Balanced Ranked Set Sampling II: Parametric

Abstract

We turn to balanced parametric RSS in this chapter. It is assumed that the underlying distribution is known to belong to a certain distribution family up to some unknown parameters. From an information point of view, intuitively, the amount of information contained in a ranked set sample should be larger than that contained in a simple random sample of the same size, since a ranked set sample contains not only the information carried by the measurements but also the information carried by the ranks. We shall deal with the Fisher information of an RSS sample and make this assertion rigorous. In Section 3.1, we consider the Fisher information of a ranked set sample first for the special case of perfect ranking and then for general cases when the assumption of perfect ranking is dropped. In the case of perfect ranking, we derive the result that the information matrix based on a balanced ranked set sample is the sum of the information matrix based on a simple random sample of the same size and a positive definite information gain matrix. In general cases, it is established that the information matrix based on a balanced ranked set sample minus the information matrix based on a simple random sample of the same size is always non-negative definite. It is also established that the positive-definiteness of the difference holds as long as ranking in the RSS is not a purely random permutation. Conditions for the difference of the two information matrices to be of full rank are also given. In Section 3.2, we discuss maximum likelihood estimation (MLE) based on ranked set samples and its relative efficiency with respect to MLE based on simple random samples.