نشریه اندیشه آماری
سال بیست و دوم شماره 2 (پیاپی 44، پاییز و زمستان 1396)

Robust regression is an appropriate alternative for ordinal regression, when outliers exist in a given data set. If we have fuzzy observations, using ordinal regression methods can’t model them; In this case, using fuzzy regression is a good method. When observations are fuzzy and there are outliers in the data sets, using robust fuzzy regression methods are appropriate alternatives. In this paper, we propose a fuzzy least square regression analysis. When independent variables are crisp, the dependent variable is fuzzy number and outliers are present in the data set. In the proposed method, the residuals are ranked as the comparison of fuzzy sets. In the proposed method, the residuals are ranked as the comparison of fuzzy sets, and the weight matrix is defined by the membership function of the residuals. Weighted fuzzy least squares estimators (WFLSE) are obtained by using weight matrix. Two examples are discussed and results of these examples are presented.

One of the main goal in the mixture distributions is to determine the number of components. There are different methods for determination the number of components, for example, Greedy-EM algorithm which is based on adding a new component to the model until satisfied the best number of components. The second method is based on maximum entropy and finally the third method is based on nonparametric. In this manuscript it is considered the mixture distributions with Skew-t-Normal components.

There are two main approches to the fuzzy regression (more precisely: regression in fuzzy environment): the least of sum of distances (including two methods of least squared errors and least absolute errors) and the possibilistic method (the method of least whole vaguness under some restrictions). Beside, some heuristic methods have been proposed to deal with fuzzy regression. Some of them are based on a combination of two mentioned approaches. Some of them are based on computational algorithmes. A few of heuristic methods use the fuzzy inference systems. Also, there are some methods based on clustering, artificial neural networks, evolutionary algorithms, and nonparametric procedures.
In this paper, a history and basic ideas of the two main approaches to fuzzy regression are reveiwed, and some heuristic methods in this topic are investigated. Moreover, 10 criterion are proposed by which one can evaluate and compare fuzzy regression models.

Today's manufacturers face increasingly intense competition and to remain profitable they needed to design, develop and produce high reliable products. One way by which manufacturers attract consumers to their products is by providing warranties on the products. Consumers are willing to purchase a longer warranty period product. While maintaining such a policy needs very high cost for manufacturers. Determining the appropriate warranty length becomes an important decision problem for manufacturers. In this article, by Bayesian approach and using an appropriate utility function, we determine the optimal warranty lengths for product with exponential life time distribution.

In this paper, first, we investigate probability density function and the failure rate function of some families of exponential distributions. Then we present their features such as expectation, variance, moments and maximum likelihood estimation and we identify the most flexible distributions according to the figure of probability density function and the failure rate function and finally we offer practical examples of them.

ýýIn this paperý, ýwe introduce a new five-parameter is called Beta-Gompertz Geometric (BGG) distributioný. ýýýýIt can have a decreasingý, ýincreasingý, ýand bathtub-shaped failure rate function depending on its parametersý. ýSome mathematical properties of the new distributioný, ýsuch as the density ýandý ýhazard rate ýfunctionsýý, ýmomentsý, ýmoment generating functioný, ý$Racute{e}nyi$ and Shannon entropyý, ýBonferroni and Lorenz curves and the mean deavations are providedý. ýWe discuss maximum likelihood estimation of the BGG parameters from one observed sampleý. ýAt the endý, ýin order to show the BGG distribution flexibilityý, ýan application using a real data set is ýpresented.ý

ýOne of the factors affecting the statistical analysis of the data is the presence of outliersý. ýThe methods which are not affected by the outliers are called robust methodsý. ýBesides outliersý, ýthe linear dependency of regressor variablesý, ýwhich is called multicollinearityý, ýthe large number of regressor variables with respect to sample sizeý, ýspecially in high dimensional sparse modelsý, ýare problems which result in efficiency reduction of inferences in classical regression methodsý. In this paper, ýwe first study the disadvantages of classical least squares regression methodý, ýwhen facing with outliersý, ýmulticollinearity and high dimensional sparse modelsý. ýThený, ýwe introduce and study robust and penalized regression methodsý, ýas a solution to overcome these problemsý. ýFurthermoreý, ýconsidering outliers and multicollinearity or sparse modelsý, ýsimultaneouslyý, ýwe study penalized-robust regression methodsý. We examine the performance of different estimators introdused in this paper through three different simulation studies. A real data set is also analyzed using the proposed methods.

Maximum likelihood estimation of multivariate distributions needs solving a optimization problem with large dimentions (to the number of unknown parameters) but twoý- ýstage estimation divides this problem to several simple optimizations. ýIt saves significant amount of computational timeý. ýTwo methods are investigated for estimation consistency checký. ýWe revisit Sankaran and Nair's bivariate Pareto distribution as an exampleý. ýTwo data sets (simulated data and real data) have been analyzed for illustrative purposesý.

In this paper, a new probability distribution based on the family of hyperbolic cosine distributions is proposed, and its various statistical and reliability characteristics are investigated. The new category of HCF distributions is obtained by combining a baseline F distribution with the hyperbolic cosine function (Kharazmi and Saadatinik [9]). Based on the base log-logistics distribution, we introduce a new distribution called HCLL and we derive the various properties of the proposed distribution including the moments, quantiles, moment generating function, failure rate function, mean residual lifetime, order statistics and stress-strength parameter. Estimation of the parameters of HCLL for a real data set is investigated by using three maximum likelihood, Bayesian and bootstrap (parametric and non-parametric). In addition, in the application section, by using a realistic data set, the superiority of HCLL model to generalized exponential, Weibull, hyperbolic cosine exponential, gamma, weighted exponential distributions is shown through the different criteria of selection model.