Statistical Topics and Stochastic Models for Dependent Data with Applications

11Goodness-of-Fit Tests Based on Divergence Measures for Frailty Models

In this chapter, we first review results from Vonta and Karagrigoriou (2014) where new goodness-of-fit tests based on (φ-divergence measures were defined for censored data. These results are then extended to the case where there is heterogeneity in the data described by a random effect called frailty. The goodness-of-fit tests are extended in order to test under the null hypothesis the fit of a frailty model to right censored data. The performance of the tests is assessed through simulations.

11.1. Introduction

Measures of divergence between two probability distributions have a very long history initiated by the pioneering work of Pearson, Mahalanobis, Lévy and Kolmogorov. Among the most popular measures of divergence are the Kullback–Leibler measure of divergence (Kullback and Leibler 1951) and the Csiszár’s φ-divergence family of measures (Csiszár 1964; Ali and Silvey 1966). A unified analysis has been provided by Cressie and Read (1984) who introduced the power divergence family of statistics that depends on a parameter a and is used for goodness-of-fit tests for multinomial distributions. The Cressie and Read family includes among others the well-known Pearson’s χ² divergence measure and for multinomial models the loglikelihood ratio statistic. The BHHJ divergence measure was proposed by Basu et al. (1998) and generalized to the BHHJ family of measures by Mattheou et al. (2009). The BHHJ family depends ...