TY - JOUR

T1 - Nonparametric estimation of multivariate elliptic densities via finite mixture sieves

AU - Battey, Heather

AU - Linton, Oliver

PY - 2014/1

Y1 - 2014/1

N2 - 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 etal. (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.

AB - 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 etal. (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.

UR - http://www.scopus.com/inward/record.url?scp=84884126048&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2013.08.013

DO - 10.1016/j.jmva.2013.08.013

M3 - Article

AN - SCOPUS:84884126048

VL - 123

SP - 43

EP - 67

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

ER -