10386 0.25106 −0.08502 0.23365 −0.06712 0.20289
3 1 7 0.01639 0.10237 0.07606 0.00826 0.09758 −0.03684
2 8 −0.03113 0.07180 −0.01265 0.03685 −0.00555 0.02024
3 9 −0.03212 0.06953 −0.06420 0.15982 −0.06212 0.17108
4 1 10 −0.00505 0.09211 0.03286 0.03744
2 11 −0.01806 0.04347 −0.00934 0.02677
3 12 −0.01808 0.04147 −0.04280 0.10953
5 1 13 −0.01018 0.07756
2 14 −0.01164 0.02900
3 15 −0.01148 0.02753
Table 5

Coefficients aNj(i)(i) and bNj(i)(i) for computing μ^GRRSS and σ^GRRSS when three order statistics are selected from each set.

N Varμ^GRRSSσ2 Varσ^GRRSSσ2 Covμ^GRRSS,σ^GRRSSσ2 GVarμ^GRRSS,σ^GRRSSσ4 K Varσ^GRRSS,NNσ2
3 0.37765 0.39677 −0.24682 0.08892 1.00000 0.39677
6 0.09507 0.12966 −0.06823 0.00767 0.99999 0.12963
9 0.04107 0.06454 −0.03083 0.00170 0.99996 0.06444
12 0.02243 0.03875 −0.01734 0.00057 0.99998 0.03870
15 0.01397 0.02587 −0.01102 0.00024 1.00011 0.02610

Note: 1 ≤ i ≤ N/3.

Table 6

Variances, covariances and generalized variances for ranked-set sampling estimators when three order statistics are selected from each set.

As the estimates of σ can be negative a nonnegative unbiased estimator σ^GRRSS,NN is obtained by taking the absolute value of σ^GRRSS multiplied by an unbiasing constant, i.e. let σ^GRRSS,NN=K|σ^GRRSS| where K is chosen such that Eσ^GRRSS,NN=σ.

Then Varσ^GRRSS,NN=K2Varσ^GRRSS+K21σ2

Tables 4 and 6 also includes the values of K and Varσ^GRRSS,NN. The values of K were obtained by simulation. In each case 104 generalized ranked-set samples were generated.

2.4. Best Linear Unbiased Estimators

BLUEs of location and scale parameters of the half logistic distribution have been obtained by Balakrishnan and Puthenpura [14]. Table 7 provides variances, covariances and generalized variances of the estimators for N = 2(2)12 and 3(3)15.

N Varμ^BLUEσ2 Varσ^BLUEσ2 Covμ^BLUE,σ^BLUEσ2 GVarμ^BLUE,σ^BLUEσ4
2 1.00170 0.81616 −0.68028 0.35473
3 0.37765 0.39677 −0.24682 0.08892
4 0.20545 0.25962 −0.13084 0.03622
6 0.09163 0.15210 −0.05648 0.01075
8 0.05253 0.10704 −0.03175 0.00461
9 0.04190 0.09314 −0.02514 0.00327
10 0.03424 0.08240 −0.02042 0.00240
12 0.02416 0.06691 −0.01427 0.00141
15 0.01576 0.05214 −0.00921 0.00074
Table 7

Variances, covariances, generalized variances for μ^BLUE and σ^BLUE.

3. SAMPLE DESIGN

Table 8 tabulates for each N (N = 3(3)15 and N = 4(2)12), the number of units selected and the relative efficiency of the generalized ranked-set sampling for each sample design. This will enable researchers to explore the relationship between efficiency of the estimators and the number of units sampled and thus the corresponding cost. It would also enable researchers to decide which sample design to select at what cost.

N Design Number of Order Statistics Selected Number of Units Selected Var(μ^GRRSS)σ2 Var(σ^GRRSS)σ2 Var(μ^BLUE)Var(μ^GRRSS) Var(σ^BLUE)Var(σ^GRRSS)
4 4 × 1 2 8 0.22636 0.24452 0.90763 1.06175
2 × 2 1 8 0.88712 0.65808 0.23159 0.39451
6 6 × 1 3 12 0.09507 0.12966 0.96384 1.17307
3 × 2 1 18 0.26033 0.23421 0.35199 0.64943
6 × 1 2 18 0.09334 0.11768 0.98170 1.29249
2 × 3 1 12 0.59141 0.43872 0.15494 0.34669
8 8 × 1 2 32 0.04959 0.06927 1.05933 1.54526
4 × 2 1 32 0.11867 0.12092 0.44269 0.88525
2 × 4 1 16 0.44356 0.32904 0.11843 0.32531
4 × 2 2 16 0.11318 0.12226 0.46415 0.87551
9 9 × 1 3 27 0.04107 0.06454 1.02026 1.44314
3 × 3 1 27 0.17355 0.15614 0.24144 0.59653
10 10 × 1 2 50 0.03031 0.04562 1.12979 2.12650
5 × 2 1 50 0.06623 0.07375 0.79318 1.66217
2 × 5 1 20 0.35485 0.26323 0.14804 1.40044
12 12 × 1 3 48 0.02243 0.03875 1.07704 1.72679
12 × 1 2 72 0.020254 0.032299 1.19275 2.07167
6 × 2 1 72 0.04162 0.04958 0.58051 1.34973
6 × 2 2 36 0.04667 0.05884 0.51763 1.13720
6 × 2 3 24 0.04754 0.06483 0.50821 1.03213
4 × 3 1 48 0.07911 0.08061 0.30537 0.83008
3 × 4 1 36 0.13016 0.11710 0.18560 0.57141
15 15 × 1 3 75 0.01397 0.02587 1.12785 2.01562
5 × 3 1 75 0.04415 0.04917 0.35685 1.06055
3 × 5 1 45 0.10413 0.02950 0.15131 1.76759
Table 8

