Annals of Mathematical Statistics

Use of Maximum Likelihood for Estimating Error Variance from a Collection of Analysis of Variance Mean Squares

R. Gnanadesikan and M. B. Wilk

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Abstract

Given a collection of analysis of variance mean squares, not all of which necessarily have the same degrees of freedom, the present paper describes a method of "mapping" them so as to facilitate the statistical structuring of the mean squares. Even under a null model of no real effects, the mean squares do not have the same distribution because their degrees of freedom may differ, and the ordered mean squares cannot be regarded as the usual order statistics of a sample from a single common distribution. If the ordered mean squares in a general orthogonal analysis of variance are $0 < S_1 \leqq S_2 \leqq \cdots \leqq S_K$ with corresponding degrees of freedom, $\nu_1,\nu_2, \cdots, \nu_K$, then the inferential reference set in the present approach is one obtained by so-called complete conditioning, i.e., repeated sampling from a set of $K$ populations such that the $i$th ordered mean square will be considered to have come from the population associated with $v_i$ degrees of freedom, for $i = 1,2, \cdots, K$. The approach consists of obtaining from each of the ordered mean squares, in turn, a maximum likelihood estimate of a presumed common error variance based on an order statistics formulation which employs complete conditioning of the mean squares. Methods of obtaining the sequence of maximum likelihood estimates as well as two graphical modes of displaying them are described. Illustrative examples are included.

Article information

Source
Ann. Math. Statist., Volume 41, Number 1 (1970), 292-304.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177697208

Digital Object Identifier
doi:10.1214/aoms/1177697208

Mathematical Reviews number (MathSciNet)
MR261759

Zentralblatt MATH identifier
0188.50401

JSTOR
links.jstor.org

Citation

Gnanadesikan, R.; Wilk, M. B. Use of Maximum Likelihood for Estimating Error Variance from a Collection of Analysis of Variance Mean Squares. Ann. Math. Statist. 41 (1970), no. 1, 292--304. doi:10.1214/aoms/1177697208. https://projecteuclid.org/euclid.aoms/1177697208


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