Studentization Versus Variance Stabilization: A Simple Way Out of an Old Dilemma

Abstract

Assume ${\hat{\mathit{\theta }}}_{\mathit{n}}$ is a statistic used to estimate a parameter θ on the basis of data ${\mathit{X}}_1,\dots ,{\mathit{X}}_{\mathit{n}}$. Further assume that ${\hat{\mathit{\theta }}}_{\mathit{n}}$ is consistent and asymptotically normal, with asymptotic variance given by ${\mathit{\sigma }}^2(\mathit{\theta })$. Even if the functional form of ${\mathit{\sigma }}^2(·)$ is known, its dependence on the unknown parameter θ creates a dilemma as regards the construction of a confidence interval for θ. Should the interval be based on the normal quantiles with estimated variance, that is, studentization, or shall we transform the statistic ${\hat{\mathit{\theta }}}_{\mathit{n}}$ to ${\mathit{Y}}_{\mathit{n}}=\mathit{g}({\hat{\mathit{\theta }}}_{\mathit{n}})$ such that the asymptotic variance of ${\mathit{Y}}_{\mathit{n}}$ does not depend on θ, that is, variance stabilization? We show how this dilemma can be bypassed by a straightforward construction that applies rather generally, and just hinges on solving simple algebraic equations. We illustrate the new approach on a host of numerical examples, including two examples in nonparametric function estimation. In the latter, a different sort of dilemma arises: employing undersmoothing versus an explicit bias correction. This paper is dedicated to the memory of Dr. Dimitrios Gatzouras (1962–2020).