J. R. Statist. Soc. B (1994) 56, No.2, pp. 313-326 Empirical Saddlepoint Approximations for Multivariate M-estimators By ELVEZIO RONCHETTI University of Geneva, Switzerland and A. H. WELSHt Australian National University, Canberra, Australia [Received October 1991. Revised November 1992) SUMMARY In this paper. we investigate the use of the empirical distribution function in place of the underlying distribution function F to construct an empirical saddlepoint approximation to the density In of a general multivariate M-estimator. We obtain an explicit form for the error term in the approximation, investigate the effect of renormalizing the estimator, carry out some numerical comparisons and discuss the regression problem. Keywords: BOOTSTRAP; EMPIRICAL DISTRIBUTION FUNCTION; REGRESSION PROBLEM; RENORMALIZATION; SMALL SAMPLE ASYMPTOTIC APPROXIMATION 1. INTRODUCTION The sampling distribution of a statistic is a basic tool for evaluating the properties of the statistic and for constructing inference procedures based on the statistic. The density of the sampling distribution may be used to compute moments, to develop approximations or to compute the distribution function and hence various tail prob- abilities which can be used to construct tests or confidence intervals. When the exact density is intractable, it may be possible to approximate it by a normal density, an Edgeworth expansion or a saddlepoint approximation (Daniels, 1954). The saddle- point approximation is derived from an asymptotic expansion but is often very accurate even in quite small samples. To highlight this property, Hampel (1973) described these and related techniques as small sample asymptotic approximations. As can be seen from equation (3) later, the main property of this approximation and its major advantage over Edgeworth expansions is that it is always non-negative and the relative error is uniformly of order n -1. For recent reviews see Reid (1988), Barndorff-Nielsen and Cox (1989) and Field and Ronchetti (1990). The construction of these approximations typically requires knowledge of the underlying distribution. However, just as we can estimate the variance of a normal approxima- tion or estimate the terms in an Edgeworth expansion, we can estimate the saddle- point approximation. In this paper, we examine the properties of an estimate of the saddlepoint approximation. Our investigation is motivated by three considerations. Firstly, the estimated approximation may be useful for comparing various estimators on a given data set. If the sampling distribution is non-normal, at least in principle, we need to compare the whole distribution and comparisons based on the estimated asymptotic variance alone may be misleading. Secondly, the estimated approximation may be useful for evaluating the quality of simpler approximations including the normal approxima- tion. Finally, it may be useful for developing inference procedures. Methods like tAddressfor correspondence: Department of Statistics, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia. © 1994 Royal Statistical Society 0035-9246/94/56313