Estimation of scale parameter of a Bivariate Lomax distribution by Ranked Set Sampling

Abstract

Concept of Ranked Set Sampling is applicable whenever ranking on a set of sampling units can be done easily by a judgment method or based on an auxiliary variable. In this work, we consider a study variable $Y$ correlated with auxiliary variable $X$ which is used to rank the sampling units. Further $(X,Y)$ is assumed to have bivariate Lomax distribution. We obtain an unbiased estimator of scale parameter of study variable $Y$ based on ranked set sample and censored ranked set sample. Efficiency comparison of these estimators with usual SRS estimator is also performed.

Keywords

Ranked set sampling censored Ranked Set Sampling bivariate Lomax distribution concomitants of order statistics Best linear unbiased estimator

1. Introduction

The concept of Ranked Set Sampling (RSS) was first introduced by McIntyre (1952) for improving the precision of the sample mean, an estimator of population mean. This concept is applicable whenever variable of interest is difficult or expensive to measure but ranking on a small set of measurement is easily available. McIntyre (1952) uses judgement method for ranking a set of sample units. For a detailed discussion on the theory and applications of RSS see, Chen et al. (2004). Judgement method is not suitable when there is ambiguity in discriminating the rank of one unit with another. Judgement method is not mathematically much relevant with the problem of study. To remedy this problem Stokes (1977) uses an auxiliary variable for ranking of the sampling units. Stokes (1977) considers a situation where the variable of interest, say $Y$ is difficult or expensive to measure, but an auxiliary variable $X$ correlated with $Y$ is easily measurable and can be ordered exactly. The procedure of RSS using auxiliary variable described by Stokes (1977) is as follows.

Choose $n$ independent bivariate samples each of size $n$ . Arrange them randomly into $n$ samples each with $n$ units and observe the value of the auxiliary variable $X$ on each of these units. For the first sample, select that unit for which the measurement on the auxiliary variable is the smallest and measure the $Y$ variate associated with smallest $X$ variate. In the second sample, choose $Y$ associated to the second smallest $X$ . This procedure is repeated until the last sample associated with the largest $X$ is measured. The resulting set of $n$ units chosen by one from each $X$ sample as described above is called RSS. Let $(X_{(r)r},Y_{[r]r})r=1,2,\ldots,n$ be the bivariate random sample selected from the $r^{\rm th}$ sample, where $X_{(r)r}$ denote the $r^{\rm th}$ order statistic of the auxiliary variable in the $r^{\rm th}$ sample and $Y_{[r]r}$ denote the measurement made on the $Y$ variate associated with $X_{(r)r}$ . David and Nagaraja (2003) referred $Y_{[r]r}$ as the concomitant of the $r^{\rm th}$ order statistic arising from the $r^{\rm th}$ sample.

Bain (1978) discusses a striking example of RSS where a study variate $Y$ represents the oil pollution of sea water and the auxiliary variable $X$ represents the tar deposit in the nearby sea shore. Clearly collecting sea water sample and measuring the oil pollution in it is strenuous and expensive. However the prevalence of pollution in the sea water is much reflected by the tar deposit in the surrounding terminal sea shore. In this example ranking the pollution level of sea water based on the tar deposit on the sea shore is more natural and scientific than ranking it visually or by judgement method.

Stokes (1977) proposes RSS mean as an estimator for the mean of the study variate, when an auxiliary variable $X$ is used for ranking the sample units, under the assumption that $(X,Y)$ follow a bivariate normal distribution. While Barnett and Moore (1997) obtained the Best Linear Unbiased Estimator (BLUE) of the mean of the study variate $Y$ based on a RSS when $\left(X,Y\right)$ follow bivariate Normal distribution. Stokes (1995) studies the estimation of parameters in location-scale family of distributions using RSS. Lam et al. (1994, 1995) have obtained the BLUE of location and scale parameter of exponential and Logistic distribution. Chacko and Thomos (2007) obtained the BLUE of the parameter involved in the study variate $Y$ based on a RSS when $\left(X,Y\right)$ follows bivariate Pareto distribution. Chacko and Thomos (2007) estimated the parameters of Morgenstern type bivariate exponential distribution by RSS and censored RSS. Singh and Mehta (2014) derived linear shrinkage estimator of scale parameter of Morgenstern type bivariate logistic distribution using RSS. Singh and Mehta (2015) considered estimation of scale parameter of a Morgenstern type bivariate uniform distribution by censored RSS. Attia et al. (2014a) estimated the parameters of the bivariate Lomax distribution of Marshall-Olkin based on right censored sample using expectation-maximization (EM) algorithm. Further Attia et al. (2014b) considered generalized bivariate Lomax distribution and maximum likelihood estimation procedure is derived for the estimation its parameters based on censored samples.

