Specification tests for the response distribution in generalized linear models

Goodness-of-fit tests are proposed for the case of independent observations coming from the same family of distributions but with different parameters. The most popular related context is that of generalized linear models (GLMs) where the mean of the distribution varies with regressors. In the proposed procedures, and based on suitable estimators of the parameters involved, the data are transformed to normality. Then any test for normality for i.i.d. data may be applied. The method suggested is in full generality as it may be applied to arbitrary laws with continuous or discrete distribution functions, provided that an efficient method of estimation exists for the parameters. We investigate by Monte Carlo the relative performance of classical tests based on the empirical distribution function, in comparison to a corresponding test which instead of the empirical distribution function, utilizes the empirical characteristic function. Standard measures of goodness-of-fit often used in the context of GLM are also included in the comparison. The paper concludes with several real-data examples.