Asymptotic Expansions

Abstract

The consistency and asymptotic normality of the MLE, BE, and MDE provide us with the first terms of some asymptotic expansions, for example, $\hat \vartheta _n = \hat \vartheta _0 + o\left( 1 \right)$. In the following we consider the problem of asymptotic expansions of these estimators by the powers of certain small parameters. We then expand their distribution functions and the moments. These expansions, sometimes called “small sample asymptotics,” allow us to apply the asymptotic theory in the case of a moderate volume of observations. The results presented here are “not asymptotic in nature”; that is, the expansions are valid for all n > n₀ and all random variables have exact representations. The constants in the inequalities can also be calculated or estimated.