Weighted Ranked Set Sampling for Skewed Distributions

Abstract

Ranked set sampling (RSS) is a useful technique for improving the estimator of a population mean when the sampling units in a study can be more easily ranked than the actual measurement. RSS performs better than simple random sampling (SRS) when the mean of units corresponding to each rank is used. The performance of RSS can be increased further by assigning weights to the ranked observations. In this paper, we propose weighted RSS procedures to estimate the population mean of positively skewed distributions. It is shown that the gains in the relative precisions of the population mean for chosen distributions are uniformly higher than those based on RSS. The gains in relative precisions are substantially high. Further, the relative precisions of our estimator are slightly higher than the ones based on Neyman’s optimal allocation model for small sample sizes. Moreover, it is shown that the performance of the proposed estimator increases as the skewness increases by using the example of the lognormal family of distributions.

1. Introduction

The ranked set sampling (RSS) procedure has been used advantageously in agriculture, forestry, environmental studies, ecological studies and recently in human studies where the exact measurement of units is either difficult or expensive. For example, in forestry, the measurement of the stem volume of standing trees is difficult but the ranking of the trees using their height and diameter at breast height is rather easy. For such situations, McIntyre (1952) [1] introduced RSS to estimate the population mean. RSS is a cost-efficient alternative to simple random sampling (SRS) if observations can be ranked according to the characteristic under investigation by means of visual inspection or other methods not requiring actual measurements. McIntyre (1952) [1] indicated that the RSS procedure is superior to the SRS procedure in estimating the population mean. Further, Dell and Clutter (1972) [2] and Takahasi and Wakimoto (1968) [3] provided a mathematical foundation for RSS. Dell and Clutter (1972) [2] also showed that the estimator for the population mean based on RSS is at least as efficient as the estimator based on SRS with the same number of measurements even when there are ranking errors. Bhoj (2001) [4] introduced RSS with unequal samples. Recently, some novel versions of RSS have been suggested, for example, except extreme ranked set sampling (Aldrabseh and Ismail, 2023 [5]); partial stratified ranked set sampling (Almanjahie et al., 2023 [6]); and dual ranked set sampling (Taconeli, 2023 [7]). Some other versions are found in Latpate et al. (2021) [8]. Some classes of estimators using RSS are presented by Bhushan et al. (2022) [9] and Yusuf et al. (2023) [10].

Substantial work is found in the area of unequal allocations of RSS (Kaur et al., 1997, 2000 [11,12]; Tiwari and Chandra 2011 [13]; Tiwari et al., 2023 [14]). Bhoj and Kushary (2016) [15] proposed RSS with unequal samples for positively skewed distributions with heavy right tails. However, in the present paper, an attempt is made by assigning unequal weights instead of unequal allocations to obtain more precision from the estimators. Further, weight assignment is rather easy and cost-effective compared with the repeated allocation of order statistics.

The selection of a ranked set sample of size k involves drawing k random samples with k units in each sample. The units in each sample are ranked using judgment or other methods not requiring actual measurements. The unit with the lowest rank is measured from the first sample, the unit with the second-lowest rank is measured from the second sample, and the procedure is continued until the unit with the highest rank is measured from the last sample. The k² ordered observations in k samples can be displayed in the matrix form as

$$\left[\begin{array}{ccc}{y}_{\left(11\right)}& y_{\left(12\right)}\dots & y_{\left(1k\right)}\\ y_{\left(21\right)}& y_{\left(22\right)}\dots & y_{\left(2\mathrm{k}\right)}\\ \dots & \dots & \dots \\ y_{\left(\mathrm{k}1\right)}& y_{\left(\mathrm{k}2\right)}\dots & y_{\left(\mathrm{k}\mathrm{k}\right)}\end{array}\right]$$

We measure only k$\left(y_{\left(ii\right)},i=\mathrm{1,2}...,k\right)$ diagonal observations, and they constitute the RSS. We note that these k observations are independently but not identically distributed. In RSS, k is usually small to reduce the ranking errors, and therefore, to increase the sample size, the above procedure is repeated $m\ge 2$ times to obtain a sample of the size $n=mk$. In this paper, we assume $m=1$; i.e., the sample size is equal to the set size.

