The Development of New Life-Time Distributions and Investigation of their Statistical Properties

Abstract

Due to the widespread application of probability calculus in various fields, such as engineering, reliability analysis, economics, medical sciences and others, many probability distributions were derived and have been extensively applied to real world problems. However, these distributions do not model the bath-tub or inverse bath-tub failure rate function. One such example is the Weibull distribution. The Weibull distribution was presented by Wallodi Weibull. This distribution is very famous due to its statistical properties and specifically for modeling the lifetime data with a monotonic failure rate but fails to model non-monotonic failure rates. Researchers have made many attempts to provide new probability distributions and the research still is going on. The main interest in the development of the probability distributions is due to modeling the hazard rate function and flexibility of the probability models. This thesis explores six new probability distributions under two sections with the aim to solve the issue of not having the option of fitting the data with non-monotonic failure rates and the model flexibility. In section one (chapter 3), new techniques have been established for generating probability distributions while the second section (chapter 4) consisting of those probability distributions which have been developed by adding an extra parameter or replacing a variable by a suitable function in the existing probability density functions. A short description of the proposed distributions is as follows The New Alpha Power Transformed Exponential distribution has two non-negative parameters, that is,  (scale) and  (shape). Different mathematical functions are discussed along with various shapes of the failure rate function. The New Flexible Exponential distribution has been described in section 3.4. This distribution can model the non-monotonic hazard rate shapes such as the bath-tub or inverted bath-tub shapes. The Gull Alpha Power Weibull distribution consists of three parameters, that is,  being the scale parameter and  , being the shape parameters. Various statistical properties have been explored such as hazard rate function, survival function, Renyi entropy, moment generating function, and some others. The flexibility of these distributions is compared using applied data sets with other existing distributions by means of different goodness of fit criteria that is AIC, BIC, CAIC, xviii HQIC, Anderson-darling, and Cramer-von Mises test. Moreover, the Simulation experiments are also carried out to test the versatility of all the proposed distributions in this section. In chapter 4 section 4.2 present the Flexible Lomax distribution with three parameters. The behavior of the hazard rate and probability density function are discussed in detail with graphs. No closed form exists for parameters, thus confidence bound is presented on the basis of sample information. The Lomax Exponential distribution is characterized by two positive parameters in section 4.3. Distinct values of the parameters are considered so as to study the behavior of the failure rate function of the Lomax Exponential distribution. The parameter estimation is carried out using the most frequently used approach which is the MLE. Since no closed form of the parameters exists, asymptotic confidence bounds are provided for these parameters. The New Weighted Lomax distribution is explored with two parameters in section 4.4. Specific properties are discussed including estimation of parameters, entropy calculation, and mean residual life function. The utility of the suggested densities in this section were achieved by considering both the real data sets as well as the simulated data.