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In this paper, some reliability concepts have been investigated based on the α-pessimistic and its relationship with the α-cut of a fuzzy number. For this purpose, if the lifetime distribution of the system components is known, using the definition of the scale fuzzy random variable, based on α-pessimistic, some reliability criteria have been investigated. Also, suppose the lifetime distribution of the components is unknown or only the fuzzy observations of the lifetime of the features are available. In that case, the empirical distribution function of the fuzzy data is used to estimate the reliability, and some examples are provided to illustrate the results.
 

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In the excess loss reinsurance risk model, the amount of insurance premium paid by the company is influential in the ruin of that company. In this paper, the premium function is presented based on the expected amount of total payments of the reinsurer to the assigning insurer, the constraint on this function is investigated, and for the claims with any arbitrary distribution, the contour plots are drawn and with presenting optimization algorithm, infinite time ruin probability function will be minimum for different values of initial capital and threshold value. Finally, the excess loss reinsurance risk model with non-exponential claims is considered, and the infinite time ruin probability is calculated with numerical examples. 

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The exact distribution of many applicable statistics could not be accessible in various statistical inference problems. To deal with such an issue in the large sample problem, an approach is to obtain the asymptotic distribution. In this article, we have expressed the asymptotic distribution of multivariate statistics class approximated by averages based on the Taylor expansion. Then, the asymptotic distribution of an empirical Mahalanobis depth-based statistic is obtained, and the statistic is applied to test the scale difference between two multivariate distributions. Simulation studies are carried out to explore the behavior of the asymptotic distribution of the test statistic. A real data example illustrating the use of the test is also presented.

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In this article, it is assumed that the arrival rate of customers to the queuing system M/M/c has an exponential distribution with parameter $lambda$ and the service rate of customers has an exponential distribution with parameter $mu$ and is independent of the arrive rate. It is also assumed that the system is active until time T. Under this stopping time, maximum likelihood estimation and bayesian estimation under general entropy loss functions and weighted error square, as well as under-informed and uninformed prior distributions, the system traffic intensity parameter M/M/c and system stationarity probability are obtained. Then the obtained estimators are compared by Monte Carlo simulation and a numerical example to determine the most suitable estimator.

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This article considers the stress-strength reliability of a coherent system in the state of stress at the component level. The coherent series, parallel and radar systems are investigated. For 2-component series or parallel systems and radar systems, this reliability based on Exponential distribution is estimated by maximum likelihood, uniformly minimum variance unbiased and Bayes methods. Also, simulation studies have been done to check estimators' performance, and real data are analyzed.
 

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In multiple regression analysis, the population multiple correlation coefficient (PMCC)  is widely used to    measure the correlation between a variable and a set of variables. To evaluate the existence or non-existence of this type of correlation, testing the hypothesis of zero  PMCC can be very useful. In high-dimensional data, due to the singularity of the sample covariance matrix, traditional testing procedures to test this hypothesis lose their applicability. A simple test statistic was proposed for zero  PMCC  based on a plug-in estimator of the sample covariance matrix inverse. Then, a permutation test was constructed based on the proposed test statistic to test the null hypothesis. A  simulation study was carried out to evaluate the performance of the proposed test in both high-dimensional and low-dimensional normal data sets. This study was finally ended by applying the proposed approach to mice tumour volumes data.

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Record values have many applications in reliability theory, such as the shock and minimal repairs models. In this regard, many works have been done based on records in the classical model. In this paper, the records are studied in the geometric random model. The concept of the mean residual of records is defined in the random record model and some of its properties are investigated in the geometric random record model. Then, it is shown that the parent distribution can be characterized by using the sequence of the mean residual of records in a geometric random model. Finally, the application of the characterization results to job search models in labor economics is mentioned.

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Cancer progression among patients can be assessed by creating a set of gene markers using statistical data analysis methods. Still, one of the main problems in the statistical study of this type of data is the large number of genes versus a small number of samples. Therefore, it is essential to use dimensionality reduction techniques to eliminate and find the optimal number of genes to predict the desired classes accurately. On the other hand, choosing an appropriate method can help extract valuable information and improve the machine learning model's efficiency. This article uses an ensemble learning approach, a random support vector machine cluster, to find the optimal feature set. In the current paper and in dealing with real data, it is shown that via randomly projecting the original high-dimensional feature space onto multiple lower-dimensional feature subspaces and combining support vector machine classifiers, not only the essential genes are found in causing prostate cancer, but also the classification precision is increased.

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The analysis of spatio-temporal series is crucial but a challenge in different sciences. Accurate analyses of spatio-temporal series depend on how to measure their spatial and temporal relation simultaneously. In this article, one-sided dynamic principal components (ODPC) for spatio-temporal series are introduced and used to model the common structure of their relation. These principal components can be used in the data set, including many spatio-temporal series. In addition to spatial relations, trends, and seasonal trends, the dynamic principal components reflect other common temporal and spatial factors in spatio-temporal series. In order to evaluate the capability of one-sided dynamic principal components, they are used for clustering and forecasting in spatio-temporal series. Based on the precipitation time series in different stations of Golestan province, the efficiency of the principal components in the clustering of hydrometric stations is investigated. Moreover, forecasting for the SPI index, an essential indicator for detecting drought, is conducted based on the one-sided principal components.

