The Annals of Statistics

Empirical process of residuals for high-dimensional linear models

Enno Mammen

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Abstract

We give a stochastic expansion for the empirical distribution function $\hat{F}_n$ of residuals in a p-dimensional linear model. This expansion holds for p increasing with n. It shows that, for high-dimensional linear models, $\hat{F}_n$ strongly depends on the chosen estimator $\hat{\theta}$ of the parameter $\theta$ of the linear model. In particular, if one uses an ML-estimator $\hat{\theta}_{ML}$ which is ML motivated by a wrongly specified error distribution function G, then $\hat{F}_n$ is biased toward G. For p^2 / n \to \infty$, this bias effect is of larger order than the stochastic fluctuations of the empirical process. Hence, the statistical analysis may just reproduce the assumptions imposed.

Article information

Source
Ann. Statist., Volume 24, Number 1 (1996), 307-335.

Dates
First available in Project Euclid: 26 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1033066211

Digital Object Identifier
doi:10.1214/aos/1033066211

Mathematical Reviews number (MathSciNet)
MR1389892

Zentralblatt MATH identifier
0853.62042

Subjects
Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 62J05: Linear regression 62J20: Diagnostics

Keywords
Empirical processes residuals linear models asymptotics with increasing dimension

Citation

Mammen, Enno. Empirical process of residuals for high-dimensional linear models. Ann. Statist. 24 (1996), no. 1, 307--335. doi:10.1214/aos/1033066211. https://projecteuclid.org/euclid.aos/1033066211


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