Consistency property of elliptical probability density functions
Journal of Multivariate Analysis
A Comparison of New and Old Algorithms for a Mixture EstimationProblem
Machine Learning - Special issue on the eighth annual conference on computational learning theory, (COLT '95)
Multivariate stable densities as functions of one dimensional projections
Journal of Multivariate Analysis
Multivariate locally adaptive density estimation
Computational Statistics & Data Analysis
A semiparametric density estimator based on elliptical distributions
Journal of Multivariate Analysis
Local convergence rates of simple evolutionary algorithms withCauchy mutations
IEEE Transactions on Evolutionary Computation
Hi-index | 0.00 |
This paper considers the class of p-dimensional elliptic distributions (p=1) satisfying the consistency property (Kano, 1994) [23] and within this general framework presents a two-stage nonparametric estimator for the Lebesgue density based on Gaussian mixture sieves. Under the on-line Exponentiated Gradient (EG) algorithm of Helmbold et al. (1997) [20] and without restricting the mixing measure to have compact support, the estimator produces estimates converging uniformly in probability to the true elliptic density at a rate that is independent of the dimension of the problem, hence circumventing the familiar curse of dimensionality inherent to many semiparametric estimators. The rate performance of our estimator depends on the tail behaviour of the underlying mixing density (and hence that of the data) rather than smoothness properties. In fact, our method achieves a rate of at least O"p(n^-^1^/^4), provided only some positive moment exists. When further moments exists, the rate improves reaching O"p(n^-^3^/^8) as the tails of the true density converge to those of a normal. Unlike the elliptic density estimator of Liebscher (2005) [25], our sieve estimator always yields an estimate that is a valid density, and is also attractive from a practical perspective as it accepts data as a stream, thus significantly reducing computational and storage requirements. Monte Carlo experimentation indicates encouraging finite sample performance over a range of elliptic densities. The estimator is also implemented in a binary classification task using the well-known Wisconsin breast cancer dataset.