Sample Design and Estimation of Parameters of Half Logistic Distribution Using Generalized Ranked-Set Sampling
- 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
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
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
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.
We note that from the generalized ranked-set sampling procedure, the balanced ranked-set sample is obtained by selecting the ordered unit
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))
Let
Then the generalized ranked-set sample is given by
Least squares estimator of the parameter based on generalized ranked-set sample when two order statistics are selected from each sample:
The variances and covariances of these estimators are given by
The generalized variance is
The variance and covariance matrix is given by
The variances and covariance of these estimators are given by
The generalized variance is
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
| N | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| 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 |
Variances, covariances, generalized variances and coefficients for
| N | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 |
Variances, covariances, generalized variances and coefficients for
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
| N = 2 |
N = 4 |
N = 6 |
N = 8 |
N = 10 |
||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| i | j | |||||||||||
| 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 | |||||||||
Coefficients
| N | K | |||||
|---|---|---|---|---|---|---|
| 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 |
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 | |||||||||||
| 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 | |||||||||
Coefficients
| N | K | |||||
|---|---|---|---|---|---|---|
| 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.
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
Then
Tables 4 and 6 also includes the values of K and
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 | ||||
|---|---|---|---|---|
| 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 |
Variances, covariances, generalized variances for
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 | ||||
|---|---|---|---|---|---|---|---|
| 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 | |
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 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| 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 |
Relative efficiencies for the estimators when two order statistics are selected from each set.
| N | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| 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 |
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.
REFERENCES
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 -