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

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.