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‎In this paper, under adaptive hybrid progressive censoring samples, Bayes estimation of the multi-component reliability, with the non-identical-component strengths, in unit generalized Gompertz distribution is considered. This problem is solved in three cases. In the first case, strengths and stress variables are assumed to have unknown, uncommon parameters. In the second case,  it is assumed that strengths and stress variables have two common and one uncommon parameter, so all of these parameters are unknown. In the third case, it is assumed that strengths and stress variables have two known common parameters and one unknown uncommon parameter. In each of these cases, Bayes estimation of the multi-component reliability, with the non-identical-component strengths, is obtained with different methods. Finally, different estimations are compared using the Monte Carlo simulation, and the results are implemented on one real data set.

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Gaussian random field is usually used to model Gaussian spatial data. In practice, we may encounter non-Gaussian data that are skewed. One solution to model skew spatial data is to use a skew random field. Recently, many skew random fields have been proposed to model this type of data, some of which have problems such as complexity, non-identifiability, and non-stationarity. In this article, a flexible class of closed skew-normal distribution is introduced to construct valid stationary random fields, and some important properties of this class such as identifiability and closedness under marginalization and conditioning are examined. The reasons for developing valid spatial models based on these skew random fields are also explained. Additionally, the identifiability of the spatial correlation model based on empirical variogram is investigated in a simulation study with the stationary skew random field as a competing model. Furthermore, spatial predictions using a likelihood approach are presented on these skew random fields and a simulation study is performed to evaluate the likelihood estimation of their parameters. 

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In this paper, fuzzy order statistics are expressed based on the concept of α-value, and some of its applications in reliability have been examined. For this purpose, if the lifetime distribution of the system components is known, some of the reliability criteria of the $i$th order statistic using the definition of a fuzzy random variable based on the α-value have been investigated. Also, if the lifetime distribution of the components is unknown or only the fuzzy observations of the lifetime of the components are available, the empirical distribution function of the fuzzy data is used to estimate the reliability based on ordinal statistics, and examples are provided to illustrate the results.

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‎In this paper‎, ‎non-parametric inference is considered for $k$-component coherent systems‎, ‎when the‎ ‎system lifetime data is progressively type-II censored‎. ‎In these coherent systems‎, ‎it is assumed that the‎ ‎system structure and system signature are known‎. ‎Based on the observed progressively type-II censored‎, ‎non-parametric confidence intervals are calculated for the quantiles of component lifetime distribution‎. ‎Also‎, ‎tolerance limits for component lifetime distribution are obtained‎. ‎Non-parametric confidence intervals for quantiles and tolerance limits are obtained based on two methods‎, ‎distribution function method and W mixed matrix method‎. ‎Two numerical‎ ‎example is used to illustrate the methodologies developed in this paper‎.

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In this paper, we first introduce semi-parametric heteroscedastic hierarchical models. Then, we define a new version of the empirical likelihood function (Restricted Joint Empirical likelihood) and use it to obtain the shrinkage estimators of the models' parameters in these models. Under different assumptions, a simulation study investigates the better performance of the restricted joint empirical likelihood function in the analysis of semi-parametric heterogeneity hierarchical models. Furthermore, we analyze an actual data set using the RJEL method.

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Often, reliability systems suffer shocks from external stress factors, stressing the system at random. These random shocks may have non-ignorable effects on the reliability of the system. In this paper, we provide sufficient and necessary conditions on components' lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two $(n-1)$-out-of-$n$ systems in two cases: (i) when components are independent, and then (ii) when components are dependent.  

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First, this article defines a meter between fuzzy numbers using the support function. Then, based on the support function, the concepts of variance, covariance, and correlation coefficient between fuzzy random variables are expressed, and their properties are investigated. Then, using the above concepts, the p-order fuzzy autoregressive model is introduced based on fuzzy random variables, and its properties are investigated. Finally, to explain the problem further, examples will be presented and compared with similar models using some goodness of fit criteria.

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In this paper, the entropy characteristics of the lifetime of coherent systems are investigated using the concept of system signature. The results are based on the assumption that the lifetime distribution of system components is independent and identically distributed. In particular, a formula for calculating the Tsallis entropy of a coherent system's lifetime is presented, which is used to compare systems with the same characteristics. Also, bounds for the lifetime Tsallis entropy of coherent systems are presented. These bounds are especially useful when the system has many components or a complex structure. Finally, a criterion for selecting the preferred system among coherent systems based on the relative Tsallis entropy is presented.

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The spatial generalized linear mixed models are often used, where the latent variables representing spatial correlations are modeled through a Gaussian random field to model the categorical spatial data. The violation of the Gaussian assumption affects the accuracy of predictions and parameter estimates in these models. In this paper, the spatial generalized linear mixed models are fitted and analyzed by utilizing a stationary skew Gaussian random field and employing an approximate Bayesian approach. The performance of the model and the approximate Bayesian approach is examined through a simulation example, and implementation on an actual data set is presented.

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Spatial regression models are used to analyze quantitative spatial responses based on linear and non-linear relationships with explanatory variables. Usually, the spatial correlation of responses is modeled with a Gaussian random field based on a multivariate normal distribution. However, in practice, we encounter skewed responses, which are analyzed using skew-normal distributions. Closed skew-normal distribution is one of the extended families of skew-normal distributions, which has similar properties to normal distributions. This article presents a hierarchical Bayesian analysis based on a flexible subclass of closed skew-normal distributions. Given the time-consuming nature of Monte Carlo methods in hierarchical Bayes analysis, we have opted to use the variational Bayes approach to approximate the posterior distribution. This decision was made to expedite the analysis process without compromising the accuracy of our results. Then, the proposed model is implemented and analyzed based on the real earthquake data of Iran.