An Order Selection Criterion for Testing Goodness of Fit

The classical problem of assessing the goodness of fit of a postulated parametric distribution is investigated using techniques from nonparametric density estimation. A new test is proposed based on the data-selected order of a Fourier series density estimator. This test has the novel feature of providing an associated nonparametric estimator that can be used to estimate the unknown density when the null hypothesis is rejected. The limiting null distribution of the proposed test statistic is derived, and the test is shown to be consistent against essentially any fixed alternative. Results are reported from a simulation experiment that compared empirical power properties of the new test with those of Cramér-von Mises and data-driven Neyman smooth-type tests.