Testing hypotheses about mixture distributions using not identically distributed data

TL;DR

This paper develops and analyzes statistical tests for goodness-of-fit of mixture distributions using independent, non-identically distributed data, focusing on asymptotic properties and finite sample performance.

Contribution

It introduces new tests based on Kolmogorov-Smirnov and Cramér-von-Mises statistics for mixture distributions with non-i.i.d. data, including methods for critical value determination.

Findings

Tests are asymptotically exact and consistent.

Simulation results demonstrate good finite sample performance.

Applications include models with non-identically distributed errors.

Abstract

Testing hypotheses of goodness-of-fit about mixture distributions on the basis of independent but not necessarily identically distributed random vectors is considered. The hypotheses are given by a specific distribution or by a family of distributions. Moreover, testing hypotheses formulated by Hadamard differentiable functionals is discussed in this situation, in particular the hypothesis of central symmetry, homogeneity and independence. Kolmogorov-Smirnov or Cram\'er-von-Mises type statistics are suggested as well as methods to determine critical values. The focus of the investigation is on asymptotic properties of the test statistics. Further, outcomes of simulations for finite sample sizes are given. Applications to models with not identically distributed errors are presented. The results imply that the tests are of asymptotically exact size and consistent.