Observations generated from real world processes are not usually crisp and There is ambiguity in the amount of ‎the observation an presented. Each fuzzy number is determined by a membership function that shows the information of fuzzy data. Then of, obtaining appropriated membership function is important the fuzzy data.‎‏‎‎‎
The membership function has significant influence on fuzzy statistical inference. As a result, the type of membership function is important because the membership function is important in bayesion methods in determining the fuzzy bayes estimator, even when information about the prior function is also inaccurate, we use the the prior function, the role of the membership function in determining the fuzzy bayes estimator is very important.‎
,  ‏‎In this dissertation a new membership function W‎hich called the membership function triangular-Gaussian is introduced and a definition for the optimal membership function is introduced also. Obviously, the value of posterior bayes risk function plays a major role in determining the bayes estimates. Therefore, the criterion for choosing the optimal membership function in fuzzy environments is the minimum posterior bayes risk for possibility bayes estimator. The results show that the triangular-Gaussian new membership function is optimal for the various types of membership functions such as triangular, trapezoidal, Gaussian, Gaussian-triangular and Gaussian-trapezoi

 The weibull distribution is one of the life time flexibility distributions that has wide application in natural phenomena such as people life time‎. ‎In this thesis one family of multi parametric compound distributions is introduced and then with embedding a log-logistic distribution as a baseline distribution‎, a new family can be generated‎. ‎For new family‎, ‎various properties such as density‎, ‎hazard rate‎, ‎moments‎, ‎order statistics and maximum likelihood estimations have studied‎. ‎In order to investigate the effective factors marriage and first birth intervals in Ahvaz city of functional form of regression model was used. of factors and for considering effect of unknown and/or immeasurable factors‎, ‎random effect with normal and skew-normal Azzalini (1985) were added to regression model and the estimation of parametors are obtained using SAS software. ‎In this thesis‎, ‎the influence of several factors on the intervals between marriage and the first birth have been studied‎. The results showed that the new introduced distribution can be employed for the lifetime data.

 ‎Generating ‎continous ‎statistical ‎distributions ‎is ‎taken ‎into ‎consideration ‎in ‎recent ‎research. Which ‎Marshall-Olkin ‎method ‎is ‎one ‎of ‎them. ‎In ‎this ‎thesis ‎Marshall-Olkin ‎distribution ‎with ‎it's ‎features, ‎uch ‎that ‎density, ‎survival ‎and ‎hazard ‎rate ‎functions ‎have ‎been ‎studied. Then‎, with using Generalized Marshall-Olkin and extended weibull, we introducing generalized Marshall-Olkin extended weibull with it's features. We considering new Gompertz generalized Marshall-Olkin family of distributins as a pecial one.The new four parameter distributin contains decreasing, increasing,bathub shape and unimodal failure rate functions. By using simulated data, the accuracy of estimating the parameters of the model is assessed. In the end, with using life time data, has been shown that new distriution has a better fit compare to his similar distributions.

Continuous univariate lifetime models and the methods to generate them have gained much attention in recent studies. Among these methods, the Azzalini’s method for making skewed distributions and generally, methods of combination are of particular importance. In this thesis, different approaches for making continuous univariate lifetime models and especially the Azzalini’s method have been introduced. Then, by using the Azzalini’s modified weighted distribution and merging it with Weibull’s modified family of distribution functions, two new families of distribution functions named as weighted inversed extended Weibull and exponentiated weighted extended Weibull have been introduced. Weighted inversed Weibull and exponentiated weighted Weibull distributions of these families and some of their statistical properties were studied as special cases. The first case is weighted inversed Weibull which is a three-parameter distribution with constant, decreasing, unimodal and bimodal failure rates. The second one is the exponentiated weighted Weibull which is a four-parameter distribution with constant, increasing, unimodal and bathtub shape failure rates. Using simulated data, the accuracy of estimating the parameters of the model is assessed. Also, by using actual data of the lifetime, it has been shown that proposed distributions have a better fitting with experimental data compared to some existing similar distributions.

