Wishart kernel density estimation for strongly mixing time series on the cone of positive definite matrices

TL;DR

This paper introduces a boundary-aware Wishart kernel density estimator for positive definite matrices, demonstrating its theoretical properties and superior performance in financial covariance matrix estimation.

Contribution

It presents the first density estimation method on the cone of positive definite matrices that is boundary-aware, consistent, and asymptotically normal, with practical implementation in finance.

Findings

Wishart KDE outperforms existing boundary-aware KDEs in simulations.

The estimator is strongly consistent and asymptotically normal.

Application to financial data demonstrates practical utility.

Abstract

A Wishart kernel density estimator (KDE) is introduced for density estimation in the cone of positive definite matrices. The estimator is boundary-aware and mitigates the boundary bias suffered by conventional KDEs, while remaining simple to implement. Its mean squared error, uniform strong consistency on expanding compact sets, and asymptotic normality are established under the Lebesgue measure and suitable mixing conditions. This work represents the first study of density estimation on this space under any metric. For independent observations, an asymptotic upper bound on the mean absolute error is also derived. A simulation study compares the performance of the Wishart KDE to another boundary-aware KDE that relies on the matrix-variate lognormal distribution proposed by Schwartzman [Int. Stat. Rev., 2016, 84(3), 456-486]. Results suggest that the Wishart KDE is superior for a selection of autoregressive coefficient matrices and innovation covariance matrices when estimating the stationary marginal density of a Wishart autoregressive process. To illustrate the practical utility of the Wishart KDE, an application to finance is made by estimating the marginal density function of a time series of realized covariance matrices, calculated from 5-minute intra-day returns, between the share prices of Amazon Corp. and the Standard & Poor's 500 exchange-traded fund over a one-year period. All code is publicly available via the R package ksm to facilitate implementation of the method and reproducibility of the findings.