Testing the distribution specification in multiparameter local likelihood models

Abstract:

Multiparameter local likelihood models have been accepted as a flexible tool for modeling the relationship between responses and covariates, and the corresponding methodology has been used to analyze data arising from climatology, environmetrics, finance, medicine, and so on. Although both point and interval estimation for the unknown parameter functions in the model have been investigated in the literature, how to formally test goodness-of-fit of the specified form of the conditional density function remains an unsolved problem.  Testing the specification of the conditional density is an important issue, the inference becomes inclusive or misleading and the estimated parameter functions become meaningless if the form of the true conditional density is different from the specified one. In this paper, we address this specification test problem. We construst  Kolmogorov-Smirnov and Cramer-von Mises type test statistics, and  show that formal tests can be constructed if undersmoothing is employed. The asymptotic null distributions of the proposed test statistics depend on the unknown parameter functions, so bootstrap approach is suggested. We conduct a simulation study to assess finite sample properties of the proposed test and apply it to validate the generalized extreme value local likelihood model for an environmental data set.