In this chapter, the authors suggest some improved versions of estimators of Morgenstern type bivariate exponential distribution (MTBED) based on the observations made on the units of ranked set sampling (RSS) regarding the study variable Y, which is correlated with the auxiliary variable X, where (X,Y) follows a MTBED. In this chapter, they firstly suggested minimum mean squared error estimator for estimation of 𝜃2 based on censored ranked set sample and their special case; further, they have suggested minimum mean squared error estimator for best linear unbiased estimator of 𝜃2 based on censored ranked set sample and their special cases; they also suggested minimum mean squared error estimator for estimation of 𝜃2 based on unbalanced multistage ranked set sampling and their special cases. Efficiency comparisons are also made in this work.
Top1.1. Introduction
Ranked set sampling (RSS) is a method of sampling that can be advantageous when quantification of all sampling units is costly but a small set of units can be easily ranked, according to the character under investigation, without actual quantification. The technique was first introduced by McIntyre (1952) for estimating means pasture and forage yields. The theory and application of ranked set sampling given by Chen et al. (2004). Suppose the variable of interest say Y, is difficult or much expensive to measure, but an auxiliary variable X correlated with Y is readily measureable and can be ordered exactly. In this case as an alternative to McIntyre (1952) method of ranked set sampling, Stokes (1977) used an auxiliary variable for the ranking of sampling units. If X(r)r is the observation measured on the auxiliary variable X from the unit chosen from the rth set then we write Y[r]r to denote the corresponding measurement made on the study variable Y on this unit, then Y[r]r, r=1,2,…,n, form the ranked set sample. Clearly Y[r]r is the concomitant of the rth order statistic arising from the rth sample.
A striking example for the application of the ranked set sampling as proposed by Stokes (1977) is given in Bain (1978, p. 99), where the 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 in the sea shore is more natural and scientific than ranking it visually or by judgment method.
Stokes (1995) has considered the estimation of parameters of location-scale family of distributions using RSS. Lam et al. (1994, 1995) have obtained the BLUEs of location and scale parameters of exponential distribution and logistics distribution. The Fisher information contained in RSS have been discussed by Chen (2000) and Chen and Bai (2000). Stokes (1980) has considered the method of estimation of correlation coefficient of bivariate normal distribution using RSS. Modarres and Zheng (2004) have considered the problem of estimation of dependence parameter using RSS. Robust estimate of correlation coefficient for bivariate normal distribution have been developed by Zheng and Modarres (2006). Stokes (1977) has suggested the ranked set sample mean as an estimator for the mean of the study variate Y, when an auxiliary variable X is used for ranking the sample units, under the assumption that (X,Y) follows a bivariate normal distribution. Barnett and Moore (1997) have improved the estimator of Stokes (1977) by deriving the Best Linear Unbiased Estimator (BLUE) of the mean of the study variate Y, based on ranked set sample obtained on the study variate Y. For current references in this context the reader is referred to Bouza (2001), Singh and Mehta (2013, 2014a, b, 2015, 2016a, b, c, 2017), Mehta and Singh (2015, 2014), and Mehta (2017, 2018a, b), Deka et. al (2021), Alawady (2021), Scaria and Mohan (2021), Abd Elgawad et. al (2020) and Irshad et. al (2019).