Density model checks via the lack-of-fitness

Parametric multivariate density estimators, such as the maximum likelihood, can be generalized by mixing them with a kernel estimator. The mixture weights can be chosen to optimize a measure of the goodness-of-fit. The optimal weight of the kernel estimator, which we call the lack-of-fitness coefficient, then provides a simple check of the parametric model. The test statistic is defined as the appropriately normalized lack-of-fitness coefficient. When the parametric density model is correct, the statistic converges in distribution to the positive part of a standard Gaussian variable, regardless of the dimension of the observations. In addition, the test has good power against alternative hypotheses approaching the density model.