In the present paper, our main interest is to estimate the population mean for positively skewed distributions with a longer right tail. We propose estimators based on weighted ranked set sampling (WRSS) and compare their performance with those of the ones based on the usual RSS procedure and Neyman’s optimal allocation model. In Section 2, we summarize the estimators of population mean based on the RSS procedure and Neyman’s optimal solution. In Section 3, we propose our WRSS procedure to estimate the population mean of skewed distributions. First, we introduce the WRSS procedure, where we assign one low weight to the highest order statistics and calculate the relative precisions of the estimator based on WRSS, RSS and Neyman’s optimal procedure with respect to the estimator based on SRS. The procedures are used to obtain the relative precisions by using the four positively skewed distributions. We also compute one set of weights for all four distributions for each k. In Section 4, we derive optimal weights for the lowest and highest order statistics for the chosen distributions for each k. We then obtain one set of weights for the lowest and highest order statistics for each k, which will maximize the sum of relative precisions of four distributions. In Section 5, we generalize the use of all optimal weights for all order statistics for k = 4 and k = 5 for each distribution. We also obtain one set of weights for each k for the four chosen distributions. In Section 6, to see the effect of increasing skewness, the relative precisions of estimators for the lognormal family of distributions are compared. In Section 7, we summarize the results and give recommendations.

2. Estimation of Mean

We consider first the usual RSS to estimate the population mean. We let $y_{\left(ii\right)}, i=\mathrm{1,2},\dots ,k$ denote the value of the characteristic under study of the ith order statistic. The mean and variance of the ith rank order statistic for set size k are denoted by ${\mu }_{\left(ii\right)}$ and ${\sigma }_{\left(ii\right)}^2$, respectively. We denote the population mean and variance with $\mu$ and ${\sigma }^2$, respectively. Then, the unbiased estimator for $\mu$ based on RSS is given by

$$\overline{\mu }=\frac{1}{k}{\sum }_{i=1}^k{y}_{\left(ii\right)},$$

with the variance

$$Var\left(\overline{\mu }\right)=\frac{1}{k^2}{\sum }_{i=1}^k{\sigma }_{\left(ii\right)}^2.$$

The relative precision of $\overline{\mu }$ compared to the estimator based on SRS with the same number of observations k (Bhoj and Chandra, 2019 [16]) is

$$R{P}_1=\frac{{\sigma }^2}{\overline{{\sigma }^2}}$$

(1)

where $\overline{{\sigma }^2}=\frac{1}{k}{\sum }_{i=1}^k{\sigma }_{\left(ii\right)}^2$ is the average within-rank variance.

For the skewed distribution, Neyman’s allocation $m_i=\frac{n{\sigma }_{\left(ii\right)}}{{\sum }_{i=1}^k{\sigma }_{\left(ii\right)}}$ provides the optimal allocation and the relative precision of the unbiased estimator of $\mu$ based on this model with respect to SRS with the same number of observations n and is given by the following equation (Bhoj and Chandra, 2019 [16]):

$$R{P}_3=\frac{{\sigma }^2}{{\overline{\sigma }}^2}$$

(2)

where $\overline{\sigma }=\frac{1}{k}{\sum }_{i=1}^k{\sigma }_{\left(ii\right)}$ is the average within-rank standard deviation.

There are some unequal allocation models for the skewed distributions in the literature (see “t” and “s, t” model (Kaur et al., 1997 [11]), systematic model (Tiwari and Chandra, 2011 [13]) and simple model (Chandra et al., 2018 [17] and Bhoj and Chandra, 2019 [16])). The Neyman’s allocation does not provide the integer values of $m_i$ which are necessary for any application. The procedure of making them integers is shown in Bhoj and Chandra (2019) [16] and used in this paper. It is noted that the inequality $R{P}_3>R{P}_1$ always holds for the skewed distributions.

