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

2025-12-08

AI Summary

Density estimation for strongly mixing time series on the cone of positive-definite matrices suffers from severe boundary bias under conventional kernel methods. Method: This paper introduces, for the first time, a boundary-aware Wishart kernel density estimator on this space: it mitigates boundary effects via matrix logarithmic transformation and rigorously establishes its strong consistency and asymptotic normality under strong mixing conditions. Contribution/Results: We derive upper bounds for mean squared error and mean absolute error. Simulations under Wishart autoregressive models demonstrate substantial accuracy gains over log-normal kernel methods. Empirical analysis of high-frequency covariance matrices from Amazon and the S&P 500 confirms the method’s effectiveness in financial covariance density modeling. This work fills a theoretical gap in density estimation for mixing sequences on the positive-definite matrix manifold and provides a novel tool for high-dimensional dynamic covariance modeling.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Data Compression
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Estimates densities on positive definite matrix cones
• Addresses boundary bias in kernel density estimation
• Applies to strongly mixing time series data

Innovation

• Wishart kernel density estimator for positive definite matrices
• Boundary-aware design reduces bias in density estimation
• Establishes asymptotic normality under mixing conditions