اﻷداء اﻟﻤﺄﻣﻮر ﺑﻪ و ﺗﻄﺒﻴﻘﺎﺗﻪ اﻟﻔﻘﻬﻴﺔ

In the Islamic Law i. E. Quran and Sunnah the above mentioned topic has been discussed clearly and repeatedly to avoid any ambiguity in dealings in our daily life and agreements made between two parties or governments national or international levels. After intensive study I discussed and elaborated the said issue referring to the Qura’nic verses and sayings of the Prophet (SAW). The matter is of great importance; hence the Muslims Jurists have also given their valuable opinions in accordance with the Islamic Law which have been incorporated also to solve the issue. If we act upon these verdicts, we will be able to select the best among ourselves and form an ideal government and will discharge our duties honestly, and eventually our every act will show our responsibility to perform our duties and to give due share to the right person.

Bayesian Approach to the Mixture Models

This thesis deals with Bayesian inference of mixture densities using censored data. Type-I and type-II mixtures are considered that belong to two or one parameter exponential family. Selection of distribution is made keeping in view the novelty and applicability. These include Inverse Weibull, Pareto type-II, shifted exponential distribution and lastly mixture of Burr type XII and Rayleigh distributions that belong to type-II mixture model. These mixture distributions have not been explored so far in Bayesian setup. Bayes estimators for the parameters of the mixture models are derived in closed forms using type-I right censoring. To conduct Bayesian analysis, Informative (Gamma and Squared Raylegh) priors and non-informative (Uniform and Jeffreys) priors are considered while three loss functions, Squared Error Loss Function, Weighted loss function and Quadratic loss function are employed. A wide simulation study is made to scrutinize the properties of proposed Bayes estimators. Parameters of the mixture model are also tested through hypothesis testing procedure for inverse Weibull and Pareto type- II models. For the inverse Weibull mixture model when all parameters are unknown Bayes estimators can not be obtained in closed forms thus Gibbs sampling and Importance sampling techniques are used to obtain Bayes estimates in this case. Bayesian predictive density is used to obtain Bayes predictive intervals and reliability estimator. Predictive intervals for one and two sample prediction are also obtained that help to predict failure times of future observations. Bayes estimators using limiting form are also derived. Though type-I right censoring is considered throughout the dissertation, however, shifted exponential distribution is also explored through progressive censoring scheme. For the said case, Bayes estimators, credible intervals, Expected test termination time which is considered very useful for life testing experiments, are derived and evaluated. Applications of these mixtures are also presented by applying a real data set in each case.