Asymptotic Distribution of Cramer-von Mises Statistic When Contamination Exists

In this paper, asymptotic distribution of Cram\'er-von Mises goodness-of-fit test statistic is investigated when contamination exists.<br />We first derive the asymptotic distribution of the Cram\'er-von Mises statistic when the observations are contaminated with noise as a mixture.<br />The result is extended to the case where the parameters are estimated by the minimum distance estimator,<br />which minimizes the Cram\'er-von Mises statistic.<br />In both cases the asymptotic distribution of the Cram\'er-von Mises statistic is given by that of the weighted infinite sum of non-central $\chi^2_1$ variables and the effect of contamination appears only in the non-centrality of the variables.<br />We also demonstrate the robustness of the goodness-of-fit test by Monte Carlo simulations when the parameters are estimated<br />by the minimum distance estimator and the maximum likelihood estimator.<br />Numerical experiments indicate that the use of the minimum distance estimator makes the test insensitive to contamination whereas the power is retained almost the same as that of the maximum likelihood estimator.