Regression analysis for exponential family data in a finite population setup using two-stage cluster sample

Over the last four decades, the cluster regression analysis in a finite population (FP) setup for an exponential family such as linear or binary data was done by using a two-stage cluster sample chosen from the FP but by treating the sample as though it is a single-stage cluster sample from a super-population (SP) which contains the FP as a hypothetical sample. Because the responses within a cluster in the FP are correlated, the aforementioned sample mis-specification makes the sample-based so-called GLS (generalized least square) estimators design biased and inconsistent. In this paper, we demonstrate for the exponential family data how to avoid the sampling mis-specification and accommodate the cluster correlations to obtain unbiased and consistent estimates for the FP parameters. The asymptotic normality of the regression estimators is also given for the construction of confidence intervals when needed.