The Annals of Statistics

Bootstrapping $M$-Estimators of a Multiple Linear Regression Parameter

Soumendra Nath Lahiri

Abstract

Consider a multiple linear regression model $Y_i = x'_i\beta + \varepsilon_i$, where the $\varepsilon_i$'s are independent random variables with common distribution $F$ and the $x_i$'s are known design vectors. Let $\bar\beta_n$ be the $M$-estimator of $\beta$ corresponding to a score function $\psi$. Under some conditions on $F, \psi$ and the $x_i$'s, two-term Edgeworth expansions for the distributions of standardized and studentized $\bar\beta_n$ are obtained. Furthermore, it is shown that the bootstrap method is second order correct in the studentized case when the bootstrap samples are drawn from some suitable weighted empirical distribution or from the ordinary empirical distribution of the residuals.

Article information

Source
Ann. Statist., Volume 20, Number 3 (1992), 1548-1570.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348784

Digital Object Identifier
doi:10.1214/aos/1176348784

Mathematical Reviews number (MathSciNet)
MR1186265

Zentralblatt MATH identifier
0792.62058

JSTOR
Lahiri, Soumendra Nath. Bootstrapping $M$-Estimators of a Multiple Linear Regression Parameter. Ann. Statist. 20 (1992), no. 3, 1548--1570. doi:10.1214/aos/1176348784. https://projecteuclid.org/euclid.aos/1176348784