The above overview highlights the role of Ranked Set Sampling scheme in estimation of parameters associated with a study variable correlated with auxiliary variable and having different bivariate distributions. In this paper, our main objective is to estimate the parameter associated with $Y$ (study variable) which is correlated with $X$ (auxiliary variable) by RSS and censored RSS scheme where $(X,Y)$ are assumed to have bivariate Lomax distribution. The paper is organized as follows.

In Section 2, we obtain an unbiased estimator of scale parameter of bivariate Lomax distribution based on RSS when shape parameter is known. While in Section 3, we derive its BLUE based on RSS. In Section 4, we obtain censored RSS estimator of scale parameter. We perform efficiency comparison for obtained RSS estimators with that of estimator based on simple ransom sample (SRS). Section 5, concludes the paper with final remarks.

2. Estimator of scale parameter based on RSS

Consider the bivariate Lomax distribution (Sankaran and Nair (1993)) with parameters $\left(a,b,p\right)$ and probability density function (pdf) given by,

$f(x,y)=\frac{p(p+1)}{{ab\left(1+\frac{x}{a}+\frac{y}{b}\right)}^{p+2}};∼{}x,y>% 0,∼{}p,a,b>0.$ (1)

Each marginal of the above bivariate Lomax distribution $\left(BL\left(a,b,p\right)\right)$ , is univariate Lomax (Lomax (1954)). This distribution is also known as Pareto distribution of type II. Particularly, marginal distribution of $Y$ is $L\left(b,p\right)$ , a univariate Lomax distribution and is given by,

$f_{2}\left(y\right)=\frac{p}{b\left(1+\frac{y}{b}\right)^{p+1}};∼{}y>0,∼{}b,p>0.$

The mean and variance of $Y$ is given by,

$E(Y)=\frac{b}{p-1};∼{}p>1,\quad Var(Y)=\frac{b^{2}p}{(p-1)^{2}(p-2)};∼{}p>2.$

Note that the distribution of $Y$ is free of scale parameter $a$ . Our objective is to propose different RSS estimators for scale parameter $b$ when shape parameter $p$ is known.

Suppose $n$ random samples each of size $n$ are drawn from $BL\left(a,b,p\right)$ . Let $X_{(r)r}$ be the $r^{\rm th}$ order statistic of the auxiliary variable $X$ in the $r^{\rm th}$ sample and $Y_{[r]r}$ be the measurement on the $Y$ variate associated with $X_{(r)r},r=1,2,\ldots,n$ . Here $Y_{[r]r}$ is referred as the concomitant of $r^{\rm th}$ order statistics, $1\leqslant r\leqslant n$ and $Y_{\left[1\right]1},Y_{\left[2\right]2},\ldots,Y_{\left[n\right]n}$ is referred as RSS from $BL\left(a,b,p\right)$ distribution. The expressions for mean and variance of $Y_{[r]r}$ given by Begum (2003) are as follows.

