Nonparametric estimation of multivariate density with direct and auxiliary data and application

We consider the problem of multivariate density estimation, using samples from the distribution of interest as well as auxiliary samples from a related distribution. We assume that the data from the target distribution and the related distribution may occur individually as well as in pairs. Using nonparametric maximum likelihood estimator of the joint distribution, we derive a kernel density estimator of the marginal density. We show theoretically, in a simple special case, that the implied estimator of the marginal density has smaller integrated mean squared error than that of a similar estimator obtained by ignoring dependence of the paired observations. We establish consistency of the marginal density estimator under suitable conditions. We demonstrate small sample superiority of the proposed estimator over the estimator that ignores dependence of the samples, through a simulation study with dependent and non-normal populations. The application of the density estimator in nonparametric classification is also discussed. It is shown that the misclassification probability of the resulting classifier is asymptotically equivalent to that of the Bayes classifier. We also include a data analytic illustration.