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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.

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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.

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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).

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‎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.

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‎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.

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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.

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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.

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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.

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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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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.

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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.