Bayesian inference in anova models

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.


Adaptive estimation and hypothesis testing methods
Dönmez, Ayça; Tiku, Moti Lal; Department of Statistics (2010)
For statistical estimation of population parameters, Fisher’s maximum likelihood estimators (MLEs) are commonly used. They are consistent, unbiased and efficient, at any rate for large n. In most situations, however, MLEs are elusive because of computational difficulties. To alleviate these difficulties, Tiku’s modified maximum likelihood estimators (MMLEs) are used. They are explicit functions of sample observations and easy to compute. They are asymptotically equivalent to MLEs and, for small n, are equal...
Effect of estimation in goodness-of-fit tests
Eren, Emrah; Sürücü, Barış; Department of Statistics (2009)
In statistical analysis, distributional assumptions are needed to apply parametric procedures. Assumptions about underlying distribution should be true for accurate statistical inferences. Goodness-of-fit tests are used for checking the validity of the distributional assumptions. To apply some of the goodness-of-fit tests, the unknown population parameters are estimated. The null distributions of test statistics become complicated or depend on the unknown parameters if population parameters are replaced by ...
A computational approach to nonparametric regression: bootstrapping cmars method
Yazıcı, Ceyda; Batmaz, İnci; Department of Statistics (2011)
Bootstrapping is a resampling technique which treats the original data set as a population and draws samples from it with replacement. This technique is widely used, especially, in mathematically intractable problems. In this study, it is used to obtain the empirical distributions of the parameters to determine whether they are statistically significant or not in a special case of nonparametric regression, Conic Multivariate Adaptive Regression Splines (CMARS). Here, the CMARS method, which uses conic quadr...
Pairwise multiple comparisons under short-tailed symmetric distribution
Balcı, Sibel; Akkaya, Ayşen; Department of Statistics (2007)
In this thesis, pairwise multiple comparisons and multiple comparisons with a control are studied when the observations have short-tailed symmetric distributions. Under non-normality, the testing procedure is given and Huber estimators, trimmed mean with winsorized standard deviation, modified maximum likelihood estimators and ordinary sample mean and sample variance used in this procedure are reviewed. Finally, robustness properties of the stated estimators are compared with each other and it is shown that...
Multiple frame sampling theory and applications
Dalçık, Aylin; Ayhan, Hüseyin Öztaş; Department of Statistics (2010)
One of the most important practical problems in conducting sample surveys is the list that can be used for selecting the sample is generally incomplete or out of date. Therefore, sample surveys can produce seriously biased estimates of the population parameters. On the other hand updating a list is a difficult and very expensive operation. Multiple-frame sampling refers to surveys where two or more frames are used and independent samples are taken respectively from each of the frames. It is assumed that the...
Citation Formats
P. Özbozkurt, “Bayesian inference in anova models,” Ph.D. - Doctoral Program, Middle East Technical University, 2010.