3. WRSS with One Optimal Weight

In this section, we propose a weighted ranked set sampling (WRSS) with the optimal weight for the largest order statistic since the largest order statistic has the highest variance and higher bias of the estimator for the mean when we deal with the positively skewed distributions. We define the weights $w_i$ $\left(\mathrm{with} {0\le w}_i\le 1 \mathrm{a}\mathrm{n}\mathrm{d} {\sum }_{i=1}^k{w}_i=1\right)$ as

$$w_i\propto 1, \mathrm{for}i=1, 2, \dots ,k-1.$$

$$w_k\propto \frac{1}{C_k}.$$

The exact values of the weights are proposed as follows:

$$w_i=\frac{1}{\left(k-1\right)+\frac{1}{C_k}}, \mathrm{for}i=1, 2, \dots ,k-1.$$

$$w_k=\frac{1}{C_k\left(\left(k-1\right)+\frac{1}{C_k}\right)}.$$

Our weighted estimator for the population mean $\mu$ is

$${\overline{\mu }}_{W_1}={\sum }_{i=1}^k{w_{i}y}_{\left(ii\right)}$$

(3)

The relative precision of our biased estimator ${\overline{\mu }}_{W_1}$ with respect to the estimator based on SRS is

$$R{P}_2=\frac{{\sigma }^2{\left[1+C_k\left(k-1\right)\right]}^2}{k\left[C_k^2{\sum }_{i=1}^{k-1}{\sigma }_{\left(ii\right)}^2+{\sigma }_{\left(kk\right)}^2+{\left\{\mu \left(1+C_k\left(k-1\right)\right)-C_k{\sum }_{i=1}^{k-1}{\mu }_{\left(ii\right)}-{\mu }_{\left(kk\right)}\right\}}^2\right]}$$

(4)

The value of $C_k$ is to be chosen such that the $R{P}_2$ is maximum. To find the optimum value of $C_k$ (for each k), a program using the “Solver Tool” of MS Excel version 2016 for $R{P}_2$ was developed, and using the different iterations on $C_k$, the value of $R{P}_2$ was tested until it reached its maximum. For all the other values above and below this optimal $C_k$, $R{P}_2$ started decreasing.

We computed $C_k$ for all four chosen distributions, lognormal (LN(0,1)), Pareto (P(3.5) and P(4.5)) and Weibull (W(0.5)), and k = 2(1)5. The values of $R{P}_2$, $C_k$, $R{P}_3$ and $R{P}_1$ for these distributions and k = 2(1)5 are presented in

Table 1. The RPs ($R{P}_1$,$R{P}_2$, $R{P}_3$) at an individual optimal $C_k$ of each distribution for k = 2(1)5.

Set Size (k)		2	3	4	5
LN(0,1)	$C_k$	4.3798	3.2859	2.8028	2.5263
	$R{P}_1$	1.1872	1.3393	1.4711	1.5891
	$R{P}_2$	2.5946	2.7278	2.8083	2.8845
	$R{P}_3$	1.5765	2.1182	2.6219	3.1347
P(3.5)	$C_k$	4.7900	3.5693	3.0417	2.7427
	$R{P}_1$	1.1707	1.3073	1.4238	1.5269
	$R{P}_2$	2.7528	2.8579	2.9189	2.9805
	$R{P}_3$	1.5834	2.1273	2.6370	3.1434
P(4.5)	$C_k$	3.8151	2.8990	2.4962	2.2678
	RP1	1.2134	1.3901	1.5451	1.6847
	$R{P}_2$	2.3679	2.5338	2.6535	2.7676
	$R{P}_3$	1.5544	2.0810	2.5995	3.0878
W(0.5)	$C_k$	6.7391	4.4803	3.5837	3.0972
	$R{P}_1$	1.1268	1.2362	1.3345	1.4250
	$R{P}_2$	3.6271	3.3698	3.2166	3.1379
	$R{P}_3$	1.6306	2.2105	2.7913	3.3840

. The values of $R{P}_2$ are much higher than those of $R{P}_1$, i.e., the relative precisions of the estimator based on the RSS procedure. Furthermore, the $R{P}_2$ values are higher than those of $R{P}_3$, i.e., those based on Neyman’s optimal allocation model for all four distributions when $k\le 4$. All relative precisions increase as k increases for LN(0,1), P(3.5) and P(4.5). However, for W(0.5), $R{P}_2$ decreases as k increases. This may be because the distribution W(0.5) has extremely large skewness and kurtosis.

