Open Access
February 2017 Flexible results for quadratic forms with applications to variance components estimation
Lee H. Dicker, Murat A. Erdogdu
Ann. Statist. 45(1): 386-414 (February 2017). DOI: 10.1214/16-AOS1456


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.


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Lee H. Dicker. Murat A. Erdogdu. "Flexible results for quadratic forms with applications to variance components estimation." Ann. Statist. 45 (1) 386 - 414, February 2017.


Received: 1 September 2015; Revised: 1 February 2016; Published: February 2017
First available in Project Euclid: 21 February 2017

zbMATH: 1364.62040
MathSciNet: MR3611496
Digital Object Identifier: 10.1214/16-AOS1456

Primary: 62F99
Secondary: 62E17 , 62F12

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

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 1 • February 2017
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