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.
(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.