The Annals of Statistics

An Ancillarity Paradox Which Appears in Multiple Linear Regression

Lawrence D. Brown

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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

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

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


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

Admissibility ancillary statistics multiple linear regression


Brown, Lawrence D. An Ancillarity Paradox Which Appears in Multiple Linear Regression. Ann. Statist. 18 (1990), no. 2, 471--493. doi:10.1214/aos/1176347602.

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