Approximations for densities of sufficient estimators

Abstract

SUMMARY A simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described. It is an extension of the one developed by Barndorff-Nielsen & Cox (1979) for exponential families. A series expansion in powers of n-1 is derived of which the first term has an error of order n-1 which can effectively be reduced to n-3/2 by renormalization. The results obtained are similar to those given by Daniels's (1954) saddlepoint method but the derivations are simpler. A brief treatment of approximations to conditional densities is given. Theorems are proved which extend the validity of the multivariate Edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors. These extensions permit the treatment of problems arising in time series analysis. The technique is used in another paper (Durbin, 1980) to obtain approximations to the densities of partial serial correlation coefficients. Some key word8: Asymptotic expansion; Circular autoregressive process; Conditional density; Edgeworth series; Saddlepoint approximation; Time series. The main purpose of this paper is to describe a simple technique for obtaining an asymptotic expansion for the density of a sufficient estimator. The expansion is a series which is effectively in powers of n-1, where n is sample size, as compared with the Edgeworth expansion which is in powers of n-4. The basic approximation is just the first term of this series. This has an error of order n-1 compared to the error of n-4 in the usual asymptotic normal approximation. The order of magnitude of the error can generally be reduced to order n-3/2 by renormalization. Integration of the basic approximation to obtain probability statements is also considered. The results obtained are similar to those given by Daniels's (1954, 1956) saddlepoint method or by the method of conjugate distributions introduced by Esscher (1932) and used recently by Hampel (1973), in as yet unpublished work by 0. A. Field and F. R. Hampel,