Ordinal data is a fundamental area in applied sciences. In some studies, it is possible that for a person there are more than one situations and each situation has its own model.
In this case we can consider a selection model for the interaction situations of a person in which each situation also has its own separate regression model.
This separate regression model can be a discrete model such as binary or ordinal. If the linear model is a continuous model, then we can use endogenous switching regression model. In addition, there may be a correlation between response variables in which we can model it using random effects.
If we add random effect to linear predictor, then the resulting model will be a model with random effect. Therefore it is important to choose the proper distribution for the random effect such that it can control the correlation between data.
In this thesis, we study the methods which have been provided for analysis of ordinal responses when the distribution of random effect is normal. As an example, we study data related to payment of premiums of insured social security organization.
The aim is to use switching regression model to investigate the factors affecting the percentage of disability of an insured during the time he/she pays the premiums. The disability may be caused by a non-occupational disease or some occupational accident

In the past decade poisson regression has been used to fit model count data response variables‎. ‎However it must be noted that it is not appropriate to use regression Poisson when a set of data bears superfluity with zero numbers‎. ‎Therefore‎, ‎zero inflated poisson regression or zero inflated generalized poisson regression model is used‎. ‎In this study‎, ‎in order to investigate the effective factors on the number of failed courses and fail grade point average semesters a sample of 2690 students who entered Shahid Chamran University in 88-89 was selected and bivariate zero inflated poisson regression and bivariate zero inflated generalized poisson regression models were applied to them‎. ‎Also‎, ‎considering the probable effects of unknown and/or immeasurable factors‎, ‎random effect is added to the aforementioned models and parameters are assessed using SAS software‎. ‎Four factors of domesticity‎, ‎gender‎, ‎ration and level of education has been taken into account‎, ‎and the degree of each factor’s effect on the number of failed courses and mashroot terms has been estimated‎. ‎Among the applied models‎, ‎the bivariate zero inflated generalized poisson regression model with skew normal random effect proved to possess the least AIC and BIC‎. ‎Finally‎, ‎factors level of education‎, ‎zone ration‎, ‎and gender proved meaningful in case of failed courses and the sole factor level of education was found meaningful in number of fail grade point average semesters.

Mixed models are a type of statisticalmodels that combine two, three or more statistical distribution with weight related.
Mixed models are used for fitting todata that collected from a hetergeneous society.
In this study the method of parameter stimation in mixture modified weibull model with bivariate skew-normal random effect is described and this model fitted to data of life time of water counter in ahvaz city.
To investigate life time of water counter , a random sample of size 3626 from 20000 water counters that was substituted by ahvaz water company in 2012 is selected and the effect of five factor included, counters installation location (city or country), the diameter of the counter, the counter manufacturer, the human factor and water saturation PH on life time of water counter were determined.
In order to consider impact unknown and not measured factors on life time of each counter. A random effect of each components of the mixed model have arrived and to investigate assumed correlation, between effective factors on life time of counters in components of the mixed model, we used random effects correlated.
The model parameters are estimated using SAS.
 

The Weibull distribution is one of the most important models commonly used for analysis of the lifetime data. The statistical inference about the reliability parameter involving independent modified Weibull lie et al (2003) stress and strength is considered. Assuming the shape parameter and one of the scale parameter of the modified Weibull are known when other scale parameter, α, of the Modified Weibull unknown. The Maximum likelihood and uniformly minimum variance unbiased estimators of R are obtained. The performances of the uniformly minimum variance unbiased estimator and Maximum likelihood estimator are compared numerically based on extensive Monte Carlo simulation. Simulation studies indicate that the performance of uniformly minimum variance unbiased estimator is better than the Maximum likelihood estimator. However in practical maximum likelihood estimator is preferred because of the simplicity of its computation. We also read two real data sets of time servicing to customer in two banks and two type of companies, to show the importance of reliability parameter.