Likelihood Prediction for Generalized Linear Mixed Models under Covariate Uncertainty

Abstract

This article presents the techniques of likelihood prediction for the generalized linear mixed models. Methods of likelihood prediction are explained through a series of examples; from a classical one to more complicated ones. The examples show, in simple cases, that the likelihood prediction (LP) coincides with already known best frequentist practice such as the best linear unbiased predictor. This article outlines a way to deal with the covariate uncertainty while producing predictive inference. Using a Poisson errors-in-variable generalized linear model, it has been shown in certain cases that LP produces better results than already known methods.

Keywords:

• Coverage interval
• Credit risk prediction
• Future value prediction
• Predictive likelihood
• Profile predictive likelihood
• Stochastic covariate

Mathematics Subject Classification:

• Primary 62M20
• Secondary 60G25, 62J12, 62J99

Notes

Note. , NA stands for not available. The predictive variance is the variance of the predictive distribution if the distribution is available, otherwise it is the variance of the point predictor.

If σ² is unknown, the above mathematical derivation becomes very tedious therefore, we omit the latter case. Interested readers are referred to Bjørnstad (1990) for further results.

See Bolfarine (1991) and Buzas and Stefansky (1996) for further discussion on the problems induced by unknown σ and σ_δ.

Note. The results of the PsL are quoted from Huwang and Hwang (2002).