Adaptive estimation and hypothesis testing methods

Dönmez, Ayça
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 equally efficient. Moreover, MLEs and MMLEs are numerically very close to one another. For calculating MLEs and MMLEs, the functional form of the underlying distribution has to be known. For machine data processing, however, such is not the case. Instead, what is reasonable to assume for machine data processing is that the underlying distribution is a member of a broad class of distributions. Huber assumed that the underlying distribution is long-tailed symmetric and developed the so called M-estimators. It is very desirable for an estimator to be robust and have bounded influence function. M-estimators, however, implicitly censor certain sample observations which most practitioners do not appreciate. Tiku and Surucu suggested a modification to Tiku’s MMLEs. The new MMLEs are robust and have bounded influence functions. In fact, these new estimators are overall more efficient than M-estimators for long-tailed symmetric distributions. In this thesis, we have proposed a new modification to MMLEs. The resulting estimators are robust and have bounded influence functions. We have also shown that they can be used not only for long-tailed symmetric distributions but for skew distributions as well. We have used the proposed modification in the context of experimental design and linear regression. We have shown that the resulting estimators and the hypothesis testing procedures based on them are indeed superior to earlier such estimators and tests.


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 ...
Bayesian inference in anova models
Özbozkurt, Pelin; Tiku, Moti Lal; Department of Statistics (2010)
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 eng...
Estimation and hypothesis testing in multivariate linear regression models under non normality
İslam, Muhammed Qamarul (Informa UK Limited, 2017-01-01)
This paper discusses the problem of statistical inference in multivariate linear regression models when the errors involved are non normally distributed. We consider multivariate t-distribution, a fat-tailed distribution, for the errors as alternative to normal distribution. Such non normality is commonly observed in working with many data sets, e.g., financial data that are usually having excess kurtosis. This distribution has a number of applications in many other areas of research as well. We use modifie...
Robust estimation and hypothesis testing under short-tailedness and inliers
Akkaya, Ayşen (Springer Science and Business Media LLC, 2005-06-01)
Estimation and hypothesis testing based on normal samples censored in the middle are developed and shown to be remarkably efficient and robust to symmetric short-tailed distributions and to inliers in a sample. This negates the perception that sample mean and variance are the best robust estimators in such situations (Tiku, 1980; Dunnett, 1982).
Regression analysis with a dtochastic design variable
Sazak, HS; Tiku, ML; İslam, Muhammed Qamarul (Wiley, 2006-04-01)
In regression models, the design variable has primarily been treated as a nonstochastic variable. In numerous situations, however, the design variable is stochastic. The estimation and hypothesis testing problems in such situations are considered. Real life examples are given.
Citation Formats
A. Dönmez, “Adaptive estimation and hypothesis testing methods,” Ph.D. - Doctoral Program, Middle East Technical University, 2010.