Divergence-based tests for model diagnostic

Pearson's χ2 test, and more generally, divergence-based tests of goodness-of-fit are asymptotically χ2-distributed with m−1 degrees of freedom if the numbers of cells m is fixed, the observations are i.i.d and the cell probabilities and model parameters are completely specified. Jiang [Jiang, J., 2001. A nonstandard χ2-test with application to generalized linear model diagnostics. Statistics and Probability Letters 53, 101-109] proposed a nonstandard χ2 test to check distributional assumptions for the case of observations not identically distributed. Under the same setup, in this paper a family of divergence-based tests are introduced and their asymptotic distributions are derived. In addition bootstrap tests based on the given divergence test statistics are considered. Applications to generalized linear models diagnostic are proposed. A simulation study is carried out to investigate performance of several power-divergence tests.