Estimating a bivariate density when there are extra data on one or both components
1 Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia. halpstat{at}maths.anu.edu.au, 2 Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany. natalie.neumeyer{at}rub.de
The objective of this paper is to estimate a bivariate density nonparametrically from a dataset from the joint distribution and datasets from one or both marginal distributions. We develop a copula-based solution, which has potential benefits even when the marginal datasets are empty. For example, if the copula density is sufficiently smooth in the region where we wish to estimate it, the joint density can be estimated with a high degree of accuracy. Similar improvements in performance are available if the marginals are close to being independent. We use wavelet estimators to approximate the copula density, which in cases of statistical interest can be unbounded along boundaries. Our techniques are also useful for solving recently-considered related problems, for example where the marginal distributions are determined by parametric models. The methodology is also readily extended to more general multivariate settings.
Key Words: Copula; Dimension reduction; Independence; Kernel method; Prediction; Threshold; Wavelet.
Received August 2005. Revised November 2005.