Covariance Matrix Estimation with Non Uniform and Data Dependent Missing Observations

TL;DR

This paper develops unbiased covariance matrix estimators for data with non-uniform, dependent missingness, providing bounds that match or improve upon existing results for various missing data mechanisms.

Contribution

It introduces new bounds for covariance estimation under complex missing data mechanisms, extending the effective rank concept to account for missingness.

Findings

Bounds are comparable to state-of-the-art for uniform missing data.

New bounds are derived for non-uniform, dependent missing data.

Estimators achieve the same asymptotic rate across mechanisms.

Abstract

In this paper we study covariance estimation with missing data. We consider missing data mechanisms that can be independent of the data, or have a time varying dependency. Additionally, observed variables may have arbitrary (non uniform) and dependent observation probabilities. For each mechanism, we construct an unbiased estimator and obtain bounds for the expected value of their estimation error in operator norm. Our bounds are equivalent, up to constant and logarithmic factors, to state of the art bounds for complete and uniform missing observations. Furthermore, for the more general non uniform and dependent cases, the proposed bounds are new or improve upon previous results. Our error estimates depend on quantities we call scaled effective rank, which generalize the effective rank to account for missing observations. All the estimators studied in this work have the same asymptotic convergence rate (up to logarithmic factors).