Edgeworth expansions for studentized statistics under weak dependence

TL;DR

This paper develops Edgeworth expansions for studentized statistics under weak dependence, accounting for complex lag-covariance structures, and reveals a superposition of three series in the asymptotic approximation.

Contribution

It introduces a novel Edgeworth expansion framework for studentized statistics with dependent data, incorporating infinite lag-covariance estimators and their bias.

Findings

Derived a three-series Edgeworth expansion under dependence.

Showed the expansion differs from the independent case by including bias and lag-covariance terms.

Provided theoretical insights into the asymptotic behavior of studentized statistics with dependent data.

Abstract

In this paper, we derive valid Edgeworth expansions for studentized versions of a large class of statistics when the data are generated by a strongly mixing process. Under dependence, the asymptotic variance of such a statistic is given by an infinite series of lag-covariances, and therefore, studentizing factors (i.e., estimators of the asymptotic standard error) typically involve an increasing number, say, $\ell$ of lag-covariance estimators, which are themselves quadratic functions of the observations. The unboundedness of the dimension $\ell$ of these quadratic functions makes the derivation and the form of the expansions nonstandard. It is shown that in contrast to the case of the studentized means under independence, the derived Edgeworth expansion is a superposition of three distinct series, respectively, given by one in powers of $n^{-1/2}$, one in powers of $[n/\ell]^{-1/2}$ (resulting from the standard error of the studentizing factor) and one in powers of the bias of the studentizing factor, where $n$ denotes the sample size.