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Consider two parallel systems with their component lifetimes following a generalized exponential distribution. In this paper, we introduce a region based on existing shape and scale parameters included in the distribution of one of the systems. If another parallel system's vector of scale parameters lies in that region, then the likelihood ratio ordering between the two systems holds. An extension of this result to the case when the lifetimes of components follow exponentiated Weibull distribution is also presented. 

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One of the most critical challenges in progressively Type-II censored data is determining the removal plan. It can be fixed or random so that is chosen according to a discrete probability distribution. Firstly, this paper introduces two discrete joint distributions for random removals, where the lifetimes follow the two-parameter Weibull distribution. The proposed scenarios are based on the normalized spacings of exponential progressively Type-II censored order statistics. The expected total test time has been obtained under the proposed approaches. The parameters estimation are derived using different estimation procedures as the maximum likelihood, maximum product spacing and least-squares methods. Next, the proposed random removal schemes are compared to the discrete uniform, the binomial, and fixed removal schemes via a Monte Carlo simulation study in terms of their biases; root means squared errors of estimators and their expected experiment times. The expected experiment time ratio is also discussed under progressive Type-II censoring to the complete sampling plan. 

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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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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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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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‎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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‎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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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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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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This paper introduces the dynamic weighted cumulative residual extropy criterion as a generalization of the weighted cumulative residual extropy criterion. The relationship of the proposed criterion with reliability criteria such as weighted mean residual lifetime, hazard rate function, and second-order conditional moment are studied. Also, characterization properties, upper and lower bounds, inequalities, and stochastic orders based on dynamic weighted cumulative residual extropy and the effect of linear transformation on it will be presented. Then, a non-parametric estimator based on the empirical method for the introduced criterion is given, and its asymptotic properties are studied. Finally, an application of the dynamic weighted cumulative residual extropy in selecting the appropriate data distribution on a real data set is discussed.

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Adding parameters to a known distribution is a valuable way of constructing flexible families of distributions. In this paper, we introduce a new model, the modified additive hazard rate model, by replacing the additive hazard rate distribution in the general proportional add ratio model. Next, when two sets of random variables follow the modified additive hazard model, we establish stochastic comparisons between the series and parallel systems comprising these components.

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It was proved about 60 years ago that if a continuous random variable X has an increasing failure rate  then its order statistics will also be increasing failure rate, and this problem remained unproved for the discrete case until recently a proof method using an integral inequality was provided. In this article, we present a completely different method to solve this problem.