The concept of R-symmetry, which provides a more applicable framework for nonnegative right-skewed populations, was introduced and developed by Mudholkar and Wang (2007) as an alternative to the mathematically well defined but conceptually opaque notion of IG-symmetry related to the inverse Gaussian distribution. A random variable X ≥ 0 is said to be R-symmetric about R-center &thetas; if its density satisfies f( x&thetas;) = f(&thetas;/x); &thetas; is the mode if f(·) is unimodal. Mudholkar and Wang (2007) noted some moment relationships satisfied by R-symmetric random variables which are useful in estimating the R-center. They also gave some results depending on sums, products and mixtures which may be used to generate R-symmetric densities. Recently, Baker (2008) applied the Cauchy-Schlomilch transform to obtain R-symmetric densities from some common distributions.
An important distribution obtainable by applying either the Cauchy-Schlomilch or the Mudholkar-Wang product method to the half-normal density is the CoGaussian distribution. This distribution, which is central in the class of R-symmetric distributions, shares many analogies with the Gaussian distribution and it is also related to the inverse Gaussian family. In this dissertation, we first study its properties and introduce its generalized version, called the Power CoGaussian distribution.
The moment relationships satisfied by R-symmetric random variables are analogous to the vanishing of odd-ordered central moments of symmetric distributions and yield a coefficient of R-skewness which measures departure from R-symmetry. The properties of this coefficient and its use in testing the R-symmetry assumption are investigated.
The mode, which is the most likely value in the sense of having the highest probability content neighborhood, is a natural parameter of unimodal R-symmetric distributions. We apply some tools of robustness theory and estimation procedures in the presence of threshold parameters to the moment estimator of R-symmetric modes. This moment estimator, which also coincides with the maximum likelihood estimator of the CoGaussian mode, is adapted for the general purpose of estimating modes of right-skewed populations.
The last chapter of the thesis is given to an outline of some ongoing projects involving CoGaussian analogs of the well known Tukey (1957) and Box-Cox (1964) transformations. Preliminary Bayesian and decision theoretic results in the context of CoGaussian modes are also discussed.
|Advisor:||Mudholkar, Govind S., McDermott, Michael P.|
|Commitee:||Groenevelt, Harry, Liang, Hua, Wang, Hongyue|
|School:||University of Rochester|
|Department:||School of Medicine and Dentistry|
|School Location:||United States -- New York|
|Source:||DAI-B 73/07(E), Dissertation Abstracts International|
|Keywords:||Mode, R-center, R-symmetry, Random variables, Unimodality|
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