A loss function approach to model specification testing and its relative efficiency

TL;DR

This paper introduces a loss function-based model specification test that outperforms the generalized likelihood ratio test in power, providing a more decision-relevant approach for nonparametric inference.

Contribution

It proposes a new loss function approach for model testing that is asymptotically more powerful than the traditional GLR test, regardless of smoothing parameters.

Findings

The new test is asymptotically more powerful than the GLR test.

The efficiency gain is consistent across different smoothing parameters and kernel functions.

The proposed method is applicable even when the true likelihood is known.

Abstract

The generalized likelihood ratio (GLR) test proposed by Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] and Fan and Yao [Nonlinear Time Series: Nonparametric and Parametric Methods (2003) Springer] is a generally applicable nonparametric inference procedure. In this paper, we show that although it inherits many advantages of the parametric maximum likelihood ratio (LR) test, the GLR test does not have the optimal power property. We propose a generally applicable test based on loss functions, which measure discrepancies between the null and nonparametric alternative models and are more relevant to decision-making under uncertainty. The new test is asymptotically more powerful than the GLR test in terms of Pitman's efficiency criterion. This efficiency gain holds no matter what smoothing parameter and kernel function are used and even when the true likelihood function is available for the GLR test.