Joint distribution and marginal distribution methods for checking assumptions of generalized linear model

In this article, we consider the model diagnostic plot and test of the generalized linear model. There exist several commonly used plotting methods and tests for checking the regression model assumptions. However, the existing plots and tests require certain constraints on the joint cumulative distribution function of the response variable Y and the covariate Z and thus are invalid when the real data set does not satisfy those constraints. In particular, in the latter case, the p-values provided by these tests are false. In this article, we propose a new method to check the model assumptions. This method compares two estimators of the marginal distribution of Y (or the joint distribution of (Y, Z)): one is the non-parametric maximum likelihood estimator and the other is an estimator based on the null hypothesis. This method is called the marginal distribution (MD) method or the joint distribution (JD) method. Their asymptotic properties are studied. The simulation results suggest both the diagnostic plots and the hypothesis tests using the new methods provide satisfactory results and the JD method is always consistent even when the existing methods fail.