The Annals of Statistics

Flexible results for quadratic forms with applications to variance components estimation

Lee H. Dicker and Murat A. Erdogdu

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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.

Article information

Ann. Statist., Volume 45, Number 1 (2017), 386-414.

Received: September 2015
Revised: February 2016
First available in Project Euclid: 21 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F99: None of the above, but in this section
Secondary: 62E17: Approximations to distributions (nonasymptotic) 62F12: Asymptotic properties of estimators

Hanson–Wright inequality random-effects models model misspecification Stein’s method uniform concentration bounds


Dicker, Lee H.; Erdogdu, Murat A. Flexible results for quadratic forms with applications to variance components estimation. Ann. Statist. 45 (2017), no. 1, 386--414. doi:10.1214/16-AOS1456.

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Supplemental materials

  • Supplement to “Flexible results for quadratic forms with applications to variance components estimation”. The supplementary document Dicker and Erdogdu (2016b) contains proofs of Propositions 1–4, along with statements and proofs of additional auxiliary results.