Chapter 9 - Multivariate density estimation

Abstract

The local parametric approach was originally used for density estimation in the beginning for the univariate case. In the present chapter, we use the local Gaussian approximation for multivariate density estimation. To avoid the curse of dimensionality, we use a pairwise simplification of the local Gaussian approximation. As a first step, we transform the marginals to standard normals (in practice, to pseudo-standard normals) by using the cumulative distribution function (in practice, the empirical distribution function) for each of the variables involved. We use a pairwise simplification for the transformed variables, and as a further simplification, we set the mean and the standard deviation of the transformed variables to 0 and 1, respectively. As a final step, we transform the density of the transformed variables back to the original variables. We also develop an asymptotic theory. The convergence rate is the same as in an additive regression model, and we show that the difference between the empirical and true cumulative distribution functions can be neglected. To obtain error limits, we use the bootstrap. We compare the method to the kernel estimator on a number of examples with increasing dimension. The new method fares well for these examples.