Resampling methods for ranked set samples

Abstract

When measuring units are expensive or time consuming, while ranking them can be done easily, it is known that ranked set sampling (RSS) is preferred to simple random sampling (SRS). Available results for RSS are developed under specific parametric assumptions or are asymptotic in nature, with few results available for finite size samples when the underlying distribution of the observed data is unknown. We investigate the use of resampling techniques to draw inferences on population characteristics. To obtain standard error and confidence interval estimates we discuss and compare three methods of resampling a given ranked set sample. Chen et al. (2004. Ranked Set Sampling: Theory and Applications. Springer, New York) suggest a natural method to obtain bootstrap samples from each row of a RSS. We prove that this method is consistent for a location estimator. We propose two other methods that are designed to obtain more stratified resamples from the given sample. Algorithms are provided for these methods. We recommend a method that obtains a bootstrap RSS from the observations. We prove several properties of this method, including consistency for a location parameter. We define two types of L-estimators for RSS and obtain expressions for their exact moments. We discuss an application to obtain confidence intervals for the Winsorized mean of a RSS.

Introduction

Ranked set sampling (RSS) introduced by McIntyre (1952) has many applications in ecological and environmental studies (e.g., Dell and Clutter, 1972, Al-Saleh and Zheng, 2002), reliability theory (Kvam and Samaniego, 1994) and medical studies (Samawi and Al-Sagheer, 2001). RSS is useful when a measurement of interest is difficult or expensive to obtain, but its value can be easily ordered by some means without actual quantification. The RSS procedure is briefly described as follow.

RSS is a two-stage sampling procedure. In the first stage, units are identified and ranked, and in the second stage, measurements are taken from a fraction of the ranked elements. Let ${\mathit{mk}}^2$ units be randomly identified from the population. The units are then randomly divided into k groups of mk units each. In the first group, units are further randomly divided into m subgroups of size k. Units in each of the m subgroups are ordered by any means other than actual quantification, and actual measurement is taken from the unit having the lowest rank within each subgroup. The resulting measurements from the first group are labeled as $\left\{x_{(1)1},\dots ,x_{(1)m}\right\}$. This step is repeated for the second group; but this time, actual measurement is taken from the unit having the second lowest rank within each subgroup. Continuing this procedure until all k groups are processed such that, in the $r$th group, actual measurement is taken from the unit having the $r$th lowest rank within each subgroup, yielding measurements $\left\{x_{(r)1},\dots ,x_{(r)m}\right\}$. The resulting ranked set sample is denoted by $\left\{x_{(r)j};r=1,\dots ,k;j=1,\dots ,m\right\}$. Note that the $n=\mathit{mk}$ resulting measurements are independently distributed. However, they are not identically distributed.

Existing results show that one can often improve the accuracy of the analysis using RSS. The available results for RSS focus on inferences on population characteristics either under specific parametric assumptions or asymptotic results, with few results available for finite size samples when the underlying distribution of the observed data is unknown. For applications with small m, however, asymptotic inference may not be valid and for complex statistics ${\widehat{\theta }}_n$ with sampling distribution $H_{n,F}(t)$, their standard errors may not be known. Bootstrap offers an alternative approach to estimate $H_{n,F}(t)$ by replacing F with its estimate $F_n$, the bootstrap estimate of $H_{n,F}(t)$ is $H_{n,F_n}(t)$.

Bootstrap is a viable approach to obtain the sampling distribution of test statistics in SRS. It is important to study its use with RSS. Several methods of bootstrapping a RSS suggest themselves. What are the algorithms of these methods? Which algorithm has better coverage probability and produces more accurate confidence intervals? We study the use of bootstrap to draw inference under RSS. Chen et al. (2004) introduced the method of bootstrapping a RSS row-wise. Hui et al. (2005) considered bootstrapping as a way to construct confidence interval for estimation of the population mean via linear regression under RSS. To motivate the use of resampling in RSS and to illustrate these resampling methods, we focus our investigation on the trimmed mean as members of a class of L-estimators in RSS. Simulation results are given on the 20% sample trimmed mean for symmetric distributions and 10% one-sided sample trimmed mean for a skewed distribution.

The article is arranged as follow. Section 2 describes and discusses the properties of three methods of bootstrap: BRSSR (bootstrap RSS by row), BRSS (bootstrap RSS), and MRBRSS (mixed row bootstrap RSS). Section 3 describes linear estimators under RSS and presents the results of a simulation study to compare the three methods. An application using these resampling methods are discussed in Section 4. Summary and concluding remarks are given in Section 5.