Integrated Cumulative Error (ICE) distance for non-nested mixture model selection: Application to extreme values in metal fatigue problems

Abstract

In this paper, we consider the problem of selecting the most appropriate model, amongst a given collection of mixture models, to describe datasets likely drawn from mixture of distributions. The proposed method consists of finding the quasi-maximum likelihood estimators (QMLEs) of the various models in competition, using the Expectation-Maximization (EM) type algorithms, and subsequently estimating, for every model, a statistical distance to the true model based on the empirical cumulative distribution function (cdf) of the original dataset and the QMLE-fitted cdf. To evaluate the goodness of fit, a new metric, the Integrated Cumulative Error ($ICE$) is proposed and compared with other existing metrics for accuracy of detecting the appropriate model. We state, under mild conditions, that our estimator of the $ICE$ distance converges at the rate $\sqrt{n}$ in probability along with the consistency of our model selection procedure (ability to detect asymptotically the right model). The $ICE$ criterion shows, over a set of benchmark examples, numerically improved performance from the existing distance-based criteria in identifying the correct model. The method is applied in a material fatigue life context to model the distribution of indicators of the fatigue crack formation potency, obtained from numerical experiments.