Bayesian Inference For The Mean of a Stratified Population When There Are Order Restrictions

Abstract

Consider a stratified random sample of firms with the strata defined by a firm’s number of employees. We wish to make inference about the mean sales and receipts (SR) for all such firms, and the proportion of firms belonging to each of several classes defined by a firm’s SR. Let P _i. denote the known proportion of firms belonging to stratum i, π _ij the proportion of firms in i belonging to SR class j, and Y _j the mean SR in class j. We use Bayesian methods to make inference about $P._j = \sum\limits_i {P_{i.} \pi _{ij} }$ and $\mu = \sum\limits_j {Y_j P._j }$. In particular, specifications of smoothness, expressed as unimodal order relations among the π _ij (within and between the rows of the two-way table), are incorporated into the prior distributions. With computations facilitated by using the Gibbs sampler, we show that the smoothness conditions provide substantial gains in precision when the P _.j and µ are estimated.