Thesis (Ph.D.)--University of Rochester. School of Medicine & Dentistry. Dept. of Biostatistics and Computational Biology, 2016.
Most statistical applications involve distributional assumptions such as the Exponential,
Weibull, Gamma and the most common Gaussian distribution. In practice,
the Gaussian assumption is often unsatisfactory because the data call for a
model which is unimodal, non-negative and right-skewed. Furthermore, the entire
Gaussian theory is closely tied to the mean. However, for a right-skewed data set,
the mode better represents the rational anticipation than the mean because it is the
most likely value and more appropriate for predictive purposes. Therefore, Mudholkar
et al. (2015) proposed a new distribution family, namely the Mode Centric
M-Gaussian family, which is a right-skewed Gaussian twin distribution with support
(0;1) and uses the population mode as the centrality parameter. In this dissertation,
we extensively discuss the theory and applications of the newly proposed
M-Gaussian distribution. The structure of the dissertation is as follows:
In Chapter 1, the essentials, namely the concept of R-symmetry, the definition
of M-Gaussian distribution, the roles of the mode and harmonic variance as,
respectively, the centrality and dispersion parameters of the M-Gaussian distribution,
are introduced. The pivotal role of the M-Gaussian family in the class of Rsymmetric
distributions and the estimation, testing and characterization properties
are discussed. The similarities between the Gaussian and the M-Gaussian distribution,
namely the G-MG analogies, are summarized.
In Chapter 2, the coefficient of R-skewness, an R-symmetric analogue of the
coefficient of skewness, is introduced and used to develop a testing procedure for the hypothesis of R-symmetry based on the previous work of Awadalla (2011). The
empirical evaluation of the testing method is conducted, and its potential in testing
the composite goodness-of-fit hypothesis for the M-Gaussian distribution is also
discussed.
In Chapter 3 and 4 of this dissertation, we study the application of the mode
centric M-Gaussian distribution in the framework of sequential analysis when the
distribution is nonnegative, right-skewed and unimodal. We propose a sequential
probability ratio test (SPRT) for both simple and composite hypotheses regarding
the population mode of an M-Gaussian distribution. As compared to the existing
studies for the Inverse Gaussian distribution, we demonstrate that the corresponding
results of the SPRT procedure for the M-Gaussian distribution are mathematically
tractable, analytically simpler and more robust with respect to the possible misspecification
in the assumptions of dispersion parameter.
In Chapter 5 we develop a parametric modal linear regression model based on
the M-Gaussian distributional assumption, which models the conditional mode of a
response Y given a set of predictors X. It differs from standard generalized linear
models (GLM) in that the latter model the conditional mean (as opposed to mode)
of Y as a linear function of X. The estimation, the asymptotic properties and the
significance test procedures of the regression coefficients are discussed. Chapter 6
is given to conclusion and future work.
