## Hiroshima Mathematical Journal

### An unbiased $C_{p}$ type criterion for ANOVA model with a tree order restriction

Yu Inatsu

#### 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.

#### Article information

Source
Hiroshima Math. J., Volume 47, Number 2 (2017), 181-216.

Dates
Revised: 19 December 2016
First available in Project Euclid: 7 July 2017

https://projecteuclid.org/euclid.hmj/1499392825

Digital Object Identifier
doi:10.32917/hmj/1499392825

Mathematical Reviews number (MathSciNet)
MR3679889

Zentralblatt MATH identifier
06775347

Subjects
Primary: 62F30: Inference under constraints
Secondary: 62F07: Ranking and selection

#### Citation

Inatsu, Yu. An unbiased $C_{p}$ type criterion for ANOVA model with a tree order restriction. Hiroshima Math. J. 47 (2017), no. 2, 181--216. doi:10.32917/hmj/1499392825. https://projecteuclid.org/euclid.hmj/1499392825