A comparison of the maximum likelihood estimators under ranked set sapling some of its modifications

Abstract

This paper compares the maximum likelihood estimators (MLEs) of the parameters of the location-scale parameter family of distributions, under the methods of ranked set sampling (S.L. Stokes, Annals of the Institute of Statistical Mathematics 47 (3) (1995) 465), median ranked set sampling and extreme ranked set sampling (A.-B Shaibu, H.A. Muttlak, Annals of the Institute of Statistical Mathematics, submitted), and percentile ranked set sampling. Several distributions have been considered and the estimators are compared based on their asymptotic relative precision with respect to the corresponding simple random sample estimators.

Introduction

As a cost effective and efficient alternative to the method of simple random sampling (SRS), [3] intuitively proposed and used the method of ranked set sampling (RSS) in the estimation of population mean. From a population of interest, the method draws n random samples each of size n. The members of each random sample are ranked with respect to the characteristic under study. From the first ordered sample, the smallest unit is chosen for measurement as is the second smallest from the second ordered sample. This continues until the largest element from the last ordered sample is measured. The set of measured elements then constitutes the ranked set sample of size n. This process may be repeated m times (i.e., m cycles) to yield a sample of size nm. Patil et al. [6] classified and extensively reviewed work done in RSS. However, we will briefly review some of the literature relevant to this study.

Takahasi and Wakimoto [12] independently described the same sampling method and presented a sound mathematical argument, which supports McIntyre's intuitive assertion. Dell and Clutter [2] showed that errors in ranking reduce the precision of the RSS mean relative to the SRS mean. However, the RSS mean remains dominant over the SRS mean until ranking is so poor as to yield a random sample when it performs just as well as the SRS mean.

Stokes [10] proposed a RSS estimator of population variance analogous to the SRS unbiased estimator and showed it to be asymptotically unbiased and demonstrated its dominance over the SRS unbiased estimator. Stokes [11] studied the maximum likelihood estimators (MLEs) under RSS of the parameters of the location-scale family having cumulative distribution function (c.d.f.) of the form F(x−μ/σ) with F known. Assuming the usual regularity conditions, Stokes [11] considered several examples and demonstrated the dominance of the MLEs under RSS over other estimators. BLUEs of the location and scale parameters were proposed in the same study and shown to do nearly as well as their maximum likelihood counterparts in most cases. Sinha et al. [9] proposed some best linear unbiased estimators (BLUEs) of the parameters of the normal and exponential distributions under RSS and some modifications of it.

The method of extreme ranked set sampling (ERSS) as studied by Samawi et al. [7] draws n random samples each of size n from a population under consideration. For even set size n, the largest and smallest units are alternately taken from the first to the nth random sample. The resulting sample of n/2 each of first and nth order statistics forms the extreme ranked set sample (ERSS). On the other hand, if n is odd, the largest and smallest units are alternately selected from the first random sample to the (n−1)th random sample. From the nth random sample, either the mean of the largest and smallest unit is chosen or the median of the whole set. In this study, we will consider taking the median from the nth sample. This method may also be repeated m times to give an ERSS of size mn.

Muttlak [4] proposed median ranked set sampling (MRSS), a modification of ranked set sampling, which upon picking and ordering the n random samples selects the median element of each ordered set if n is odd. However, if n is even, it selects the (n/2)th smallest observation from each of the first n/2 ordered sets and the ((n+2)/2)th smallest observation from the second n/2 sets. This selection procedure yield a median ranked set sample (MRSS) of size n, and may be repeated m times to give a MRSS of size mn. Muttlak [4] showed the MRSS mean to be an unbiased estimator when the underlying distribution is symmetric and biased otherwise. In both cases, his estimator has been shown under various distributions, to dominate the RSS sample mean.

Following the methods of Stokes [11], Shaibu and Muttlak [8] considered the maximum likelihood estimators of the parameters of the location-scale parameter family under MRSS and ERSS and compared their results with other methods of estimating the same parameters. They also proposed some unbiased estimators of the location and scale parameters based on MRSS and demonstrated their efficiency with respect to the MLEs under each of MRSS and RSS.

In performing percentile ranked set sampling (PRSS) as proposed by Muttlak [5], n random samples each of size n are drawn from a population under consideration and ranked. If n is odd, then in the first (n−1)/2 sets, the s(n+1)th order statistics (correcting to the nearest integer if necessary) are selected, where 0<s⩽0.5. From the ((n+1)/2)th ranked set, which is the middle set (the median) is selected. The t(n+1)th order statistics are chosen from the remaining (n−1)/2 ranked sets, where t=1−s. However, if n is even, the s(n+1)th order statistics are chosen for measurement from the first (n/2) ordered sets and the t(n+1)th from the rest.

In the next section, maximum likelihood estimation of the location and scale parameters under PRSS is discussed. Section 3 compares the results of the asymptotic relative precision of the maximum likelihood estimators under RSS and the modifications considered. Conclusions of the study are presented in Section 4.