Hypothesis Testing for a Functional Parameter via Self-normalization

2026-05-30

AI Summary

This study addresses the limitations of traditional nonparametric methods in hypothesis testing for functional parameters, which often rely on bandwidth selection and consequently suffer from poor finite-sample performance. The authors propose a sample-splitting self-normalization (SS-SN) approach, extending the tuning-free self-normalization framework—previously unavailable for functional parameter testing—to a broad range of applications, including tests for marginal distributions, time reversibility, and spectral distribution change points. By integrating sample splitting with self-normalization, the method constructs a test statistic with a pivotal limiting distribution, and the paper derives its asymptotic null distribution as well as its power function under local alternatives. Numerical experiments demonstrate that SS-SN accurately controls type I error while achieving testing power comparable to or better than existing methods.

Abstract

Testing simple or composite hypothesis on a functional parameter has attracted considerable attention in time series analysis. To accommodate for the unknown temporal dependence, classical nonparametric approaches such as block bootstrapping and subsampling all involve a bandwidth parameter, the choice of which can substantially affect the finite sample performance. The self normalization (SN) method is tuning parameter free when applied to the inference of a finite-dimensional parameter but its applicability to a functional parameter is unknown. In this paper, we propose a sample splitting based approach to generalize the SN method to hypothesis testing of a functional parameter. Our SS-SN (sample splitting plus self-normalization) idea is broadly applicable to many testing problems for functional parameters, including testing for simple/composite hypothesis on marginal cumulative distribution function, testing for time-reversibility and testing for a change point on the spectral distribution of a multivariate time series. Specifically, we derive the pivotal limiting distributions of our SS-SN test statistics under the null for both simple and composite null hypothesis, and derive the limiting power function under the local alternatives. Numerical simulations show that our new tests tend to yield accurate size with competitive power performance as compared to many existing ones.

Problem

• functional parameter
• hypothesis testing
• self-normalization
• time series
• bandwidth selection

Innovation

• self-normalization
• functional parameter
• sample splitting
• hypothesis testing
• time series