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

Flexible results for quadratic forms with applications to variance components estimation