Performance of U-Statistics Having Kernels of Degree Higher Than Two in Inference Problems with Applications

Date of Completion

January 2012

Keywords

Statistics

Degree

Ph.D.

Abstract

This dissertation addresses some interesting inferential problems for the location and scale using the role of U-Statistics. It primarily focuses on using the U-statistics for estimating population standard deviation for the distributions where certain moments exist. We develope new test statistics for a one-sample problem which in some cases have better efficiency than the customary sign test. Next, we obtain percentile points of Gini's mean difference (GMD) based test statistics which can be used for testing the population mean when observations come from a normal distribution with suspect outliers. ^ We implement the full spectrum of Hoeffding's (1948, 1961) U-statistics for different inference problems. In the first case, we consider the unbiased estimation of a population standard deviation σ using GMD in the face of observing possible outliers when random samples arrive from a normal population. Nair (1936) proposed a suitable multiple of GMD that could be used to estimate a unbiasedly and explored its role with regard to robustness issues. ^ We introduce a technique based on U-statistics of higher orders, by modifying the GMD to come up with new unbiased estimators of σ. They are more efficient than the unbiased form of GMD. Next, we have indicated how a similar extension of the degree of a kernel for sample variance does not lead to a new estimator of σ2. In the process, we come up with an entirely new interpretation of S2. ^ This new technique is interesting in its own right and we demonstrate this by additionally constructing new competing nonparametric tests in the case of both one-sample and two-sample location problems. ^ We have provided the percentile points of GMD based test statistics which can be utilized for testing the mean when observations come from a normal distribution with suspect outliers and have successfully implemented our methodology on a realistic dataset. ^ We conclude that the newly developed methodologies have worked remarkably well, both theoretically as well as practically. Such methodologies are rich enough to feed into a wide range of future research problems of practical significance. ^

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