Assessment of Distributional Goodness-of-Fit for Modeling the Superposition of Renewal Process Data

Abstract

Parametric maximum likelihood methods can be used to estimate the renewal distribution based on aggregate data from superpositions of a group of renewal processes (SRP). The traditional distributional assessment approaches are, however, unable to be applied to the SRP data directly as the actual locations where the renewal events occurred are unknown. In this paper, we present two graphical distributional assessment procedures to evaluate how well alternative parametric models fit the observed SRP data. The first procedure provides a flexible semi-parametric estimate of the renewal distribution which is based on a piecewise exponential (PEX) model. Corresponding nonparametric simultaneous confidence bands (NPSCBs) are given to assess the amount of statistical uncertainty of the semi-parametric estimate. The results show that the PEX model with NPSCBs provides a flexible framework for comparing different parametric distributions. The second procedure is based on a comparison of parametric and nonparametric estimates of the mean cumulative function (MCF) for the SRP recurrence data. These two procedures are illustrated by applications to both real and simulated SRP data.