Nicholas Eugene

Bio: Nicholas Eugene is an academic researcher from Central Michigan University. The author has contributed to research in topics: Normal distribution & Generalized beta distribution. The author has an hindex of 2, co-authored 2 publications receiving 933 citations.

Beta-normal distribution and its applications

Abstract: This paper introduces a general class of distributions generated from the logit of the beta random variable. A special case of this family is the beta-normal distribution. The shape properties of the beta-normal distribution are discussed. Estimation of parameters of the beta-normal distribution by the maximum likelihood method is also discussed. The beta-normal distribution provides great flexibility in modeling not only symmetric heavy-tailed distributions, but also skewed and bimodal distributions. The flexibility of this distribution is illustrated by applying it to two empirical data sets and comparing the results to previously used methods.

Beta-Normal Distribution: Bimodality Properties and Application

Abstract: The beta-normal distribution is characterized by four parameters that jointly describe the location, the scale and the shape properties. The beta-normal distribution can be unimodal or bimodal. This paper studies the bimodality properties of the beta-normal distribution. The region of bimodality in the parameter space is obtained. The beta-normal distribution is applied to fit a numerical bimodal data set. The beta-normal fits are compared with the fits of mixture-normal distribution through simulation.

Reliability Engineering and System Safety

Abstract: This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].

A new family of generalized distributions

Abstract: Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with a...

A new method for generating families of continuous distributions

Abstract: In this paper, a new method is proposed for generating families of continuous distributions. A random variable $$X$$ , “the transformer”, is used to transform another random variable $$T$$ , “the transformed”. The resulting family, the $$T$$ - $$X$$ family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable $$X$$ . Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.

Families of distributions arising from distributions of order statistics

Abstract: Consider starting from a symmetric distributionF on ℜ and generating a family of distributions from it by employing two parameters whose role is to introduce skewness and to vary tail weight. The proposal in this paper is a simple generalisation of the use of the collection of order statistic distributions associated withF for this purpose; an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution. General properties of the proposed family of distributions are explored. It is argued that two particular special cases are especially attractive because they appear to provide the most tractable instances of families with power and exponential tails; these are the skewt distribution and the logF distribution, respectively. Limited experience with fitting the distributions to data in their four-parameter form, with location and scale parameters added, is described, and hopes for their incorporation into complex modelling situations expressed. Extensions to the multivariate case and to ℜ+ are discussed, and links are forged between the distributions underlying the skewt and logF distributions and Tadikamalla and Johnson'sLU family.