Stochastic Differentiability in Maximum Likelihood Theory

In the classical asymptotic likelihood theory, there exists a vast number of results concerning asymptotic normality of maximum likelihood estimators (see for example [2]). Proofs in asymptotic maximum theory begin with a careful application of Taylor's theorem. In the past, generalizations were focused on weakening conditions on the remainder term (for example, LeCam [3]). In the last ten years, the remainder term has been treated stochastically by Pollard (1984) and Hoffmann-Jorgensen (1990). Based on the most recent developments in the theory of infinite dimensional laws of large numbers and infinite dimensional central limit theorems, Pollard [4] has condensed many of the technicalities that arise in the asymptotic normality of maximum estimators into a single stochastic equicontinuity condition imposed on a remainder term. These generalizations have been pursued further by Hoffmann-Jorgensen [1] who developed the idea of stochastic differentiability which requires that a remainder term satisfies an even weaker condition than stochastic equicontinuity.