Summary: Ann. Inst. Statist. Math.
Vol. 00, No. 0, 1­000 (0000)
c 0000 The Institute of Statistical Mathematics
LARGE DEVIATIONS FOR M­ESTIMATORS
MIGUEL A. ARCONES
Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902, USA.
E-mail: arcones@math.binghamton.edu
(Received August 9, 2004; revised November 29, 2004)
Abstract. We study the large deviation principle for M­estimators (and maximum
likelihood estimators in particular). We obtain the rate function of the large deviation
principle for M­estimators. For exponential families, this rate function agrees with
the Kullback­Leibler information number. However, for location or scale families this
rate function is smaller than the Kullback­Leibler information number. We apply our
results to obtain confidence regions of minimum size whose coverage probability con-
verges to one exponentially. In the case of full exponential families, the constructed
confidence regions agree with the ones obtained by inverting the likelihood ratio test
with a simple null hypothesis.
Key words and phrases: M­estimators, maximum likelihood estimators, large devi-
ations, empirical processes, Kullback­Leibler information.
1. Introduction

Source: Arcones, Miguel A. - Department of Mathematical Sciences, State University of New York at Binghamton

Collections: Mathematics