Although, the baseline distributions such as Lindley, Weibull, logistic, gamma, exponential etc. have been widely used in statistical modeling, they however have shortcomings when it comes to modeling some new emerging data sets. Some of their shortcomings are: • Limited flexibility: Classical distributions assume specific shapes for the density and hazard rate functions, limiting their flexibility in modeling various types of survival or failure times data. These means that they mail fail to accurately describe or char- acterize data that exhibit different hazard rate patterns such as bathtub, upside- down bathtub and a combination of bathtub and upside-down bathtub. • Lack of tail variation and qualification: Most of the standard distributions often as- sume either exponential decay or normal tail behavior. This may limit them to model data that may exhibit heavier or lighter tails. • Inadequate modeling of dependence: Classical or standard distributions are typically used to model independent observations. However, in many real-world scenarios, data may have some form of dependence such as autocorrelation, spatial dependence, or temporal dependencies. Thus, standard distributions may fail to accurately model such forms of data sets. • Difficulty in handling missing or censored data: Classical or standard distributions are typically used to model complete and continuous data. However, data may some- times be missing or censored, bringing challenges for fitting standard distributions. • Limited support for skewed or asymmetric data: In real world phenomena, data sets may be skewed or asymmetric. In that case, classical distribution such as the normal distribution that only assume symmetry around the mean may bring biased estimates or inappropriate inferences when characterizing such data sets. • Lack of interpretability: Classical or standard distributions often have well-defined mathematical properties that enable straightforward interpretation and facilitate statistical inference. However, new genarated classes and families of distributions may provide better fit to the data but lack the same level of interpretability. This bring challenges in drawing meaningful conclusions. To address these limitations, different transformation methods are utilized to generate new distributions. These new distributions can accommodate various shapes of the hazard rate function commonly observed in real world data sets, hence providing more flexibility in capturing a wide range of data patterns and characteristics. This increased flexibility can help practitioners in selecting appropriate distributions for specific modeling scenar- ios, leading to better decision-making and more accurate predictions. To address the lack ii of interpretability issue, careful model selection and comparison may be used to assess whether the goodness-of-fit justifies the loss in interpretability, this can be done by em- ploying various model selection criteria, such as information criteria e.g., Akaike Informa- tion Criterion, Bayesian Information Criterion, Consistent Akaike Information Criterion or likelihood ratio tests. Another approach to address this issue is through the use of vi- sualization techniques, such as density plots, or probability plots to visually compare the characteristics and behavior of new generated classes and families of distributions to equi- parameter distributions as well as classical distributions. Researchers can also extend these new distributions via different transformation techniques or study them for different reasons, this may lead to improved modeling methodologies and techniques. Additionally, educators can incorporate these new distributions into their curriculum, equipping stu- dents with a broader understanding of statistical modeling techniques and providing them with practical tools for data analysis. In this work, several new families of distributions are introduced. These new families of distributions were developed via the help of different methods of generating distributions. Chapters 2 and 3 involve extensions of Topp-Leone-G, Marshall-Olkin-G and Gompertz-G families of distributions using Type II Exponentiated Half Logistic generator. Chapter 4 uses Odd Burr III and Topp-Leone generators to obtain the new family of distributions. Chapter 5 uses compounding on the Type II Exponential Half Logistic-Topp-Leone-G to obtain a new generalized class of distributions. Chapter 6 are new heavy tailed families of distributions that are obtained via the extensions of the odd Power Generalized Weibull and Generalized Exponentiated Half Logistic generators. These new heavy-tailed families of distributions are particularly useful in finance, insurance, economic and related areas. Statistical properties of these new families of distributions, such as moments, moment generating funtion,incomplete moments, hazard function, quantile function, series repre- sentation of the probability density function, distribution of order statistics, R´enyi entropy, probability weighted moments and moment of residual and reversed residual life are stud- ied in detail. The unknown parameters of these new distributions are estimated using the maximum likelihood estimation technique. The consistency property of the maximum likelihood estimators is examined by considering a Monte Carlo simulation study. Some special distributions from these new family of distributions are also derived by specify- ing the baseline distribution. Applications to real-life data sets from different fields were considered to show the importance and adaptability of these new families of distributions. For the heavy-tailed families, actuarial and risk measures were derived/computed and nu- merically studied. Comparisons with several other heavy-tailed distributions was done to establish that they have heavier tails than those compared to or competing distributions, and then applied to real life data in finance, insurance and economics.

Thesis (PhD of Mathematics and Statistical sciences)---Botswana International University of Science and Technology, 2024