Approximations for densities of sufficient estimators

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–1which can effectively be reduced to n– / 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.