Testing Nested Distributions

A number of criteria, test statistics and diagnostic plots have been developed in order to test the adapted distribution assumption. Usually, a simpler distribution is tested against a more complicated one (by adding an extra parameter), which include the first distribution as a special case (nested distributions). A characteristic example of such cases is the Burr XII distribution which can be obtained under the proportional hazards frailty model by assuming a Weibull baseline function and a Gamma frailty distribution with mean frailty equal to 1 and variance equal to theta. In this work, two new easy to construct and to interpret tests, a diagnostic plot and an asymptotic test, are presented in order to test nested distributions. The asymptotic test is based on the approximation of the difference of the two estimated nested distribution functions using the first two terms of the Taylor's expansion while the diagnostic plot is constructed using the exact difference of the two fitted distribution functions. Simulation results, using data sets with and without censored observations, demonstrate that the proposed tests perform, in most of the cases, better than other test statistics such as the LR and the Wald.