Sample design, variances and relative efficiencies for generalized ranked-set sampling.

4. COMPARISONS

In this section comparison has been made between GR-MVUEs, GR-RSSs and BLUEs. Tables 9 and 10 provide the relative efficiencies of the estimators and show that GR-MVUEs are more efficient than ranked-set samples and BLUEs, when two or three order statistics are selected from each sample. Tables 9 and 10 also show that the ranked-set sample estimators are more efficient than the BLUEs.

N Var(μ^BLUE)Var(μ^GRMVUE) Var(σ^BLUE)Var(σ^GRMVUE) GVarμ^BLUE,σ^BLUEGVarσ^GRMVUE,σ^GRMVUE Var(μ^GRRSS)Var(μ^GRMVUE) Var(σ^GRRSS)Var(σ^GRMVUE) GVarμ^GRRSS,σ^GRRSSGVarμ^GRMVUE,σ^GRMVUE Var(μ^BLUE)Var(μ^GRRSS) Var(σ^BLUE)Var(σ^GRRSS) GVarμ^BLUE,σ^BLUEGVarμ^GRRSS,σ^GRRSS
2 1.00004 1.00000 1.00000 1.00000 1.00000 1.00000 1.00004 1.00000 1.00000
4 1.94242 1.68311 3.34256 2.14012 1.58522 2.55629 0.90763 1.06175 1.30758
6 2.83638 2.20444 6.57832 2.88925 1.70558 3.59919 0.98170 1.29249 1.82772
8 3.85981 2.95275 11.72574 3.64364 1.91084 4.85301 1.05933 1.54526 2.41618
10 4.86296 3.58941 17.80073 4.30430 1.98719 5.84795 1.12979 1.80627 3.04392
Table 9

Relative efficiencies for the estimators when two order statistics are selected from each set.

N Var(μ^BLUE)Var(μ^GRMVUE) Var(σ^BLUE)Var(σ^GRMVUE) GVarμ^BLUE,σ^BLUEGVarσ^GRMVUE,σ^GRMVUE Var(μ^GRRSS)Var(μ^GRMVUE) Var(σ^GRRSS)Var(σ^GRMVUE) GVarμ^GRRSS,σ^GRRSSGVarμ^GRMVUE,σ^GRMVUE Var(μ^BLUE)Var(μ^GRRSS) Var(σ^BLUE)Var(σ^GRRSS) GVarμ^BLUE,σ^BLUEGVarμ^GRRSS,σ^GRRSS
3 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
6 1.98531 1.88017 3.75113 2.05980 1.60278 2.67714 0.96384 1.17307 1.40117
9 2.98426 2.73130 8.15433 2.92501 1.89261 4.23834 1.02026 1.44314 1.92394
12 3.97551 3.55655 14.16312 3.69115 2.05964 5.71417 1.07704 1.72679 2.47860
15 4.95378 4.35259 21.69725 4.39225 2.15943 7.06818 1.12785 2.01562 3.06971
Table 10

Relative efficiencies for the estimators when three order statistics are selected from each set.

5. CONCLUSION

The GR-MVUEs and GR-RSSs are both more efficient than the BLUEs. The GR-MVUE are the most efficient estimators. In applications where ranked-set sampling is useful, the cost involved in sampling and ranking cannot be completely ignored. Therefore, the choice of the estimators based on two or three units selected from each set would provide a reasonable alternative to the balanced ranked-set sample.

Journal
Journal of Statistical Theory and Applications
Volume-Issue
19 - 1
Pages
109 - 117
Publication Date
2020/03/10
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.200303.001How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - A. Adatia
AU  - A.K.MD. Ehsanes Saleh
PY  - 2020
DA  - 2020/03/10
TI  - Sample Design and Estimation of Parameters of Half Logistic Distribution Using Generalized Ranked-Set Sampling
JO  - Journal of Statistical Theory and Applications
SP  - 109
EP  - 117
VL  - 19
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200303.001
DO  - 10.2991/jsta.d.200303.001
ID  - Adatia2020
ER  -