Large Sample Confidence Regions Based on Subsamples under Minimal Assumptions

TLDR

The method has been studied by Wu in regular situations where the statistic is asymptotically normal. This article studies the construction of confidence regions by approximating the sampling distribution of a statistic and aims to prove the method yields asymptotically valid confidence regions under minimal conditions. The true sampling distribution is estimated by normalizing statistic values computed over subsamples, requiring only that the normalized statistic has a limit distribution, and the method adapts to parameters of stationary time series or homogeneous random fields. Unlike the bootstrap, convergence to the limit distribution need not be uniform, and the method immediately applies to constructing confidence intervals for the spectral density of a homogeneous random field. The method is applicable in i.i.d.

Abstract

In this article, the construction of confidence regions by approximating the sampling distribution of some statistic is studied. The true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data. In the i.i.d. context, the method has been studied by Wu in regular situations where the statistic is asymptotically normal. The goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions. Essentially, all that is required is that the statistic, suitably normalized, possesses a limit distribution under the true model. Unlike the bootstrap, the convergence to the limit distribution need not be uniform in any sense. The method is readily adapted to parameters of stationary time series or, more generally, homogeneous random fields. For example, an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field.