Now we attempt to compute one set of values of $C_k$ for four values of sample sizes which work well for all four chosen distributions. In these computations, $C_k$ was determined so that the sum of $R{P}_2$ for the four distributions is close to the maximum. This optimum value of $C_k$ was found using the same iteration procedure in the developed Excel program.

The values of optimum $C_k$, and $R{P}_2$ for the chosen four distributions and four sample sizes are presented in

Table 2. The RP values ($R{P}_1$, $R{P}_2$, $R{P}_3$ and total $R{P}_2$) at a combined optimal $C_k$ for k = 2(1)5.

Set Size (k)		2	3	4	5
$C_k$		5.0655	3.6004	2.9955	2.6618
Total $R{P}_2$		11.2063	11.3956	11.5215	11.7050
Total maximum $R{P}_2$ *		11.3424	11.4893	11.5973	11.7705
LN(0,1)	$R{P}_1$	1.1872	1.3393	1.4711	1.5891
	$R{P}_2$	2.5809	2.7207	2.8036	2.8811
	$R{P}_3$	1.5765	2.1182	2.6219	3.1347
P(3.5)	$R{P}_1$	1.1707	1.3073	1.4238	1.5269
	$R{P}_2$	2.7507	2.8578	2.9186	2.9794
	RP3	1.5834	2.1273	2.6370	3.1434
P(4.5)	$R{P}_1$	1.2134	1.3901	1.5451	1.6847
	$R{P}_2$	2.3205	2.4963	2.6199	2.7363
	$R{P}_3$	1.5544	2.0810	2.5995	3.0878
W(0.5)	$R{P}_1$	1.1268	1.23617	1.3345	1.4250
	$R{P}_2$	3.5542	3.3208	3.1793	3.1082
	$R{P}_3$	1.6306	2.2105	2.7913	3.3840

* Total maximum $R{P}_2$ is the sum of $R{P}_2$ of all the distributions at their respective optimum $C_k$ values.

. The values of $R{P}_2$ in Table 2 are slightly lower than the ones in Table 1, as is expected. This is due to the values of $C_k$ in Table 2 slightly differing from the optimum $C_k$ given in Table 1. However, the pattern of $R{P}_2$ remains the same in both the tables (Table 1 and Table 2).

4. WRSS with Two Optimal Weights

In this section, we propose a WRSS with two optimal weights for the two extreme order statistics. Here, the weights $w_i$ $\left(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {0\le w}_i\le 1 \mathrm{a}\mathrm{n}\mathrm{d} {\sum }_{i=1}^k{w}_i=1\right)$ for $k>2$ are defined as

$$w_1\propto \frac{1}{C_1}.$$

$$w_i\propto 1, \mathrm{for}i=2, 3, \dots ,k-1.$$

$$w_k\propto \frac{1}{C_k}.$$

The proposed exact weights are as follows:

$$w_1=\frac{1}{{DC}_1}.$$

$$w_i=\frac{1}{D}, \mathrm{for} i=2,3,\dots ,k-1.$$

$$w_k=\frac{1}{{DC}_k}.$$

where $D=\left(k-2\right)+\frac{1}{C_1}+\frac{1}{C_k}$.

Our estimator of population mean is

$${\overline{\mu }}_{W_2}={\sum }_{i=1}^k{w_{i}y}_{\left(ii\right)}$$

(5)

The relative precision of ${\overline{\mu }}_{W_2}$ with respect to the estimator based on SRS is

$$R{P}_2=\frac{{\sigma }^2{D}^2}{k\left[{\sum }_{i=2}^{k-1}{\sigma }_{\left(ii\right)}^2+\frac{{\sigma }_{\left(11\right)}^2}{C_1^2}+\frac{{\sigma }_{\left(kk\right)}^2}{C_k^2}+{\left\{\mu D-{\sum }_{i=2}^{k-1}{\mu }_{\left(ii\right)}-\frac{{\mu }_{\left(11\right)}}{C_1}-\frac{{\mu }_{\left(kk\right)}}{C_k}\right\}}^2\right]}$$

(6)

We calculate the optimal values of $C_k$ and $C_1$ using the iteration method. Based on these values, we computed $R{P}_2$ along with $R{P}_1$ and $R{P}_3$ for the chosen four distributions and sample sizes k = 3, 4 and 5; the results are presented in

Table 3. The RPs ($R{P}_1$,$R{P}_2,$ $R{P}_3$) at individual optimal $C_k$, $C_1$ values of each distribution for k = 3(1)5.

