Approximations for Densities of Sufficient Estimators

Introduction

Durbin (1980a) proposed a simple method for obtaining asymptotic expansions for the densities of sufficient estimators. The expansion is a series which is effectively in powers of n ^{− 1}, where n is the sample size, as compare with the Edgeworth expansion which is in powers of n ^{− 1 ∕ 2}. The basic approximation is just the first term of this series. This has an error of order n ^{− 1} compare to the error of n ^{− 1 ∕ 2} in the usual asymptotic normal approximation (see Asymptotic Normality). The order of magnitude of the error can generally be reduced to order n ^{− 3 ∕ 2} by renormalization.

Suppose that the real m-dimensional random vector S _n = (S _1n , S _2n , …, S _mn )^′ has a density with respect to Lebesgue measure which depends on integer n > N for some positive N and on θ ∈ Θ, where Θ is a subset of ℝ^q for q an arbitrary positive integer.

Let

$${ \mathbf{D}}_{n}(\theta ) = {n}^{-1}E\left \{{\mathbf{S}}_{ n} - E({\mathbf{S}}_{n})\right \}{\left \{{\mathbf{S}}_{n} -...