A loss function approach to model specification testing and its relative efficiency
Yongmiao Hong, Yoon-Jin Lee(Submitted on 20 Jun 2013)
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.
Comments: Published in at this http URL the Annals of Statistics (this http URL) by the Institute of Mathematical Statistics (this http URL)Subjects: Statistics Theory (math.ST)Journal reference: Annals of Statistics 2013, Vol. 41, No. 3, 1166-1203DOI: 10.1214/13-AOS1099Report number: IMS-AOS-AOS1099Cite as: arXiv:1306.4864 [math.ST] (or arXiv:1306.4864v1 [math.ST] for this version)