An important inequality for distribution of maximum independent random variables is Levy inequality. In this paper, a version of this inequality for weakly negative dependent random variables will be provided. The strong law for dependent random variables has been studied by different authors. In this research, also, the weighted complete convergence for arrays of rowwise negatively dependent random variables that are stochastically bounded will be obtained. complete convergence and strong law for such random variables will result.

Prediction of spatial variability is one of the most important issues in the analysis of spatial data. So predictions are usually made by assuming that the data follow a spatial model. In General, the spatial models are the spatial autoregressive (SAR), the conditional autoregressive and the moving average models. In this paper, we estimated parameter of SAR(2,1) model by using maximum likelihood and obtained formulas for predicting in SAR models, including the prediction within the data (interpolation) and outside the data (extrapolation). Finally, we evaluate the prediction methods by using image processing data.

Two stage linear models are applicable when the data of some dependent and independent variables was obtained at to time stage, and we want to use from the data of two stage for linear model fitting. In this article we introduce multistage and, as a special case, two-stage linear models. Then we obtain the parameter estimation by two methods and show that the estimation are the same for methods. Since the expression of estimations are very complicated we give some R program for computing the parameter estimation of two-stage linear models, then show its application in an illustrative example. Also we propose a very simple computational methods for parameter estimation which did not need to complicated expression and give and R program for it.

This paper first introduces the Kerridge inaccuracy measure as an extension of the Shannon entropy and then the measure of past inaccuracy has been rewritten based on the concept of quantile function. Then, some characterizations results for lifetimes with proportional reversed hazard model property based on quantile past inaccuracy measure are obtained. Also, the class of lifetimes with increasing (decreasing) quantile past inaccuracy property and some of its properties are studied. In addition, via an example of real data, the application of quantile inaccuracy measure is illustrated.

‎Before analyzing a time series data‎, ‎it is better to verify the dependency of the data‎, ‎because if the data be independent‎, ‎the fitting of the time series model is not efficient‎. ‎In recent years‎, ‎the power divergence statistics used for the goodness of fit test‎. ‎In this paper‎, ‎we introduce an independence test of time series via power divergence which depends on the parameter λ‎. ‎We obtain asymptotic distribution of the test statistic‎. ‎Also using a simulation study‎, ‎we estimate the error type I and test power for some λ and n‎. ‎Our simulation study shows that for extremely large sample sizes‎, ‎the estimated error type I converges to the nominal α‎, ‎for any λ‎. ‎Furthermore‎, ‎the modified chi-square‎, ‎modified likelihood ratio‎, ‎and Freeman-Tukey test have the most power‎.

‎The performance of a system depends not only on its design and operation but also on the servicing and maintenance of the item during its operational lifetime‎. ‎Thus‎, ‎the repair and maintenance are important issues in the reliability‎. ‎In this paper‎, ‎a repairable k-out-of-n system is considered that starts operating at time 0‎. ‎If the system fails‎, ‎then it undergoes minimal repair and begins to operate again‎. ‎The reliability function‎, ‎hazard rate function‎, ‎mean residual life function and some reliability properties of the system are obtained by using the connection between the concepts of minimal repair and record values‎. ‎Some known stochastic orders are also used to compare the lifetimes and residual lifetimes of two repairable k-out-of-n systems‎. ‎Finally‎, ‎based on the given information about the lifetimes of k-out-of-n systems‎, ‎some prediction intervals for the lifetime of the proposed repairable system are obtained‎.

Redundancy and reduction are two main methods for improving system reliability. In a redundancy method, system reliability can be improved by adding extra components  to some original components of the system. In a reduction method, system reliability increases by reducing the failure rate at all or some components of the system. Using the concept of reliability equivalence factors, this paper investigates equivalence between the reduction and redundancy methods. A closed formula is obtained for computing the survival equivalence factor. This factor determines the amount of reduction in the failure rate of a system component(s) to reach the reliability of the same system when it is improved. The effect of component importance measure is also studied in our derivations. 

‎Modeling and efficient estimation of the trend function is of great importance in the estimation of variogram and prediction of spatial data. In this article, the support vector regression method is used to model the trend function. Then the data is de-trended and the estimation of variogram and prediction is done. On a real data set, the prediction results obtained from the proposed method have been compared with Spline and kriging prediction methods through cross-validation.  The criterion for choosing the appropriate method for prediction is to minimize the root mean square of the error. The prediction results for several positions with known values were left out of the data set (for some reason) and were obtained for new positions. The results show the high accuracy of prediction (for all positions and elimination positions) with the proposed method compared to kriging and spline.

In this paper, based on the concept of $alpha$-values of fuzzy random variables, the fuzzy moving average model of order $q$ is introduced. In this regard, first, the definitions of variance, covariance, and correlation coefficient between fuzzy random variables are presented, and their properties are investigated. In the following, while introducing the fuzzy moving average model of order $q$, this model's autocovariance and autocorrelation functions are calculated. Finally, some examples are presented for the obtained results.