# Bayesian inference in anova models

2010
Özbozkurt, Pelin
Estimation of location and scale parameters from a random sample of size n is of paramount importance in Statistics. An estimator is called fully efficient if it attains the Cramer-Rao minimum variance bound besides being unbiased. The method that yields such estimators, at any rate for large n, is the method of modified maximum likelihood estimation. Apparently, such estimators cannot be made more efficient by using sample based classical methods. That makes room for Bayesian method of estimation which engages prior distributions and likelihood functions. A formal combination of the prior knowledge and the sample information is called posterior distribution. The posterior distribution is maximized with respect to the unknown parameter(s). That gives HPD (highest probability density) estimator(s). Locating the maximum of the posterior distribution is, however, enormously difficult (computationally and analytically) in most situations. To alleviate these difficulties, we use modified likelihood function in the posterior distribution instead of the likelihood function. We derived the HPD estimators of location and scale parameters of distributions in the family of Generalized Logistic. We have extended the work to experimental design, one way ANOVA. We have obtained the HPD estimators of the block effects and the scale parameter (in the distribution of errors); they have beautiful algebraic forms. We have shown that they are highly efficient. We have given real life examples to illustrate the usefulness of our results. Thus, the enormous computational and analytical difficulties with the traditional Bayesian method of estimation are circumvented at any rate in the context of experimental design.

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Citation Formats
P. Özbozkurt, “Bayesian inference in anova models,” Ph.D. - Doctoral Program, Middle East Technical University, 2010. 