Nonparametric multivariate density estimation using mixtures

作者：Wang Xuxu; Wang Yong^*

来源：Statistics and Computing, 2015, 25(2): 349-364.

DOI：10.1007/s11222-013-9436-y

• Multivariate density estimation
• Nonparametric mixture
• Semiparametric mixture
• Covariance matrix decomposition
• Kernel-based density estimation
• Bandwidth selection

A new method is proposed for nonparametric multivariate density estimation, which extends a general framework that has been recently developed in the univariate case based on nonparametric and semiparametric mixture distributions. The major challenge to a multivariate extension is the dilemma that one can not maximise directly the likelihood function with respect to the whole component covariance matrix, since the likelihood is unbounded for a singular covariance matrix, and that one can not leave the covariance matrix or a large part of it to be determined by a selection method, since it would be computationally infeasible. To overcome it, we consider using a volume parameter h to enforce a minimal restriction on the covariance matrix so that, with h fixed, the likelihood function is bounded and its maximisation can be successfully carried out with respect to all the remaining parameters. The role played here by the scalar h is just the same as by the bandwidth in the univariate case and its value can be determined by a model selection criterion, such as the Akaike information criterion. New efficient algorithms are also described for finding the maximum likelihood estimates of these mixtures under various restrictions on the covariance matrix. Empirical studies using simulated and real-world data show that the new multivariate mixture-based density estimator performs remarkably better than kernel-based density estimators.

• 出版日期2015-3