Flexible results for quadratic forms with applications to variance components estimation

TL;DR

This paper develops flexible finite-sample bounds and normal approximation results for quadratic forms, enhancing variance components estimation in linear random-effects models, especially in non-standard and modern applications like genomics.

Contribution

It introduces novel non-asymptotic bounds and a uniform Hanson-Wright inequality for quadratic forms, applicable to complex variance components estimation scenarios.

Findings

Derived new finite-sample concentration bounds for variance estimators

Provided a uniform Hanson-Wright inequality for quadratic forms

Achieved normal approximation results using Stein's method

Abstract

We derive convenient uniform concentration bounds and finite sample multivariate normal approximation results for quadratic forms, then describe some applications involving variance components estimation in linear random-effects models. Random-effects models and variance components estimation are classical topics in statistics, with a corresponding well-established asymptotic theory. However, our finite sample results for quadratic forms provide additional flexibility for easily analyzing random-effects models in non-standard settings, which are becoming more important in modern applications (e.g. genomics). For instance, in addition to deriving novel non-asymptotic bounds for variance components estimators in classical linear random-effects models, we provide a concentration bound for variance components estimators in linear models with correlated random-effects. Our general concentration bound is a uniform version of the Hanson-Wright inequality. The main normal approximation result in the paper is derived using Reinert and R\"{o}llin's (2009) embedding technique and multivariate Stein's method with exchangeable pairs.