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July 2017 An unbiased $C_{p}$ type criterion for ANOVA model with a tree order restriction
Yu Inatsu
Hiroshima Math. J. 47(2): 181-216 (July 2017). DOI: 10.32917/hmj/1499392825

Abstract

In this paper, we consider a $C_{p}$ type criterion for ANOVA model with a tree ordering ($\mathrm{TO}$) $\theta_{1}\leq\theta_{j}, (j=2,\ldots,l)$ where $\theta_{1},\ldots\theta_{l}$ are population means. In general, under ANOVA model with the $\mathrm{TO}$, the usual $C_{p}$ criterion has a bias to a risk function, and the bias depends on unknown parameters. In order to solve this problem, we calculate a value of the bias, and we derive its unbiased estimator. By using this estimator, we provide an unbiased $C_{p}$ type criterion for ANOVA model with the $\mathrm{TO}$, called $\mathrm{TO}C_{p}$. A penalty term of the $\mathrm{TO}C_{p}$ is simply defined as a function of an indicator function and maximum likelihood estimators. Furthermore, we show that the $\mathrm{TO}C_{p}$ is the uniformly minimum-variance unbiased estimator (UMVUE) of a risk function.

Citation

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Yu Inatsu. "An unbiased $C_{p}$ type criterion for ANOVA model with a tree order restriction." Hiroshima Math. J. 47 (2) 181 - 216, July 2017. https://doi.org/10.32917/hmj/1499392825

Information

Received: 12 October 2016; Revised: 19 December 2016; Published: July 2017
First available in Project Euclid: 7 July 2017

zbMATH: 06775347
MathSciNet: MR3679889
Digital Object Identifier: 10.32917/hmj/1499392825

Subjects:
Primary: 62F30
Secondary: 62F07

Rights: Copyright © 2017 Hiroshima University, Mathematics Program

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Vol.47 • No. 2 • July 2017
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