Weighted Means of Likelihood Ratio as Indices Measuring Similarity Between Densities

Criteria for similarity between probability density functions are important in the field of statistics such as density estimation. In this short paper, a set of indices measuring similarity between probability densities is proposed using the weighted means of the likelihood ratio function. Numerical simulations demonstrate that the estimates of these indices are easily obtained from observations and could be useful for both parametric and nonparametric density estimation with numerical optimization.

Multiscale stick-breaking mixture models

Bayesian nonparametric density estimation is dominated by single-scale methods, typically exploiting mixture model specifications, exception made for Pólya trees prior and allied approaches. In this paper we focus on developing a novel family of multiscale stick-breaking mixture models that inherits some of the advantages of both single-scale nonparametric mixtures and Pólya trees. Our proposal is based on a mixture specification exploiting an infinitely deep binary tree of random weights that grows according to a multiscale generalization of a large class of stick-breaking processes; this multiscale stick-breaking is paired with specific stochastic processes generating sequences of parameters that induce stochastically ordered kernel functions. Properties of this family of multiscale stick-breaking mixtures are described. Focusing on a Gaussian specification, a Markov Chain Monte Carlo algorithm for posterior computation is introduced. The performance of the method is illustrated analyzing both synthetic and real datasets consistently showing competitive results both in scenarios favoring single-scale and multiscale methods. The results suggest that the method is well suited to estimate densities with varying degree of smoothness and local features.

ACCURACY OF NONPARAMETRIC DENSITY ESTIMATION FOR UNIVARIATE GAUSSIAN MIXTURE MODELS: A COMPARATIVE STUDY

Flexible and reliable probability density estimation is fundamental in unsupervised learning and classification. Finite Gaussian mixture models are commonly used for this purpose. However, the parametric form of the distribution is not always known. In this case, non-parametric density estimation methods are used. Usually, these methods become computationally demanding as the number of components increases. In this paper, a comparative study of accuracy of some nonparametric density estimators is made by means of simulation. The following approaches have been considered: an adaptive bandwidth kernel estimator, a projection pursuit estimator, a logspline estimator, and a k-nearest neighbor estimator. It was concluded that data clustering as a pre-processing step improves the estimation of mixture densities. However, in case data does not have clearly defined clusters, the pre-preprocessing step does not give that much of advantage. The application of density estimators is illustrated using municipal solid waste data collected in Kaunas (Lithuania). The data distribution is similar (i.e., with kurtotic unimodal density) to the benchmark distribution introduced by Marron and Wand. Based on the homogeneity tests it can be concluded that distributions of the municipal solid waste fractions in Kutaisi (Georgia), Saint-Petersburg (Russia), and Boryspil (Ukraine) are statistically indifferent compared to the distribution of waste fractions in Kaunas. The distribution of waste data collected in Kaunas (Lithuania) follows the general observations introduced by Marron and Wand (i.e., has one mode and certain kurtosis).