Exact methods for comparison of estimation strategies in survey sampling.

Abstract

Many strategies in survey sampling depend on large sample approximation formulae for design-based inference on finite population parameters, which are not valid for small samples. We develop an approach using matrix algebra to tackle many problems for samples of any size. Poststratification under a general unequal probability sampling design has received little attention and is an area that we will consider. We demonstrate that inference should be made conditional on the observed sampling allocation rather than unconditionally and examine different types of probability weights. For certain strategies we give results that provide sufficient conditions for the superiority of one strategy over another. These methods are based on the exact mean square errors and are used to compare estimators under poststratification both conditionally and unconditionally. We also present a result that gives an exact upper bound on the absolute bias ratio of a strategy which can be used at the design stage to assess the magnitude of the bias. A general problem for unbiased variance estimators under unequal probability sampling is the possibility of obtaining a negative estimate. We show how the eigenvalues of the matrices given by a variance estimator for the ratio estimator under probability proportional to aggregate size sampling can be used to construct a class of nonnegative definite unbiased variance estimators. Our empirical studies show that estimators from this class are generally more efficient than the standard estimator, especially when the coefficient of variation of the size variable is large.