On asymptotic normality of likelihood and conditional analysis

Abstract

Abstract The likelihood function from a large sample is commonly assumed to be approximately a normal density function. The literature supports, under mild conditions, an approximate normal shape about the maximum; but typically a stronger result is needed: that the normalized likelihood itself is approximately a normal density. In a transformation‐parameter context, we consider the likelihood normalized relative to right‐invariant measure, and in the location case under moderate conditions show that the standardized version converges almost surely to the standard normal. Also in a transformation‐parameter context, we show that almost sure convergence of the normalized and standardized likelihood to a standard normal implies that the standardized distribution for conditional inference converges almost surely to a corresponding standard normal. This latter result is of immediate use for a range of estimating, testing, and confidence procedures on a conditional‐inference basis.