Application of the delta method to functions of the sample mean when observations are dependent

Abstract

Let $\mu$ be the expected value of a random variable and $\bar{X}_n$ the corresponding sample mean of n observations. If the transformed expectation $f(\mu )$ is to be estimated by $f\left( \bar{X}_n\right)$ then the delta method is a widely used tool to describe the asymptotic behaviour of $f\left( \bar{X}_n\right)$. Regarding bias and variance, however, conventional theorems require independent observations as well as boundedness conditions of f being violated even by “simple” functions such as roots or logarithms. It is shown that asymptotic expansions for bias and variance still hold if restrictive boundedness conditions are replaced by considerably weaker requirements upon the global growth of f. Moreover, observations are allowed to be dependent.