Flexible results for quadratic forms with applications to variance components estimation

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 nonstandard 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 and discuss an application involving sparse random-effects models. 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 Röllin [Ann. Probab. (2009) 37 2150–2173] embedding technique for Stein’s method of exchangeable pairs.