A Bayes Sampling Allocation Scheme for Stratified Finite Populations With Hyperbinomial Prior Distributions

When sampling is carried out independently for the K strata of a finite stratified dichotomous population (defectives vs. standard items), and the number Z_i of defectives per stratum sample is observed, the corresponding probability function for X = (X_i , …, x_K ) is the product of hypergeometric functions which depend on the sample sizes n_i , the stratum sizes N_i , and the number of defectives M_i in the stratum (i = 1, …, K). It is assumed that prior information is available about the M_i 's which can be expressed, by suitable choice of the parameters a_i and b_i , as the product of independent hyperbinomial functions.

In each stratum the cost per observation is a known constant. Using squared error loss function, the prior Bayes risk is found for the linear function of interest,

and the optimum allocation of sample sizes is found, the one for which the prior Bayes risk is minimum when the total sampling budget is fixed.