Approximation by a Normal Distribution

Abstract

In this second chapter we find sufficient conditions that must be satisfied by the function g(j, θ) and the r.v.-s ε _j in order to ensure that as n → ∞ the distributions of the normalised differences $\hat \theta _n - \theta and\mathop \theta \limits^ \vee _n - \theta$ tend uniformly to Gaussian distributions in the proximity of the parameter. The basic result of the chapter consists in obtaining the asymptotic expansion (a.e.) of the distribution of l.s.e.-s $\hat \theta _n$ (see Sections 10–11), significantly revising the usual statements about the asymptotic normality of statistical estimators.