Thesis (M.Sc.) - Cairo University - Faculty of Graduate Studies for Statistical Research - Department of Mathematical Statistics

Statistical probability distributions are commonly applied to describe real world phenomena. They play a prominent role in reliability and survival analysis. The interest in developing more flexible statistical distributions remains strong in statistics. One of the methods for developing a new distribution is studying the inverse of a distribution. The inverted distributions have a wide range of applications; in problems related to econometrics, biological sciences, survey sampling, engineering sciences, medical research and life testing problems. In addition, it is employed in financial literature, environmental studies, survival and reliability theory. On the other hand, inverted (or inverse) distributions have been proposed in literature by using the inverse transformation to well-known random variables, which exhibit different characteristics of the density and hazard rate shape, also they allow applicability to the (lifetime) phenomenon to which the non-inverted distribution cannot explore In the current thesis, a new probability distribution referred to the inverted Topp-Leone distribution is proposed. Some of its statistical properties such as; quantile function, mode, moments, probability weighted moments, incomplete moments, stress-strength model, moments of residual life function, Rényi entropy, and stochastic ordering are provided. Maximum likelihood estimation method based on complete, Type I, and Type II censored samples is considered. Simulation issues are provided to assess the results of the study. Moreover, the results are applied to a real data set.Additionally, a new family of distributions is introduced, called the odd inverted Topp-Leone-H family of probability distributions with new statistical models capable of fitting data in various fields. Different density function shapes include symmetric, unimodal, right skewed, U -shaped, or J-shaped as well as different shapes of hazard rate are observed. Explicit expressions of structural properties of the family, which hold for any baseline model, are studied

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