Generating a New Family of Distributions

Abstract

Probability distributions depict the variability in random phenomena and is the basis for the possible inferences about them in almost all the fields of life. Probability models explain the impact of the understudy variables and the parameters with different aspects providing useful predictions which ultimately help to take calculated moves. With the development of the everyday life, several complex situations emerge that strive for the compatible models. In parallel with the construction of new models, the modifications either in the classical models or in their methodologies is an integral part of the research. In this research, we suggest a new technique to develop a new family of distributions by nesting one distribution into another distribution. The resulting models have capability to deal with complex scenarios in a better way as compared to the standard models. Another attractive feature of our contributed distribution is that it is expressed as the sum of weighted exponentiated distributions. This thesis is categorized into seven chapters. Chapter 1 is preparatory in nature and stats provenance about probability distributions. A concise review of the literature on extant techniques of distributions generation and their applications are given. In Chapter 2, a new procedure for generating a new family is available and its general statistical characteristics are gleaned. Some families of generalized distributions for different generating distributions are tabulated. In Chapter 3, Rayleigh-Lomax and Logistic-Lomax distributions are derived by utilizing suggested technique. The statistical properties including moments, moment generating function, entropy and order statistics are derived. The maximum likelihood estimation technique is used to estimate parameters. The presented models are conformed on real-life data sets to highlight its compatibility. In Chapter 4, discrete analogue of an expanded family based upon some truncated probability distributions is developed by using the described methodology and its various statistical properties have been studied. The parameters are again estimated by using maximum likelihood estimation technique. Simulation study is conducted to explore the efficiency of suggested distribution. In Chapter5, bivariate distribution using two independent models developed by derived methodology is constructed to study the availability of system. Reliability measures for the speculated bivariate model are derived. Parameter estimation using maximum likelihood xi estimation and Bayesian technique is determined. In Chapter 6, Weibull-X family of distributions is obtained using projected methodology. An acceptance sampling plan based upon run length of confirming items is designed. A simulation study along with real data analysis is performed to justify the efficacy of constructed plan. In Chapter 7 the thesis is summarized and some new directions for future research are highlighted.