Estimating the First Selected PSU Mean in a Two Stage Cluster Sample with Unequal Cluster Sizes based on an Expanded PSU Model

Scott and Smith (1969) develop estimators for linear functions from a two stage cluster sample, with their discussion repeated in many places. This discussion was reviewed in c01ed13.doc. It is note-worthy that Scott and Smith allow expressions for the variance to depend on the cluster. A similar development is given by Vallient et al.. We repeated the derivation of Vallient et al using the finite population variance in c01ed15.doc. These results were extended to settings with response error in c01ed25.doc. Both derivations are based on a ‘collapsed’ version of a vector of random variables, similar to the vector of random variables usually considered. In the absence of response error, we developed results like those in c01ed15.doc in the context of an expanded PSU model with equal cluster sizes in c01ed27.doc. The results in c01ed27.doc have equal cluster sizes in the population, and produce the same estimates as the results in c01ed15.doc. However, new ideas are introduced in c01ed27.doc that we anticipate make it possible to expand the development to unequal size PSUs. The new ideas involve somewhat ‘arbitrary’ projections of terms in estimating equations to make the resulting equations of full rank and invertable. The projections are selected to make the estimating equations reduce to the estimating equations in c01ed15.doc. The purpose of this document is to extend the results in c01ed27.doc to unequal size PSUs. The documents c01ed15.doc and c01ed27.doc assume that the number of SSUs in a PSU is equal for all PSUs. In an additional document, the expected value and variance is developed for a two stage problem with unequal size PSUs with response error (see c01ed26.doc). However, the document c01ed26.doc stops short of estimation. In order to clearly construct the estimating equations, the un-balance in PSU sizes has to be accounted for. It seems that this accounting will be more straightforward in an expanded population framework. With this in mind, we develop an estimator for a cluster mean under two stage sampling with unequal size PSUs in an expanded population framework. We consider here the simpler setting where we have no response error. We expand the representation of the PSUs, but not of the SSUs. The basic problem that we consider is estimation of a linear combination of population values based on the observed response for subjects and SSUs selected via two stage cluster sampling. The problem can be summarized with a model for the vector of random variables that represents a random permutation of the population. Detailed development and notation is given elsewhere, in particular in documents c00ed57.doc, c01ed15.doc, and c01ed27.doc.