$E\left(Y_{[r]r}\right)=b\xi_{r},\quad Var\left(Y_{[r]r}\right)=b^{2}\delta_{r},$

where

$\displaystyle\xi_{r}=\frac{\left\lceil(n+1)\right.}{p\left\lceil(n+1-r)\right.% }\frac{\left\lceil\left(n+1-\frac{1}{p}-r\right)\right.}{\left\lceil\left(n+1-% \frac{1}{p}\right)\right.},∼{}∼{}p>\left(n-r+1\right)^{-1},$ (2) $\displaystyle\delta_{r}=\frac{\left\lceil(n+1)\right.}{p\left\lceil(n+1-r)% \right.}\left\{\frac{2}{(p-1)}\frac{\left\lceil\left(n+1-\frac{2}{p}-r\right)% \right.}{\left\lceil\left(n+1-\frac{2}{p}\right)\right.}-\frac{1}{p}{\frac{% \left\lceil(n+1)\right.}{\lceil\left(n-r+1\right)}\left(\frac{\left\lceil\left% (n+1-\frac{1}{p}-r\right)\right.}{\left\lceil\left(n+1-\frac{1}{p}\right)% \right.}\right)}^{2}\right\},$ $\displaystyle\qquad∼{}p>2(n-r+1)^{-1}.$ (3)

Note that $\delta_{r}>\xi_{r}^{2}$ since $\delta_{r}=\frac{2\left\lceil(n+1)\right.}{p(p-1)\left\lceil(n+1-r)\right.}% \frac{\left\lceil\left(n+1-\frac{2}{p}-r\right)\right.}{\left\lceil\left(n+1-% \frac{2}{p}\right)\right.}-\xi_{r}^{2}$ and $\xi_{r},\delta_{r}$ are non-negative for all $r=1,2,\ldots n$ .

Observe that $cov\left(Y_{[r]r},Y_{\left[s\right]s}\right)=0$ , for $r\neq s$ , since $Y_{[r]r}$ and $Y_{\left[s\right]s}$ are drawn from two independent sets of samples. Some other expressions for the concomitant of $r^{\rm th}$ order statistic for the $BL\left(a,b,p\right)$ distribution are given by Begum (2003) and Nayabuddin (2013).

Based on RSS $Y_{\left[1\right]1},Y_{\left[2\right]2},\ldots,Y_{\left[n\right]n}$ from $BL\left(a,b,p\right)$ distribution, we propose an unbiased estimator of scale parameter $b$ under the assumption that the shape parameter $p$ is known. The estimator $\hat{b}_{\textit{RSS}}$ is defined as,

$\hat{b}_{\textit{RSS}}=\frac{1}{n}\sum\nolimits_{r=1}^{n}\frac{Y_{[r]r}}{\xi_{% r}}.$

Clearly $\hat{b}_{\textit{RSS}}$ is an unbiased estimator of $b$ . Variance of $\hat{b}_{\textit{RSS}}$ is,

$Var\left(\hat{b}_{\textit{RSS}}\right)=\frac{b^{2}}{n^{2}}\sum\nolimits_{r=1}^% {n}{\delta_{r}/}\xi_{r}^{2}$

where $\xi_{r}$ and $\delta_{r}$ are as defined in Eqs 2 and 3 respectively.

The unbiased estimator $\hat{b}_{\textit{SRS}}$ of $b$ based on a SRS $y_{1},y_{2},\ldots,y_{n}$ from $L\left(b,p\right)$ is given by,

$\hat{b}_{\textit{SRS}}=(p-1)\overline{y},$

where $\overline{y}=\frac{\sum_{r=1}^{n}y_{r}}{n}$ .

The variance of $\hat{b}_{\textit{SRS}}$ is given by,

$Var\left(\hat{b}_{\textit{SRS}}\right)=\frac{b^{2}p}{n(p-2)};∼{}p>2.$

The Efficiency of $\hat{b}_{\textit{RSS}}$ over $\hat{b}_{\textit{SRS}}$ is,

$E\left(\hat{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{SRS}}\right% )=\frac{np}{(p-2)\sum_{r=1}^{n}{\delta_{r}/}\xi_{r}^{2}}.$ (4)

Observe that $\sum_{r=1}^{n}{\delta_{r}/\xi_{r}^{2}}>n$ as $\delta{}_{r}>\xi_{r}^{2}$ . Hence trivially $E(\hat{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{SRS}})>1.$ Thus $\hat{b}_{\textit{RSS}}$ is efficient than $\hat{b}_{\textit{SRS}}$ i.e. an estimator of $b$ based on RSS is more efficient than that of estimator based on SRS. This can be seen through a numerical study presented in Table 1 where we evaluate $E(\hat{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{SRS}})$ for different values of $n$ and $p>2.$ Figure 1 shows increase in $E(\hat{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{SRS}})$ with respect to $n$ for fixed $p$ . This supports to the conclusion that RSS always provides more information than SRS even if the ranking is imperfect (Chen et al. (2004)).

