The Annals of Statistics

An Ancillarity Paradox Which Appears in Multiple Linear Regression

Lawrence D. Brown

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Abstract

Consider a multiple linear regression in which $Y_i, i = 1, \cdots, n$, are independent normal variables with variance $\sigma^2$ and $E(Y_i) = \alpha + V'_i\beta$, where $V_i \in \mathbb{R}^r$ and $\beta \in \mathbb{R}^r.$ Let $\hat{\alpha}$ denote the usual least squares estimator of $\alpha$. Suppose that $V_i$ are themselves observations of independent multivariate normal random variables with mean 0 and known, nonsingular covariance matrix $\theta$. Then $\hat{\alpha}$ is admissible under squared error loss if $r \geq 2$. Several estimators dominating $\hat{\alpha}$ when $r \geq 3$ are presented. Analogous results are presented for the case where $\sigma^2$ or $\theta$ are unknown and some other generalizations are also considered. It is noted that some of these results for $r \geq 3$ appear in earlier papers of Baranchik and of Takada. $\{V_i\}$ are ancillary statistics in the above setting. Hence admissibility of $\hat{\alpha}$ depends on the distribution of the ancillary statistics, since if $\{V_i\}$ is fixed instead of random, then $\hat{\alpha}$ is admissible. This fact contradicts a widely held notion about ancillary statistics; some interpretations and consequences of this paradox are briefly discussed.

Article information

Source
Ann. Statist. Volume 18, Number 2 (1990), 471-493.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176347602

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176347602

Mathematical Reviews number (MathSciNet)
MR1056325

Zentralblatt MATH identifier
0721.62011

Subjects
Primary: 62C15: Admissibility
Secondary: 62C20: Minimax procedures 62F10: Point estimation 62A99: None of the above, but in this section 62H12: Estimation 62J05: Linear regression

Keywords
Admissibility ancillary statistics multiple linear regression

Citation

Brown, Lawrence D. An Ancillarity Paradox Which Appears in Multiple Linear Regression. The Annals of Statistics 18 (1990), no. 2, 471--493. doi:10.1214/aos/1176347602. http://projecteuclid.org/euclid.aos/1176347602.


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