ghobad barmalzan

In this paper, we discuss the hazard rate order of (n-1)-out-of-n systems arising from two sets of independent multiple-outlier modified proportional hazard rates components. Under certain conditions on the parameters and the sub-majorization order between the sample size vectors, the hazard rate order between the (n-1)-out-of-n systems from multiple-outlier modified proportional hazard rates is established.

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.

This paper deals with some stochastic comparisons of convolution of random variables comprising scale variables. Sufficient conditions are established for these convolutions' likelihood ratio ordering and hazard rate order. The results established in this paper generalize some known results in the literature. Several examples are also presented for more illustrations.

This paper discusses the hazard rate order of the fail-safe systems arising from two sets of independent multiple-outlier scale distributed components. Under certain conditions on scale parameters in the scale model and the submajorization order between the sample size vectors, the hazard rate ordering between the corresponding fail-safe systems from multiple-outlier scale random variables is established. Under certain conditions on the Archimedean copula and scale parameters, we also discuss the usual stochastic order of these systems with dependent components.

This paper discusses stochastic comparisons of the parallel and series systems comprising multiple-outlier scale components. Under uncertain conditions on the baseline reversed hazard rate, hazard rate functions and scale parameters, the likelihood ratio, dispersive and mean residual life orders between parallel and series systems are established. We then apply the results for two exceptional cases of the multiple-outlier scale model: gamma and Pareto multiple-outlier components to illustrate the found results.

In this paper, we consider series-parallel and parallel-series systems with independent subsystems consisting of dependent homogeneous components whose joint lifetimes are modeled by an Archimedean copula. Then, by considering two such systems with different numbers of components within each subsystem, we establish hazard rate and reversed hazard rate orderings between the two system lifetimes, and also discuss how these systems age relative to each other in terms of hazard rate and reversed hazard rate functions.

In this paper, we discuss stochastic comparisons of active redundancy at component level versus system level. We also consider series systems in order to compare their lifetimes using the usual stochastic, the hazard rate and the reversed hazard rate orders, for two cases: (i) the spare and parent components are independent and have the same distribution, and then (ii) the spare and parent components are dependent and have not the same distribution. 

This paper examines the problem of stochasticcomparisons of $(n-k+1)$-out-of-$n$ systems, and in the special cases, series and parallel systems with independent and heterogeneous Log-logistic components. Using concepts of majorization, weak supermajorization and p-larger order, we establish usual stochastic order, hazard rate and reversed hazard rate order between these systems. We also discuss the stochastic comparisons of two vector of order statistics arising from two independent and heterogeneous Log-logistic samples.

This paper examines the problem of stochastic comparisons of k-out-of-n systems with independent multiple-outlier scale components. In this regard, we first consider a k-out-of-n system comprising multiple-outlier scale components and then, by using a permanent function, investigate the likelihood ratio order between these systems.

‎In this paper‎, ‎we obtain the usual stochastic order of series and parallel systems comprising heterogeneous discrete Weibull (DW) components‎. ‎Suppose X1,...,Xn and Y1,...,Yn denote the independent component¢s lifetimes of two systems such that Xi ~ DW(bi‎, ‎p i) and Yi ~ DW(b*i‎, ‎p *i), i=1,...,n. We obtain the usual stochastic order between series systems‎ ‎when the vector boldsymbolb is switched to the vector b*with respect to the majorization order‎, ‎and when the vector log (1-p) is switched to the vector log (1-p *) in the sense of the weak supermajorization order‎. ‎We also discuss the usual stochastic order between series systems by using the unordered majorization between the vectors log(1-p) and log (1-p *), and the p-majorization order between the parameters boldsymbolb and b*. It is also shown that the usual stochastic order between parallel systems comprising heterogeneous discrete Weibull components when the vector log p is switched to the vector log p *in the sense of the weak supermajorization order‎. ‎These results enable us to find some lower bounds for the survival functions of a series and parallel systems consisting of independent heterogeneous discrete Weibull components.