Table 1

Efficiency of $\hat{b}_{\textit{RSS}}$ over $\hat{b}_{\textit{SRS}}$ and $\tilde{b}_{\textit{RSS}}$ over $\hat{b}_{\textit{RSS}}$

$n$	$p$	$E(\hat{b}_{\textit{RSS}}\|\hat{b}_{\textit{SRS}})$	$E(\tilde{b}_{\textit{RSS}}\|\hat{b}_{\textit{RSS}})$	$n$	$p$	$E(\hat{b}_{\textit{RSS}}\|\hat{b}_{\textit{SRS}})$	$E(\tilde{b}_{\textit{RSS}}\|\hat{b}_{\textit{RSS}})$
2	2.1	1.5782	2.3434	12	2.1	4.3010	1.5219
	2.5	1.2570	1.1466		2.5	1.8439	1.0643
	2.9	1.1498	1.0500		2.9	1.4382	1.0238
	3.3	1.0991	1.0229		3.3	1.2782	1.0115
4	2.1	2.4603	2.1435	14	2.1	4.5630	1.4579
	2.5	1.5202	1.1328		2.5	1.8767	1.0568
	2.9	1.2866	1.0471		2.9	1.4528	1.0211
	3.3	1.1856	1.0222		3.3	1.2870	1.0103
6	2.1	3.1038	1.8918	16	2.1	4.7837	1.4078
	2.5	1.6573	1.1063		2.5	1.9027	1.0508
	2.9	1.3527	1.0384		2.9	1.4643	1.0189
	3.3	1.2263	1.0184		3.3	1.2940	1.0092
8	2.1	3.5958	1.7231	18	2.1	4.9725	1.3676
	2.5	1.7425	1.0876		2.5	1.9238	1.0460
	2.9	1.3923	1.0320		2.9	1.4737	1.0172
	3.3	1.2505	1.0154		3.3	1.2996	1.0084
10	2.1	3.9850	1.6065	20	2.1	5.1357	1.3346
	2.5	1.8010	1.0742		2.5	1.9414	1.0420
	2.9	1.4189	1.0273		2.9	1.4814	1.0157
	3.3	1.2666	1.0132		3.3	1.3042	1.0077

Figure 1.

$E(\hat{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{SRS}})$ across $n$ when $p=2.1$ .

3. BLUE of scale parameter based on RSS

In this section we obtain BLUE $\tilde{b}_{\textit{RSS}}$ of $b$ based on RSS under the assumption that $p$ is known. Let $Y_{[r]r}$ be the measurement made on the $Y$ variate corresponding to $X_{(r)r}$ the observation on the auxiliary variate $X$ in the $r^{\rm th}$ unit of the RSS. Thus we have RSS $Y_{\left[1\right]1},Y_{\left[2\right]2},\ldots,Y_{\left[n\right]n}$ from $BL\left(a,b,p\right)$ distribution. Let ${\bm{Y}}_{{\bm{[}n]}}=(Y_{\left[1\right]1}∼{}$ $∼{}Y_{\left[2\right]2}∼{}$ $∼{}\ldots∼{}$ $∼{}Y_{\left[n\right]n})^{\prime}$ . Then the mean vector and the dispersion matrix of ${\bm{Y}}_{{[\bm{n}]}}$ is given by,

$\displaystyle E\left({\bm{Y}}_{\left[{\bm{n}}\right]}\right)=b{\bm{\xi}},$ $\displaystyle D\left({\bm{Y}}_{\left[{\bm{n}}\right]}\right)=b^{2}G,$

where ${\bm{\xi}}=(\xi_{1}∼{}$ $∼{}\xi_{2}∼{}$ $∼{}\ldots∼{}$ $∼{}\xi_{n})^{\prime}$ , $G=diag\left(\delta_{1},\delta_{2},\ldots,\delta_{n}\right)$ and $\xi_{r}$ and $\delta_{r}$ are as defined in Eqs 2 and 3 respectively.

