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 $\chi^2_p$.
Publié le : 1985-12-14
Classification:
$M$ estimators,
general linear model,
asymptotic normality,
consistency,
robustness,
regression,
62G35,
62E20,
62J05
@article{1176349744,
author = {Portnoy, Stephen},
title = {Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large; II. Normal Approximation},
journal = {Ann. Statist.},
volume = {13},
number = {1},
year = {1985},
pages = { 1403-1417},
language = {en},
url = {http://dml.mathdoc.fr/item/1176349744}
}
Portnoy, Stephen. Asymptotic Behavior of $M$ Estimators of $p$ Regression Parameters when $p^2 / n$ is Large; II. Normal Approximation. Ann. Statist., Tome 13 (1985) no. 1, pp. 1403-1417. http://gdmltest.u-ga.fr/item/1176349744/