Study on Zografos-Balakrishnan G-Family of Distributions

For any continuous baseline G distribution, Zografos and Balakrishnan (2009) proposed generalized gamma generated G-family of Distributions with an extra positive parameter. Nadarajah and Cordiero (2015) proposed various mathematical properties of such distributions. Lee and Famaye (2014) proposed the T-X family of distributions. As a part of these two families we propose two new gamma generated distributions such as gamma-Gumbel and gamma-normal distribution. Applications are stated as to why we need these distributions and some properties of these distributions have been derived. We have found out various properties of these distributions such as probability density function (pdf),Cumulative distribution function (cdf), survival function and hazard rate function. Expansions of pdf and cdf, asymptotes, quantile function, moment, extreme values, reliability, mean, median, mode, moment generating function, order statistics, Renyi and Shannon Entropy, bivariate generalizations and Maximum likilihood estimator (MLE). We also propose some theorems pertaining to moment of these distributions

Power Generalized KM-Transformation for Non-Monotone Failure Rate Distribution

Lifetime models with a non-monotone hazard rate $\hspace{0.12cm}$ function have a wide range of applications in engineering and lifetime data analysis. There are different bathtub shaped failure rate models that are available in reliability literature. Kavya and Manoharan (2021) introduced a new transformation called KM-transformation which was found to be more useful in reliability and lifetime data analysis. Power generalization technique would be the best approach to deal with a system whose components are connected in series, in which the distribution of the component is KM-transformation of any lifetime model. In this article, we introduce a new lifetime model, Power Generalized KM-Transformation (PGKM) for Non-Monotone Failure Rate Distribution, which shows monotone and non-monotone behavior for the hazard rate function for different choices of values of parameters. We derive the moments, moment generating function, characteristic function, quantiles, entropy etc of the proposed distribution. Distributions of minimum and maximum are obtained. Estimation of parameters of the distribution is performed via maximum likelihood method. A simulation study is performed to validate the maximum likelihood estimator (MLE). Analysis of three sets of real data are given

Study on Zografos-Balakrishnan G-Family of Distributions

For any continuous baseline G distribution, Zografos and Balakrishnan (2009) proposed generalized gamma generated G-family of Distributions with an extra positive parameter. Nadarajah and Cordiero (2015) proposed various mathematical properties of such distributions. Lee and Famaye (2014) proposed the T-X family of distributions. As a part of these two families we propose two new gamma generated distributions such as gamma-Gumbel and gamma-normal distribution. Applications are stated as to why we need these distributions and some properties of these distributions have been derived. We have found out various properties of these distributions such as probability density function (pdf),Cumulative distribution function (cdf), survival function and hazard rate function. Expansions of pdf and cdf, asymptotes, quantile function, moment, extreme values, reliability, mean, median, mode, moment generating function, order statistics, Renyi and Shannon Entropy, bivariate generalizations and Maximum likilihood estimator (MLE). We also propose some theorems pertaining to moment of these distributions

Advances in the Modeling of Heavy-tailed Distributions

Several advances are proposed in connection with the approximation and estimation of heavy-tailed distributions, some of which also apply to other types of distributions. It is first explained that on initially applying the Esscher transform to heavy-tailed density functions such as the Pareto, Student-t and Cauchy densities, one can utilize a moment-based technique whereby the tilted density functions are expressed as the product of a base density function and a polynomial adjustment. Alternatively, density approximants can be secured by appropriately truncating the distributions or mapping them onto compact supports. The validity of these approaches is corroborated by simulation studies. Extensions to the context of density estimation, in which case sample moments are employed in lieu of exact moments are discussed, and illustrative applications involving actuarial data sets are presented. Novel approaches involving making use of the Box-Cox transform in conjunction with empirical saddlepoint density estimates and generalized beta density functions are introduced for determining the endpoints of empirical distribution. Additionally, an iterative algorithm and a technique relying on approximating a function by means of Bernstein polynomials are proposed for obtaining smooth bona fide density functions. Finally, a polynomial adjustment is applied to a bivariate empirical saddlepoint estimate which is obtained from a sample estimate of the bivariate cumulant generating function. A significant contribution of this dissertation resides in the implementation of the proposed methodologies such as the constrained estimation of the four parameters of the generalized beta distribution and the adjusted bivariate empirical saddlepoint density estimation technique in the symbolic computing package Mathematica