The BLUE $\tilde{b}_{\textit{RSS}}$ of $b$ based on RSS obtained by using theory of generalized linear model for $E({\bm{Y}}_{[{\bm{n}}]})=b{\bm{\xi}}$ with $D({\bm{Y}}_{[{\bm{n}}]})=b^{2}G$ (David and Nagaraja (2003)) is given by,

$\tilde{b}_{\textit{RSS}}=\left({\bm{\xi}}^{\prime}G^{-1}{\bm{\xi}}\right)^{-1}% {\bm{\xi}}^{\prime}G^{-1}{\bm{Y}}_{\left[{\bm{n}}\right]}.$ (5)

The variance of $\tilde{b}_{\textit{RSS}}$ is given by,

$Var(\tilde{b}_{\textit{RSS}})=\left({\bm{\xi}}^{\prime}G^{-1}{\bm{\xi}}\right)% ^{-1}b^{2}.$ (6)

Simplifying Eqs 5 and 6 we get,

$\tilde{b}_{\textit{RSS}}=\sum\nolimits_{r=1}^{n}a_{r}Y_{[r]r},\quad Var(\tilde% {b}_{\textit{RSS}})=\frac{b^{2}}{\sum_{r=1}^{n}\frac{\xi_{r}^{2}}{\delta_{r}}},$

where $a_{r}=\frac{\frac{\xi_{r}}{\delta_{r}}}{\sum_{r=1}^{n}\frac{\xi_{r}^{2}}{% \delta_{r}}}$ .

The Efficiency of $\tilde{b}_{\textit{RSS}}$ over $\hat{b}_{\textit{RSS}}$ is,

$E(\tilde{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{RSS}})=\frac{1% }{n^{2}}\left[\sum\nolimits_{r=1}^{n}{\delta_{r}/\xi_{r}^{2}}\right]\left[\sum% \nolimits_{r=1}^{n}{\xi_{r}^{2}/\delta_{r}}\right].$ (7)

As $\sum_{r=1}^{n}{\delta_{r}/}\xi_{r}^{2}>n$ , $E(\tilde{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{RSS}})% \geqslant 1.$ Thus $\tilde{b}_{\textit{RSS}}$ is more efficient than $\hat{b}_{\textit{RSS}}$ . Further it can be seen that,

$\lim_{n\to\infty}{E(\tilde{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{% \textit{RSS}})}=1.$

The $E(\tilde{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{RSS}})$ for different values of $n$ and $p>2$ are reported in Table 1. Observe that $E(\tilde{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{RSS}})$ approaches to 1 for higher values of $n(p)$ when $p$ ( $n$ ) is fixed. This is presented in Figs 2 and 3.

Figure 2.

$E(\tilde{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{RSS}})$ across $n$ when $p=2.5$ .

Figure 3.

$E(\tilde{b}_{\textit{RSS}}\thinspace|\thinspace\hat{b}_{\textit{RSS}})$ across $p$ when $n=10$ .

Remark: Observe that for obtaining all estimators $\hat{b}_{\textit{SRS}},\hat{b}_{\textit{RSS}},\tilde{b}_{\textit{RSS}}$ of $b$ , $p$ is assumed to be known. In usual cases $p$ is not known. In this situation we suggest to obtain a rough estimator $\hat{p}$ of $p$ using correlation coefficient. For $(X,Y)\sim BL\left(a,b,p\right)$ the correlation coefficient between $X$ and $Y$ is given by $\rho=1/p$ for $p>2.$ If $r$ is the sample correlation coefficient between $X_{(i)i}$ and $Y_{[i]i},i=1,2,\ldots,n$ then moment estimator for $p$ is given by $\hat{p}=1/r$ The estimate $\hat{p}$ will lead to feasible value of $p$ when $0<r<1/2$ . Specifically, when $r<0,\hat{p}$ is negative and for $r\geqslant 1/2,\hat{p}<2$ which leads to non-existence of $Var(Y)$ . Other possible choice for $\hat{p}$ can be according to Begum (2003) who proposed an estimator of $p$ based on Concomitants of order statistics. There can be other choices for estimator of $p$ .

