## Annals of Statistics

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

#### 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.

#### Article information

Source
Ann. Statist., Volume 41, Number 3 (2013), 1166-1203.

Dates
First available in Project Euclid: 13 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1371150897

Digital Object Identifier
doi:10.1214/13-AOS1099

Mathematical Reviews number (MathSciNet)
MR3113807

Zentralblatt MATH identifier
1293.62100

Subjects
Primary: 62G10: Hypothesis testing

#### Citation

Hong, Yongmiao; Lee, Yoon-Jin. A loss function approach to model specification testing and its relative efficiency. Ann. Statist. 41 (2013), no. 3, 1166--1203. doi:10.1214/13-AOS1099. https://projecteuclid.org/euclid.aos/1371150897

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#### Supplemental materials

• Supplementary material: Supplementary material for a loss function approach to model specification testing and its relative efficiency. In this supplement, we present the detailed proofs of Theorems 1–4 and report the simulation results with the bandwidth $h=S_{X}n^{-1/5}$.