Density estimation using non-parametric and semi-parametric mixtures

Abstract

This article presents a general framework for univariate non-parametric density estimation, based on mixture models. Similar to kernel-based estimation, the proposed approach uses bandwidth to control the density smoothness, but each density estimate for a fixed bandwidth is determined by non-parametric likelihood maximization, with bandwidth selection carried out as model selection. This leads to simple models, yet with higher accuracy, especially in terms of the Kullback–Leibler or the Hellinger risk. The particular problem of estimating a symmetric density function is investigated. Both simulation study and real-world data examples suggest that the mixture-based estimators outperform their kernel-based counterparts.

Kernel-based estimation of individual location densities from smartphone data

“…Indeed, some apps might have further location information about the user that could be highly relevant, for example, home address, work environment and commuting plans. Semi-parametric mixture models (Wang and Chee, 2012) offer a suitable framework to move towards this extension.…”

Nonparametric estimation of species richness using discretek-monotone distributions

“…We note that Böhning and Schön (2005) considered the evaluations of candidate mixture models by the Akaike and the Bayesian information criterion in the context of species richness estimation. Wang and Chee (2012) found that while both the Akaike and the Bayesian information criterion appear to perform reasonably well for choosing an appropriate value for bandwidth in the context of mixture-based density estimation, the Akaike information criterion seems more reliable. Selecting k is a model selection problem and can also be guided by a cross-validation criterion, for example.…”

Modelling of count data using nonparametric mixtures

“…The common parameter k will be treated as a bandwidth parameter, as in the kernel-based density estimation setting, which is subject to selection. Actually, this kind of joint estimation problem is essentially similar to the problem encountered by Wang and Chee (2012) in the maximum likelihood estimation of a density function using the nonparametric normal mixture model.…”

“…See Wang and Chee (2012) for a framework for modelling continuous data based on nonparametric mixture models. Our work here, which investigates the use of the least squares criterion in nonparametric mixture modelling of count data, is distinguished from their work with respect to the estimation method and model type.…”