4. Estimator of scale parameter based on censored RSS

Consider an example of pollution study on sea samples discussed by Bain (1978). If there is no tar deposit at the seashore, then the corresponding sea sample will be treated as censored as these units can not be measured. For ranking on $X$ observations in a sample, the censored units are assumed to have distinct and consecutive lower ranks and the remaining units are ranked with the next higher ranks in a natural order. If in this censored scheme of RSS, $k$ units are censored, then we may represent the RSS observations on the study variate $Y$ in RSS as $\rho_{1}Y_{[1]1},\rho_{2}Y_{[2]2},\ldots,\rho_{n}Y_{[n]n}$ where

$\rho_{i}=\begin{cases}0&\text{if the∼{}}i^{\rm th}\text{∼{}unit is censored}\\ 1&\text{otherwise}\\ \end{cases}$

such that $\sum_{i=1}^{n}{\rho_{i}=n-k}$ .

In this case the usual censored RSS mean is,

$\bar{y}_{\textit{RSS}}^{c}=\frac{\sum_{i=1}^{n}{\rho_{i}Y_{[i]i}}}{n-k}.$

It may be noted that $\rho_{i}=0$ need not occur in a natural order for $i=1,2,\ldots,n$ . Let the integers $m_{1},m_{2},\ldots,m_{n-k}$ be such that $1\leqslant m_{1}<m_{2}<\ldots<m_{n-k}\leqslant n$ and $\rho_{m_{i}}=1$ . Then,

$E\left[\bar{y}_{\textit{RSS}}^{c}\right]=\frac{b}{n-k}\sum\nolimits_{i=1}^{n-k% }\xi_{m_{i}},$

where $\xi_{m_{i}}=\frac{\left\lceil(n+1)\right.}{p\left\lceil(n+1-m_{i})\right.}% \frac{\left\lceil\left(n+1-\frac{1}{p}-m_{i}\right)\right.}{\left\lceil\left(n% +1-\frac{1}{p}\right)\right.},i=1,2,\ldots,n-k.$

Therefore, the RSS mean in censored case is not an unbiased estimator for $b$ . However we can construct an unbiased estimator based on $\bar{y}_{\textit{RSS}}^{c}$ .

Let $Y_{[m_{i}]m_{i}},i=1,2,\ldots,n-k$ be the censored RSS observations on the study variate $Y$ when the ranking is applied to the auxiliary variate $X$ from $BL\left(a,b,p\right)$ . When $p$ is known, an unbiased estimator $\hat{b}_{\textit{RSS}}^{c}(k)$ of $b$ based on censored RSS of size $k$ is given by,

$\hat{b}_{\textit{RSS}}^{c}(k)=\frac{\left(n-k\right)\bar{y}_{\textit{RSS}}^{c}% }{\sum\nolimits_{i=1}^{n-k}\xi_{m_{i}}},$

and $Var\left(\hat{b}_{\textit{RSS}}^{c}(k)\right)=b^{2}\frac{\sum_{i=1}^{n-k}% \delta_{m_{i}}}{\left(\sum\nolimits_{i=1}^{n-k}\xi_{m_{i}}\right)^{2}},$ where $\xi_{m_{i}}=\frac{\lceil(n+1)}{p\lceil(n+1-m_{i})}\frac{\left\lceil\left(n+1-% \frac{1}{p}-m_{i}\right)\right.}{\left\lceil\left(n+1-\frac{1}{p}\right)\right% .},p>\left(n-m_{i}+1\right)^{-1}$ and $\delta_{m_{i}}=\frac{\left\lceil(n+1)\right.}{p\left\lceil(n+1-m_{i})\right.}% \left\{\frac{2}{(p-1)}\frac{\left\lceil\left(n+1-\frac{2}{p}-m_{i}\right)% \right.}{\left\lceil\left(n+1-\frac{2}{p}\right)\right.}-\frac{1}{p}{\frac{% \left\lceil(n+1)\right.}{\left\lceil(n-m_{i}+1)\right.}\left(\frac{\left\lceil% \left(n+1-\frac{1}{p}-m_{i}\right)\right.}{\left\lceil\left(n+1-\frac{1}{p}% \right)\right.}\right)}^{2}\right\},$ $p>2(n-m_{i}+1)^{-1},i=1,2,\ldots,n-k$ .