Set Size (k)		3	4	5
LN(0,1)	$C_k$	3.3322	3.8383	4.3430
	$C_1$	1.0217	1.9580	6.9907
	RP1	1.3393	1.4711	1.5891
	$R{P}_2$	2.7280	2.8982	3.1621
	$R{P}_3$	2.1182	2.6219	3.1347
P(3.5)	$C_k$	3.6003	4.1931	4.8037
	$C_1$	1.0134	2.0183	9.4583
	$R{P}_1$	1.3073	1.4238	1.5269
	$R{P}_2$	2.8579	3.0154	3.2847
	$R{P}_3$	2.1273	2.6370	3.1434
P(4.5)	$C_k$	2.8188	3.2036	3.6176
	$C_1$	0.9587	1.6485	3.9315
	$R{P}_1$	1.3901	1.5451	1.6847
	$R{P}_2$	2.5344	2.7072	2.9638
	$R{P}_3$	2.0810	2.5995	3.0878
W(0.5)	$C_k$	3.3947	3.7117	3.9996
	$C_1$	0.6463	1.0814	2.4127
	$R{P}_1$	1.2362	1.3345	1.4250
	$R{P}_2$	3.4286	3.2176	3.1936
	$R{P}_3$	2.2105	2.7913	3.3840

. The gains in precisions of the estimator ${\overline{\mu }}_{W_2}$ over ${\overline{\mu }}_{W_1}$ are marginal. The gains of $R{P}_2$ based on ${\overline{\mu }}_{W_2}$ are substantially higher than those of the estimator based on RSS. ${\overline{\mu }}_{W_2}$ is superior to the estimator based on Neyman’s optimal allocation model for all values of k for the LN(0,1) and P(3.5) distributions. The values of $R{P}_2$ are higher than those of $R{P}_3$ for the other two distributions for k = 3 and 4. The gains of $R{P}_3$ over $R{P}_2$ for k = 5 for these two distributions are marginal.

As we did in the case of ${\overline{\mu }}_{W_1},$ we attempted to compute one set of values of $C_k$ and $C_1$ for three values of sample sizes which work well for all four chosen distributions. In these computations, $C_k$ and $C_1$ were determined so that the sum of relative precisions of ${\overline{\mu }}_{W_2}$ for the four distributions was close to the maximum relative precision. The values of $C_k$ and $C_1$ and $R{P}_1$,$R{P}_2$ and $R{P}_3$ for the three sample sizes and four chosen distributions are presented in

Table 4. The RPs ($R{P}_1$,$R{P}_2$, $R{P}_3$ and total $R{P}_2$) at combined optimal $C_k$, $C_1$ values for k = 3(1)5.

Set Size (k)		3	4	5
$C_k$		3.4245	3.8499	4.2613
$C_1$		0.9241	1.7370	5.3284
Total $R{P}_2$		11.4036	11.7566	12.5655
Total maximum $R{P}_2$ *		11.5489	11.8384	12.6042
LN(0,1)	$R{P}_1$	1.3393	1.4711	1.5891
	$R{P}_2$	2.7169	2.8937	3.1605
	$R{P}_3$	2.1182	2.6219	3.1347
P(3.5)	$R{P}_1$	1.3073	1.4238	1.5269
	$R{P}_2$	2.8549	3.0112	3.2772
	$R{P}_3$	2.1273	2.6370	3.1434
P(4.5)	$R{P}_1$	1.3901	1.5451	1.6847
	RP2	2.4947	2.6841	2.9504
	$R{P}_3$	2.0810	2.5995	3.0878
W(0.5)	$R{P}_1$	1.2362	1.3345	1.4250
	$R{P}_2$	3.3371	3.1677	3.1773
	$R{P}_3$	2.2105	2.7913	3.3840

* Total maximum $R{P}_2$ is the sum of the $R{P}_2$ of all the distributions at their respective $C_k$ and $C_1$ values.

