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December, 1985 Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large; II. Normal Approximation
Stephen Portnoy
Ann. Statist. 13(4): 1403-1417 (December, 1985). DOI: 10.1214/aos/1176349744

Abstract

In a general linear model, $Y = X\beta + R$ with $Y$ and $R n$-dimensional, $X$ a $n \times p$ matrix, and $\beta p$-dimensional, let $\hat\beta$ be an $M$ estimator of $\beta$ satisfying $0 = \sum x_i\psi(y_i - x'_i\beta)$. Let $p \rightarrow \infty$ such that $(p \log n)^{3/2} /n \rightarrow 0$. Then $\max_i|x'_i(\hat{\beta} - \beta)| \rightarrow _P 0$, and it is possible to find a uniform normal approximation for the distribution of $\hat{\beta}$ under which arbitrary linear combinations $a'_n (\hat{\beta} - \beta)$ are asymptotically normal (when appropriately normalized) and $(\hat{\beta} - \beta)'(X'X)(\hat{\beta} - \beta)$ is approximately $\chi^2_p$.

Citation

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Stephen Portnoy. "Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large; II. Normal Approximation." Ann. Statist. 13 (4) 1403 - 1417, December, 1985. https://doi.org/10.1214/aos/1176349744

Information

Published: December, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0601.62026
MathSciNet: MR811499
Digital Object Identifier: 10.1214/aos/1176349744

Subjects:
Primary: 62G35
Secondary: 62E20, 62J05

Rights: Copyright © 1985 Institute of Mathematical Statistics

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Vol.13 • No. 4 • December, 1985
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