We now propose the BLUE of $b$ based on the censored RSS, resulting out of ranking of observations on $X$ . Let ${\bm{Y}}_{\left[{\bm{n}}\right]}(k)=(Y_{\left[m_{1}\right]m_{1}}∼{}∼{}Y_{\left% [m_{2}\right]m_{2}}∼{}∼{}\ldots∼{}∼{}Y_{\left[m_{n-k}\right]m_{n-k}})^{\prime}$ be the vector of observations in censored RSS. The mean vector and the dispersion matrix of ${\bm{Y}}_{\left[{\bm{n}}\right]}(k)$ are given by,

$\displaystyle E\left({\bm{Y}}_{\left[{\bm{n}}\right]}(k)\right)=b{\bm{\xi}}% \left({\bm{k}}\right),$ (8) $\displaystyle D\left({\bm{Y}}_{\left[{\bm{n}}\right]}(k)\right)=b^{2}G(k),$ (9)

where ${\bm{\xi}}\left({\bm{k}}\right)=(\xi_{m_{1}}∼{}∼{}\xi_{m_{2}}∼{}∼{}\ldots∼{}∼{% }\xi_{m_{n-k}})^{\prime}$ and $G(k)=diag(\delta_{m_{1}},\delta_{m_{2}},\ldots,\delta_{m_{n-k}})$ .

If $p$ involved in ${\bm{\xi}}\left({\bm{k}}\right)$ and $G(k)$ is known, then Eqs 8 and 9 defines a generalized Gauss-Markov set up and hence the BLUE $\tilde{b}_{\textit{RSS}}^{c}(k)$ of $b$ is obtained as,

$\tilde{b}_{\textit{RSS}}^{c}(k)=(({\bm{\xi}}({\bm{k}}))^{\prime}(G(k))^{-1}({% \bm{\xi}}({\bm{k}})))^{-1}({\bm{\xi}}({\bm{k}}))^{\prime}(G(k))^{-1}{\bm{Y}}_{% [{\bm{n}}]}(k)$ (10)

and

$Var(\tilde{b}_{\textit{RSS}}^{c}(k))=(({\bm{\xi}}({\bm{k}}))^{\prime}(G(k))^{-% 1}({\bm{\xi}}({\bm{k}})))^{-1}b^{2}.$ (11)

Simplifying the expression from Eqs 10 and 11, we get,

$\tilde{b}_{\textit{RSS}}^{c}(k)=\sum\nolimits_{i=1}^{n-k}a_{m_{i}}Y_{\left[m_{% i}\right]m_{i}},\quad Var(\tilde{b}_{\textit{RSS}}^{c}(k))=\frac{b^{2}}{\sum% \nolimits_{i=1}^{n-k}{\xi_{m_{i}}^{2}/}\delta_{m_{i}}},$

where $a_{m_{i}}=\frac{\xi_{m_{i}}/\delta_{m_{i}}}{\sum\nolimits_{i=1}^{n-k}{\xi_{m_{% i}}^{2}/}\delta_{m_{i}}}.$

5. Conclusions

In this paper, we consider a bivariate Lomax distribution and obtain an unbiased estimator and BLUE of a scale parameter associated with a study variable based on RSS. Further, BLUE of the scale parameter in case of censored RSS is also obtained. We perform efficiency comparison of proposed RSS estimators with that of estimator based on SRS. The efficiency performance of proposed estimators is studied. It is observed that, an estimator based on RSS is more efficient than that of estimator based on SRS. Moreover the efficiency based on RSS estimator increases with $n$ for fixed $p$ over SRS estimator.

Footnotes

Acknowledgments

The authors would like to thank the reviewers for their constructive comments which helped to improve the paper.

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