On the Efficiency of Generalized Likelihood Ratio Tests

The generalized likelihood ratio (GLR) test has been proposed by Fan, Zhang and Zhang (2001) as a generally applicable method to test parametric, semiparametric or nonparametric models against nonparametric alternatives. It is a natural extension of the maximum likelihood ratio test for a parametric model and fully inherits the advantages of classical likelihood ratio tests. Both true likelihood and pesudo likelihood functions can be used. Like classical likelihood ratio tests, the GLR test enjoys the appealing Wilk’s phenomena in the sense that its asymptotic distribution is independent of nuisance parameters and nuisance functions, and follows a χ2-distribution with a known large number of degree freedom. It achieves the asymptotically optimal rate of convergence for nonparametric testing problems formulated by Ingster (1993a, 1993b) and Spokoiny (1996). In this paper, we propose a class of new tests based on loss functions, which measure the discrepancies between the fitted values of the null and alternative models, and are more relevant to economic applications. Like the GLR test, the new tests are generally applicable and enjoy many appealing features of the GLR test, such as the Wilk’s phenomena. Most importantly, they are asymptotically more powerful than the GLR test in terms of Pitman’s relative efficiency criterion. This holds even when the true likelihood function is known.