LARGE DEVIATION AND OTHER RESULTS FOR MINIMUM CONTRAST ESTIMATORS

JENS LEDET JENSEN^{1} AND ANDREW T. A. WOOD^{2}

1Department of Theoretical Statistics, Institute of Mathematics, Aarhus University,

DK-8000 Aarhus C, Denmark 2School of Mathematical Sciences, University of Bath, Bath BA2 7AY, U.K.

(Received January 17, 1997; revised October 21, 1997)

Abstract. A number of authors have been concerned with constructing large deviation approximations to densities and probabilities associated with minimum contrast estimators (equivalently, M-estimators) using a tilting approach due to Field. These developments are an interesting and important extension of saddlepoint-type methodology. However, in the case of a multivariate parameter, the theoretical picture has remained incomplete in certain respects, as explained below. In this paper we present results which provide rigorous justification of the tilting argument, using conditions which it is feasible to check. These results include a new formulation and proof of Skovgaard’s theorem for the intensity of minimum contrast estimators, but under conditions which are typically straigtforward to check in practice. Our most detailed application is to multivariate location-scatter models.

Key words and phrases: Elliptical distribution, exponentially small, intensity, location-scatter model, M-estimator.