. The relative precisions of ${\overline{\mu }}_{W_2}$ in Table 4 are higher than those of ${\overline{\mu }}_{W_1}$ for each k in Table 2. The pattern of relative precisions is the same as seen in Table 3.

5. WRSS with All Optimal Weights

Now, we extend WRSS with optimal weights for all order statistics for k = 4 and k = 5. We take $C=w_1+w_k$, and determine the optimal values of C and $w_k$ by minimizing the mean square error (MSE) of the estimator by using $w_i\propto 1$, $\mathrm{for}i=2, \dots ,k-1$.

In the next step, we use

$$w_i=\left(1-C\right)f_i,i=2, \dots ,k-1 \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\sum }_{i=2}^{k-1}{f}_i=1.$$

The values of $f_i$ are chosen so that the value of $R{P}_2$ is maximized. Then we repeat the procedure of computing the optimal values of C and $w_k$ with these new $w_i$’s. The procedure is repeated until the value of $R{P}_2$ reaches the maximum. We performed this by using the developed computer program in the “Solver Tool” of MS Excel version 2016.

The values of $R{P}_2$ are presented in

Table 5. The RPs ($R{P}_1$, $R{P}_2$ and $R{P}_3$) at individual optimal C, $w_k$ and $f_i$’s values of each distribution for k = 4 and k = 5.

		LN(0,1)	P(3.5)	P(4.5)	W(0.5)
k = 4	$C$	0.2136	0.2070	0.2693	0.2191
	$w_k$	0.0937	0.0870	0.1069	0.0830
	$f_2$	0.5902	0.5827	0.5704	0.6484
	$f_3$	0.4098	0.4173	0.4296	0.3516
	$R{P}_1$	1.4711	1.4238	1.5451	1.3345
	$R{P}_2$	2.9401	3.0510	2.7304	3.2862
	$R{P}_3$	2.6219	2.7913	2.6370	2.5995
k = 5	$C$	0.0827	0.0767	0.0905	0.0799
	$w_k$	0.0702	0.0648	0.0785	0.0686
	$f_2$	0.3346	0.3260	0.3590	0.3491
	$f_3$	0.4011	0.4013	0.3774	0.4143
	$f_4$	0.2643	0.2727	0.2636	0.23660
	$R{P}_1$	1.5891	1.5269	1.6847	1.4250
	$R{P}_2$	3.2583	3.3652	3.0329	3.3215
	$R{P}_3$	3.1347	3.1434	3.0878	3.3840

. We observe that the values of $R{P}_2$ presented in Table 4 are higher than the values of $R{P}_2$ based on one or two optimal weights that are given in Table 1 and Table 3.

As we did in Section 3 and Section 4, we computed one set of values of C, $w_k$ and different fractions of $f_i$ for k = 4 and k = 5 which work well for all chosen four distributions. In these computations, these values were determined so that the sum of $R{P}_2$’s for the four distributions is close to the maximum relative precision. These values, along with $R{P}_1$,$R{P}_2$ and $R{P}_3$ for k = 4 and k = 5 and the four chosen distributions, are presented in

Table 6. The values of $R{P}_2$ and total $R{P}_2$ at combined optimal values of $C$, $w_k$ and $f_i$’s for k = 4 and 5.

Set Size (k)		4	5
$C$		0.221	0.0824
$w_k$		0.0911	0.07
$f_2$		0.5967	0.3397
$f_3$		0.4033	0.3992
$f_4$			0.2611
Total $R{P}_2$			11.9303
Total maximum $R{P}_2$ *			12.0077
LN(0,1)	$R{P}_1$	1.4711	1.5891
	$R{P}_2$	2.9367	3.2577
	$R{P}_3$	2.6219	3.1347
P(3.5)	$R{P}_1$	1.4238	1.5269
	$R{P}_2$	3.0467	3.3587
	$R{P}_3$	2.7913	3.1434
P(4.5)	$R{P}_1$	1.5451	1.6847
	$R{P}_2$	2.7018	3.0164
	$R{P}_3$	2.637	3.0878
W(0.5)	$R{P}_1$	1.3345	1.425
	$R{P}_2$	3.2452	3.3078
	$R{P}_3$	2.5995	3.384

* Total maximum $R{P}_2$ is the sum of $R{P}_2$ of all the distributions at their respective values of $C$, $w_k$ and $f_i$’s.

