الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
In the study of the robust of Bayesian estimation, the word ”robust” is referred to with many sometimes inconsistent connotations. Here, we use the word ”robust” in Bayesian estimation. from a Bayesian point of view, the concept of robustness is related to the choice of the prior density and loss function, as well as the distribution. Statistical robustness was considered in the first half of the twentieth century. [5] and Tukey [29] explained the need for robust methods. Therefore, in order to clarify the concept of robustness in our study, we must study the properties of some phenomena, such as, an observation X follows a certain probability distribution that depends on some features, and these features are those that distinguish the phenomena from each other. Often these features are unknown and we want to derive some information about them.
In many cases we have additional information from our previous experience
about the parameter and we may notice that it takes different values and there are indications that it changes, and that this change and additional information can be represented by a probabilistic distribution denoted by the symbol.
In the field of statistical inference in general, there are two basic ap- proaches to accomplish the treatments and the required results, which are Bayesian approach and the ordinary or classical approach, or the so-called non-Bayesian. The essential difference between them is that the first deals with the features of the society under study as random variables, while the second looks at the parameters as constants.
The community is summarized in the form of the prior distribution for the parameter, and undergoing the integration process, we get what is called the posterior distribution for the features of the community. from here the problems and difficulties of calculations in the Bayesian approach, pop-up. These are, mainly, that dimensional distributions are usually in a complex or non explicit form and need a lot of effort, numerical processors, and computer capabilities to obtain the required results, and accordingly many researchers resort to using some approximate numerical formulas and numerical integrals.
In the last three decades, and due to the great progress in computer tech-
nologies, it was found that Bayes’ approach was very popular with researchers in the treatment and analysis of various phenomena due to the advantages of this approach. The most important of them is that it provides the re- searcher with more information about what is contained in the sample used in classical inference. Recently, this problem has been overcome by using the Markov Chain Monte Carlo method, (MCMC).
The Markov Chain Monte Carlo method is considered one of the most important methods used in this thesis to obtain an estimate of the param- eters of the community by applying the Bayes method, in addition to some other methods that depend on numerical approximations such as the Lind- ley approximation. The other approach is the non-Bayesian approach, which depends mainly on the capabilities of the greatest possibility:
Maximum likelihood estimators is one of the most famous and important methods used in the non-Bayesian approach. The idea of making statistical inferences using incomplete samples is a very important topic in the field of statistics in general and the field of life-test studies in particular.
There are many data monitoring systems, where each of these systems deals with a case of the resulting cases in actual experiments. One of the most important and most famous control systems used is the progressive
censoring system and the subsequent modifications to this system, is the progressive type-II censoring It is one of the monitoring systems used in this thesis.
In some previous research in the field of statistical estimation, many re- searchers use the error squared function as a loss function due to the ease of obtaining the estimators based on it (since the Bayes estimator in this case is simply an average of the dimensional distribution of the unknown parameter).
However, this trend has been encountered by many researchers, as the nature of the error squared function, which is a symmetric square error loss function, gives equal importance to the cases of the highest and lowest esti- mate.
Therefore, in this thesis, we will study asymmetric loss functions such as the Linear-Exponential loss function (LINEX). One of the most important properties of this function is that it is asymmetric around the origin, as its shape approaches the shape of the exponential function on one side of the origin and approach the linear form on the other side.
We will also study the general entropy loss function. Inverted distributions are considered to have a wide range of applications; in problems related to
econometrics, survey sampling, engineering sciences, medical research prob- lems and life testing. Many researchers have developed many extensions and modified forms of inverted distributions and their applications; for exam- ple, the inverted kumaraswamy distribution, exponentiated inverted Weibull distribution, the generalized inverted exponential distribution, and the ex- ponentiated inverse Rayleigh distribution.
The main objective of this thesis is to obtain the robustness of the esti- mates of the parameters of the distributions, as well as to compare them with the exact values. This is based on samples with progressive type II censoring. Using the two approaches described above, namely the Bayes approach and the classical approach, and comparing their results