اختيار افضل تقدير للمعولية الضبابية لتوزيع فريچت

Choosing the Best Estimation for Fuzzy Reliability for Frechet Distribution

Frechet distribution considered one of probability distributions of life time models is an important distribution in the modeling of failure ratescommonly used in the reliability study which are important and effective techniques in evaluating the work of systems and units, it is defined as the probability that the unit will remain valid to work after a period of time (t) for use under normal operating conditions. In the presence of a lot of data, there is a problem of uncertainty in their measurements, which belong to different degrees of belonging to their groups. Therefore, they are called fuzzy data and expressed in fuzzy numbers; therefore, the reliability estimate under these data will lead to the inaccuracy of the estimates obtained. Therefore, it is necessary to generalize the concept of fuzzy in our study of reliability. Reliability under of fuzzy data called the foggy probability of the unit remaining valid work after (t) time. Three methods were used to estimate the parameters of the Frechet distribution, namely the maximum Likelihood, the Bayes, and moment’s method in the case of fuzzy life data, and the use of these estimates in estimating the fuzzy reliability of the distribution.

The theses included two aspects: experimental (simulation) and applied. On the experimental side, the Simulation-Monte Carlo method was adopted for the purpose of generating small sample size data (n =

10,25,35), middle (n = 50,75,100) and large (n = 150,200,500) and several hypothetical values for the shape parameter ( ) and scale parameter ( ) , and used three methods to estimating the parameters, namely the maximum likelihood method, the Bayes method and the moment’s method, were then

used to estimate the parameters obtained in estimating the reliability function of the distribution and then selecting the best estimate of the fuzzy reliability function by comparing by statistical index MSE and MAPE, and we concluded by the simulation results that the fuzzy reliability under Bayes estimates in better than another methods , because they give the lowest average MSE and MAPE and by increase the size of the sample then MSE and MAPE contrasts until it reaches a minimum of sample size n = 500, and this corresponds to the statistical theory, and the moments method is not suitable for estimating Frechet distribution parameters when . When the hypothetical parameter (α) is less than β, the maximum likelihood method is superior to the other methods at sample sizes n = 10.25, And when the hypothetical parameter (α) is greator than β, the maximum likelihood method is superior to the other methods at sample size n = 10.

On the applied side, approximate data were obtained for the duration of the linear accelerator used for the treatment of cancer tumors at the Babel Center for Tumor Therapy of the Department of Health of Babel in the size of sample (63). Four tests of Goodness of fit were applied. The linear accelerator device was more consistent with fuzzy Frechet distribution when estimating the parameters of this distribution in a Bayes method. The probability density curve for the distribution of the Frechet parameter for the parameters estimated by the Bayes method is more suitable for the representation of the linear  accelerator operation data. The curve of the cumulative distribution function of the Frechet parameter for the parameters estimated by the Bayes method is more suitable for the representation of the linear accelerator operation data. The curve of the reliability function estimated in the Bayes method is more appropriate than the other methods.