. As we expected, the values of $R{P}_2$ are smaller in Table 6 when compared to the values of $R{P}_2$ in Table 5. However, the pattern of relative precisions remains the same.

6. WRSS with Increasing Skewness

In this section, we wish to study the performance of the three methods, RSS, WRSS and Neyman’s optimum allocation model, with increasing values of skewness of a family of distributions. For this purpose, the lognormal distribution, $LN(0,b)$, is considered. The probability density function (pdf) of $LN(a,b)$ is given by

$$\begin{array}{l}f\left(x\right)=\frac{1}{xb\sqrt{2\pi }}\mathit{exp}\left[\frac{-1}{2}{\left(\frac{\mathit{log}x-a}{b}\right)}^2\right],\mathit{for}x>0,a>0,b>0,\mathrm{with}\\ \mathrm{population}\mathrm{mean}=\mathit{exp}\left(a+{\frac{b}{2}}^2\right)\mathrm{and}\mathrm{variance}=\mathit{exp}\left(2a+2b^2\right)-\mathit{exp}\left(2a+b^2\right).\end{array}$$

Then, skewness (Sk) and shape parameter (p) are given by

$$Sk=\sqrt{{\beta }_1}=\sqrt{\mathit{exp}\left(b^2\right)-1}\left(\mathit{exp}\left(b^2\right)+2\right)\mathrm{and}p=Exp\left(b^2\right).$$

The performance of these three methods relative to SRS with k = 4 is presented in

Table 7. The values of $R{P}_1$, $R{P}_2$ and $R{P}_3$ for lognormal $LN(0,b)$ distributions for k = 4.

p	$\mathit{S}\mathit{k}$	${\mathit{C}}_{\mathit{k}}$ for ${\overline{\mathit{\mu }}}_{{\mathit{W}}_1}$	$\mathbf{For} {\overline{\mathit{\mu }}}_{{\mathit{W}}_2}$		$\mathbf{For} {\overline{\mathit{\mu }}}_{{\mathit{W}}_3}$				$\mathit{R}{\mathit{P}}_1$	$\mathit{R}{\mathit{P}}_2$			$\mathit{R}{\mathit{P}}_3$
			${\mathit{C}}_{\mathit{k}}$	${\mathit{C}}_1$	C	${\mathit{w}}_{\mathit{k}}$	${\mathit{f}}_2$	${\mathit{f}}_3$		${\overline{\mathit{\mu }}}_{{\mathit{W}}_1}$	${\overline{\mathit{\mu }}}_{{\mathit{W}}_2}$	${\overline{\mathit{\mu }}}_{{\mathit{W}}_3}$
1.8	3.40	2.07	2.715	1.633	0.301	0.124	0.556	0.444	1.702	2.490	2.552	2.568	2.520
1.9	3.70	2.16	2.848	1.674	0.289	0.119	0.561	0.439	1.665	2.521	2.587	2.606	2.535
2.0	4.00	2.24	2.978	1.714	0.279	0.115	0.565	0.435	1.632	2.553	2.623	2.644	2.550
2.1	4.30	2.33	3.105	1.753	0.268	0.112	0.569	0.431	1.603	2.587	2.660	2.684	2.564
2.2	4.60	2.41	3.229	1.790	0.258	0.108	0.573	0.427	1.576	2.621	2.697	2.724	2.577
2.3	4.90	2.48	3.351	1.825	0.249	0.105	0.577	0.424	1.552	2.656	2.735	2.765	2.590
2.4	5.21	2.56	3.471	1.859	0.240	0.102	0.580	0.420	1.530	2.692	2.774	2.806	2.603
2.5	5.51	2.64	3.588	1.891	0.231	0.099	0.583	0.417	1.510	2.728	2.813	2.848	2.615
2.6	5.82	2.71	3.704	1.923	0.223	0.097	0.587	0.413	1.491	2.765	2.852	2.890	2.626
2.7	6.13	2.79	3.818	1.953	0.215	0.094	0.590	0.410	1.474	2.802	2.891	2.932	2.637
2.8	6.44	2.86	3.930	1.981	0.207	0.092	0.593	0.407	1.458	2.839	2.931	2.975	2.648
2.9	6.75	2.94	4.040	2.009	0.200	0.090	0.595	0.405	1.444	2.876	2.970	3.017	2.658
3.0	7.07	3.01	4.149	2.035	0.193	0.088	0.598	0.402	1.430	2.914	3.010	3.060	2.668

