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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.

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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.

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‎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.

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‎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.

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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. 

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In this paper, a new representation of the mean inactivity time of a coherent system with dependent identically distributed (DID) components is obtained. This representation compares the mean inactivity times of two coherent systems. Some sufficient conditions such that one coherent system dominates another system concerning ageing faster order in the reversed mean and variance residual life order are also discussed. These results are derived based on a representation of the system reliability function as a distorted function of the common reliability function of the components. Some examples are given to explain the results.

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In this article, we consider the estimation of R{r,k}= P(X{r:n1} < Y{k:n2}), when the stress X and strength Y are two independent random variables from inverse Exponential distributions with unknown different scale parameters. R{r,k} is estimated using the maximum likelihood estimation method, and also, the asymptotic confidence interval is obtained. Simulation studies and the performance of this model for two real data sets are presented.


 

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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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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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‎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.

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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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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, 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.

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In this paper, we consider the issue of data classification in which the response (dependent) variable is two (or multi) valued and the predictor (independent) variables are ordinary variables. The errors could be nonprecise and random. In this case, the response variable is also a fuzzy random variable. Based on this and logistic regression, we formulate a model and find the estimation of the coefficients using the least squares method. We will describe the results with an example of one independent random variable. Finally, we provide recurrence relations for the estimation of parameters. This relation can be used in machine learning and big data classification.

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Change point detection is one of the most challenging statistical problems because the number and position of these points are unknown. In this article, we will first introduce the concept of change point and then obtain the parameter estimation of the first-order autoregressive model AR(1); in order to investigate the precision of estimated parameters, we have done a simulation study. The precision and consistency of parameters were evaluated using MSE. The simulation study shows that parameter estimation is consistent. In the sense that as the sample size increases, the MSE of different parameters converges to zero. Next, the AR(1) model with the change point was fitted to Iran's annual inflation rate data (from 1944 to 2022), and the inflation rate in 2023  and 2024 was predicted using it.

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This article introduces a new extension of the log-logistic distribution, and its properties and parameter estimation are studied and analyzed. It is shown that adding a parameter to this distribution makes its shape more symmetric and less skewed as the parameter increases. Unlike the original distribution, the moments of the new distribution and its quantile function always exist. Furthermore, it is demonstrated that the reliability measures, such as the hazard rate function, the mean residual life function, and stochastic orderings, are more flexible in the new distribution. Additionally, the parameters of the distribution are estimated using the LLP and ML methods, and the efficiency and consistency of the estimators are evaluated through simulation studies. Finally, the practical applicability of the model is demonstrated by applying the new model to real-world data from airborne equipment and lung cancer patients.

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In the time series analysis, we may encounter situations where some elements of the model are imprecise quantities. One of the most common situations is the inaccuracy of the underlying observations, usually due to measurement or human errors. In this paper, a new fuzzy autoregressive time series model based on the support vector machine approach is proposed. For this purpose, the kernel function has been used for the stability and flexibility of the model, and the constraints included in the model have been used to control the points. In order to examine the performance and effectiveness of the proposed fuzzy autoregressive time series model, some goodness of fit criteria are used. The results were based on one example of simulated fuzzy time series data and two real examples, which showed that the proposed method performed better than other existing methods.