‎This paper studies the usual stochastic‎, ‎star ‎and ‎‎convex transform orders of both series and parallel systems comprised of heterogeneous‎ (and dependent) components‎. ‎Sufficient conditions are established for the star ordering between the lifetimes of series and parallel systems consisting of dependent‎‎components having multiple-outlier lomax model‎. ‎We also prove that‎, ‎without any restriction on the parameters‎, ‎the lifetime of a parallel or series systems‎‎with dependent heterogeneous components is smaller than that with dependent‎‎homogeneous components in the sense of the convex transform order‎.‎

The modified proportional hazard rates model, as one of the flexible families of distributions in reliability and survival analysis, and stochastic comparisons of (n-k+1) -out-of- n systems comprising this model have been introduced by Balakrishnan et al. (2018). In this paper, we consider the modified proportional hazard rates model with a  discrete baseline case and investigate ageing properties and preservation of the usual stochastic order, hazard rate order and likelihood ratio order in this family of distributions.

‎This paper examines the problem of stochastic‎ ‎comparisons of series and parallel systems with independent and heterogeneous components generalized linear failure rate‎. ‎First‎, ‎we consider two series system with possibly different parameters and obtain the usual stochastic order between the series systems‎. ‎Next‎, ‎we drive the usual stochastic order between parallel systems‎. ‎We also discuss the usual stochastic order between parallel systems by using the unordered majorization and the weighted majorization order between the parameters on the Ɗп.

‎In this paper‎, ‎under certain conditions‎, ‎the usual stochastic‎, ‎convex and dispersive orders between the smallest claim amounts with independent Weibull claims are discussed‎. ‎Also‎, ‎under conditions on some well-known common copula‎, ‎some stochastic comparisons of smallest claim amounts with dependent heterogeneous claims have been obtained‎.

The aggregate claim amount in a particular time period is a quantity of fundamental importance for proper management of an insurance company and also for pricing of insurance coverages. In this paper, the usual stochastic order between aggregate claim amounts is discussed when the survival function of claims is a increasing and concave. The results established here complete some results of Li and Li (2016).

In this paper, we discuss the usual stochasticý, ýlikelihood ratio, ýdispersive and convex transform order between two parallel systems with independent heterogeneous extended generalized exponential components. ýWe also establish the usual stochastic order between series systems from two independent heterogeneous extended generalized exponential samples. ýFinally, ýwe find lower and upper bounds for the Renyi entropy and cumulative residual entropy of series and parallel systemsý.

In this paper، series and parallel systems، when the lifetimes of their components following the scale model are studied and different stochastic orderings between them are discussed. Moreover، we apply these results to the series and parallel systems consisting of exponentiated Weibull or generalized gamma components. The presented results in this paper complete and extend some known results in the literature.

Suppose there are two groups of random variables، one with independent and non-identical distributed and another with independent and identical distributed. In this paper، for the case when the size of groups are not equal، and all of the underlying random variables have exponential distribution، the necessary and sufficient conditions are obtained for establishing the mean residual life، hazard rate and dispersive orders between the second order statistics of two groups. Moreover، when random variables follow the Weibull distribution، the hazard rate، dispersive and likelihood ratio order between the second order statistics from two groups are investigated.

Suppose there are two groups of independent exponential random variables، where the first group has different hazard rates and the second group has common hazard rate. In this paper، the various stochastic orderings between their sample spacings have studied and introduced some necessary and sufficient conditions to equivalence of these stochastic ordering. Also، for the special case of sample size two، it is shown that the hazard rate function of the second sample spacing is Shcur-concave in the inverse vector of parameters.

Suppose we have a random sample of size n of a population with true density h(.). In general, h(.) is unknown and we use the model f as an approximation of this density function. We do inference based on f. Clearly, f must be close to the true density h, to reach a valid inference about the population. The suggestion of an absolute model based on a few obsevations, as an approximation or estimation of the true density, h, results a great risk in the model selection. For this reason, we choose k non-nested models and investigate the model which is closer to the true density. In this paper, we investigate this main question in the model selection that how is it possible to gain a collection of appropriate models for the estimation of the true density function h, based on Kullback-Leibler risk.

Considering the characteristics of the bivariate normal distribution, in which uncorrelation of two random variables is equivalent to their independence, it is interesting to verify this issue in other distributions; in other words; whether or not the multivariate normal distribution is the only distribution in which uncorrelation is equivalent to independence. This paper aims to answer this question by presenting some concepts and introduce another family in which uncorrelation is equivalent to independence.