Note: Here, ${\overline{\mu }}_{W_3}$ represents the proposed estimator based on all optimal weights.

for a lognormal family of distributions for a range of values of the population’s standard deviation. The variances of the order statistics of the family of distributions were computed by using the variances of order statistics for different values of the shape parameter (p), which are readily available in Balakrishnan and Chen (1999) [18]. From Table 7, we observe that as skewness increases, the performance of (i) the RSS method decreases, and (ii) the Neyman’s and WRSS methods increases. The values of $R{P}_2$ based on all and two optimal weights are higher than those of $R{P}_3$ for all values of shape parameters. However, $R{P}_2$ based on one optimal weight is higher than $R{P}_3$ for all values of p > 1.9. The rate of increase in relative precisions of the proposed estimators based on WRSS is higher than that of the estimator based on Neyman’s method (see Figure 1).

7. Conclusions and Discussion

In this paper, we proposed a weighted ranked set sampling procedure to estimate the population mean of distributions which are positively skewed with a heavy right tail. We chose four distributions: lognormal (LN(0,1)), Pareto (P(3.5) and P(4.5)) and Weibull (W(0.5)). The means and variances of order statistics for these distributions are readily available in Harter and Balakrishnan (1996) [19]. We proposed three weighted ranked set sampling procedures. The first procedure is based on one optimal weight for the largest order statistics, the second procedure is to use the two optimal weights for the two extreme order statistics and the third is the one which is based on k optimal weights. We calculated the relative precisions for each of these four distributions by using the WRSS procedure for each sample size. These relative precisions are much higher than the relative precisions of the RSS estimator of the mean. Furthermore, the relative precisions of our estimators are higher than those that are based on Neyman’s optimal procedures for $k\le 4$. The relative precisions of our estimator are even higher than those of Neyman’s procedure for k = 5 for some distributions. Furthermore, we attempted to compute one set of weight(s) for each k for all the distributions and compared the relative precisions of our estimator with those of the RSS and Neyman’s estimators. Although there is a slight loss in the values of relative precisions, they are still higher than those of Neyman’s model for $k\le 4$ for all four distributions and either higher than or very close to Neyman’s model for k = 5. In general, as is expected, the relative precisions of our estimator based on all optimal weights are higher than the relative precisions of our estimator based on two and one optimal weight(s). The gain in relative precisions is, however, marginal.

We studied the performance of our proposed estimators for increasing skewness of a family of lognormal distributions. The relative precision of our estimator based on one optimal weight is higher than those of Neyman’s estimator when the shape parameter exceeds 1.9. The relative precisions of our estimator based on two and k optimal weights are uniformly higher than those of Neyman’s estimator for all values of the shape parameter considered in Table 7. From Figure 1, we see that with the increasing values of skewness, the rate of increase of relative precisions of our proposed estimators based on WRSS is higher than that of the estimator based on Neyman’s method.

Based on the numerical computations of relative precisions, we recommend our estimator based on WRSS procedures for estimating the population mean of skewed distributions with a heavy right tail for small values of set sizes.

Although there are studies on the allocation models of RSS for improving the precision of estimators, implementing them in actual practice is difficult and costly. The main objectives of RSS are avoiding actual measurements along with achieving higher precision. Increasing the allocations requires more actual measurements. Instead of allocating order statistics multiple times, assigning appropriate weights to each order statistic is rather easy. Further, in most of the cases, weighted RSS has higher precision than the optimal allocation model, i.e., Neyman’s method. In addition to this, there are some limitations of the weighted RSS method. This method does not work well if the distribution is symmetric in nature. For symmetric distributions, the precision is not higher than that of the optimal allocation models. For skewed distributions, if the nature of distribution is difficult to identify, the process of finding the optimal weight is difficult. In such cases, the approximation of the distribution function and therefore of the weights is one of the solutions.