A thesis submitted to the council of the college of Administration Economics\ University of Karbala as partial fulfillment of the
requirements for the degree of Master of Statistics Sciences

By
Shahad Shawkat Majid
Under supervision
Ass. Prof. Dr. Mahdi Wahab Nea’ama Naser Allah

Abstract
In order to achieve the objectives sought by the researcher, he must have all the data and information on that research, and because some data is difficult to obtain or cannot be obtained for reasons of cost, effort and time, he must choose the method of inspection, which ensures the achievement of research objectives at the minimum time Less effort and less cost. Or it may be in many practical issues cannot get the real measurements of the variable in question is expensive and therefore a waste of time, and to solve these problems must be using the method of sampling, which ensures the reduction of time and effort and the cost of data acquisition, Ranked Set Sampling (RSS) is useful in getting a lot of information. Therefore, the purpose of the thesis is to estimate the Frechet distribution parameters of the shape parameter ( ) and the scale parameter (λ) using four methods of estimation: the maximum possible method, the least square method, the Shrinkage method, and then to select the best estimate using some statistical criteria. The simulation method of Monte Carlo was used to determine the superiority of the samples used by comparing the average squares of error and the amount of bias. The results of the simulation experiments showed that the best method for estimating the reliability function in the order of the concentration of the ordered groups is the reduced estimation method of 40% ,followed by the method of least squares  with a preference of 33%, then the method of maximum of Likelihood  with a preference rate of 27%, then the method of  Tom with a preference rate of 0% , and the size of the small samples n = 12,24 The smaller squares method is superior to the other methods. The sizes of the medium and large samples n = 36,54,96 The method of quantified estimation is superior to other methods . The Tom method is not suitable for estimating the reliability function under the group observation. It is observed through the simulation experiments that the estimated reliability of the Tom method is significantly different from the default reliability. The larger the sample size, the smaller squares approach converge with the maximal method, Contracted. Real data representing the failure times of the dental restoration ceramics were also used to apply the best method. The results of the applied side showed that the shrinkage method was appropriate with the actual data of the failure times for the dental restoration ceramics.