Recurrence relations for moments of record values from generalized extreme value distribution

In this paper some recurrence relations between the moments of record values from the generalized extreme value distribution are established. It is shown that using these recurrence relations, all the single and product moments of all record values can be obtained in a simple recursive manner.     

On a Generalization of Bivariate Cauchy Distribution

This paper addresses a generalization of the bivariate Cauchy distribution discussed by Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]), derived from a trivariate normal distribution with a general correlation matrix. We obtain explicit expressions for the joint distribution function and joint density function, and show that they reduce in a special case to the corresponding expressions of Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]). Finally, we show that this generalized distribution is useful in determining the orthant probability of a bivariate skew-normal distribution of Azzalini and Dalla Valle (1996 Azzalini , A. , Dalla Valle , A. ( 1996 ). The multivariate skew-normal distribution . Biometrika 83 : 715 – 726 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Nonparametric Prediction Intervals Based on Ranked Set Samples

In this article, we develop nonparametric prediction intervals based on generalized ranked set samples using conditional as well as unconditional approaches. The predictions are developed for order statistics from a future sample as well as for order statistics from a future balanced ranked set sample. The effects of ranking errors on the coverage probabilities of these prediction intervals are also examined.     

Sign test based on k-tuple general ranked set samples

ABSTRACT

The sign test based on the k-tuple ranked set samples is discussed here. We first derive the distribution of the k-tuple ranked set sample sign test statistic, and then the asymptotic distribution is also obtained. We then compare its performance with its counterparts based on simple random sample and classical ranked set sample. The asymptotic relative efficiency and the power are then derived. Finally, the effect of imperfect ranking on the procedure is assessed.     

Necessary and sufficient conditions for stochastic orders between (n − r + 1)-out-of-n systems in proportional hazard (reversed hazard) rates model

Consider two (n ? r + 1)-out-of-n systems, one with independent and non-identically distributed components and another with independent and identically distributed components. When the lifetimes of components follow the proportional hazard rates model, we establish a necessary and sufficient condition for the usual stochastic order to hold between the lifetimes of these two systems. For the special case of r = 2, some generalized forms of this result to the hazard rate, dispersive and likelihood ratio orders are also obtained. Moreover, for the case when the lifetimes of components follow the proportional reversed hazard rates model, we derive some similar results for comparing the lifetimes of two systems . Applications of the established results to different situations are finally illustrated.     

Some new results on aggregate claim amounts from two heterogeneous Marshall–Olkin extended exponential portfolios

In this work, we discuss some stochastic comparisons of two aggregate claim amounts. Applications of our results to the value-at-risk and tail-value-at-risk are also mentioned. It is also shown that the aggregate claim amounts of risks exhibiting a weak form of dependence known as positive cumulative dependence (negatively associated) is larger (smaller) in convex order than the corresponding aggregate claim amounts under the theoretical independence assumption. The obtained results show that the correlations between individual risks increase stop-loss premiums corresponding to aggregate claim amounts. The results established here complete and extend the results of Barmalzan, Payandeh Najafabadi, and Balakrishnan (2017 Barmalzan, G., A. T. Payandeh Najafabadi, and N. Balakrishnan. 2017. Ordering results for aggregate claim amounts from two heterogeneous Marshall–Olkin extended exponential portfolios and their applications in insurance analysis. Theory of Probability and its Applications(to appear). [Google Scholar], Theory of Probability and its Applications, to appear).     

Variance-based importance analysis measure for mission reliability of phased mission system

Importance measures are used to estimate the relative importance of components to system reliability. Phased mission systems (PMS) have many components working in several phases with different success criteria, and their component structural importance is distinct in different phases. Additionally, reliability parameters of components in PMS always have uncertainty in practice. Therefore, existing component importance measures based on either the partial derivative of system structure function or component structural importance may have difficulties in PMS importance analysis. This paper presents a simulation method to evaluate the component global importance for PMS based on the variance-based method and the Monte-Carlo method. To facilitate the practical use, we further discuss the correlation relationship between the component global importance and its possible influence factors, and present here a fitting model for evaluating component global importance. Finally, two examples are given to show that the fitting model displays quite reasonable component importance.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

Simultaneous closeness among order statistics to population quantiles

We derive expressions for the probability that an individual order statistic is closest to the target parameter among the order statistics from a complete random sample. Results are given for random variables with bounded and complete support. We then apply these general results to location-scale parameter families of distributions with specific applications to estimation of percentiles. In this case, simultaneous-closeness probabilities depend upon the parameters through the value of p in the percentile and the sample size, n. Results are finally illustrated with the estimation of percentiles for normal and exponential distributions.     

Outer and inner prediction intervals for order statistics intervals based on current records

This paper considers the largest and smallest observations at the times when a new record of either kind (upper or lower) occurs. These are called the upper and lower current records and are denoted by ${R^l_m}$ and ${R^s_m}$ , respectively. The interval ${(R^s_m,R^l_m)}$ is then referred to as the record coverage. The prediction problem in the two-sample case is then discussed and, specifically, the exact outer and inner prediction intervals are derived for order statistics intervals from an independent future Y-sample based on the m-th record coverage from the X-sequence when the underlying distribution of the two samples are the same. The coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Distribution-free prediction intervals as well as upper and lower prediction limits for spacings from a future Y-sample are obtained in terms of the record range from the X-sequence.