Large Deviation Probabilities for Certain Nonparametric Maximum Likelihood Estimators

Abstract

Let $(X, \\mathscr{A})$ be a measurable space and $\\{P_{\\vartheta,\\tau}\\mid\\mathscr{A}: \\vartheta \\in \\Theta, \\tau \\in T\\}$ a family of probability measures. Given an appropriate estimator sequence for $\\vartheta$, we define a sequence of asymptotic maximum likelihood estimators for $\\tau$ and give bounds for its large deviation probabilities under conditions which are natural for the application to the estimation of mixing distributions. This paper generalizes earlier results of Pfanzagl to the following cases: (i) estimator sequences restricted to a sieve; (ii) estimator sequences using a given estimator sequence for a nuisance parameter; (iii) convergence under the "wrong model;" (iv) large deviation probabilities instead of consistency.