Bounds on convex reliability functions with known first moments

In this paper, upper and lower bounds are derived for convex reliability functions (or survival functions) with known first n moments. The case where the mean and the variance are given (n = 2) is discussed in details and explicit expressions are provided. Extensions for n ? 3 moments are described. Comparisons with existing bounds are performed.     

The generalized HNBUE (HNWUE) class of life distributions

This paper examines the properties of a new class of life distributions (and its dual class), named GHNBUE (GHNWUE) whose members have a coefficient of variation less than (greater than) or equal to one. We characterize the GHNBUE (GHNWUE) property by using the Laplace transform. Several interesting shock models leading to the GHNBUE (GHNWUE) property are studied. These include both homogeneous and nonhomogeneous Poisson processes governing the arrival of shocks. A certain cumulative damage model is also investigated. We also examine whether the GHNBUE (GHNWUE) property is preserved under the reliability operations: (i) Convolution, (ii) mixtures and (iii) formation of coherent systems.

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Zusammenfassung Diese Arbeit untersucht Eigenschaften einer neuen Klasse von Lebensdauer-Verteilungen (und deren dualen Klasse), welche GHNBUE (bzw. GHNWUE) genannt wird und deren Elemente einen Variationskoeffizienten 1 (bzw. 1) haben. Wir charakterisieren die GNBUE (GHNWUE) Eigenschaft mit Hilfe der Laplace-Transformierten der Verteilung. Es werden verschiedene interessente Schockmodelle, welche zur GHNBUE (GHNWUE) Eigenschaft führen, studiert. Als Ankunftsprozesse der Schocks verwenden wir homogene und inhomogene Poisson-Prozesse. Auch ein gewisses additives Schadensmodell wird untersucht. Wir befassen uns auch mit der Frage, ob die GHNBUE (GHNWUE) Eigenschaft unter folgenden Zuverlässigkeitsoperationen erhalten bleibt: 1. Faltung, 2. Mischungen, 3. Bildung kohärenter Systeme.

     

Reliability bounds in DFRA class with known mean and variance

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Reliability bounds for NBUE and NWUE distributions

In [5], many inequalities for NBUE and NWUE life distributions are given. But, since there are some basic observations which do not hold (see Section 1 for details), many results obtained in [5] are not true. In this note we propose some remedies and give some new results.The proect supported by the National Natural Science Foundation of China.     

Tight upper bounds for semi-online scheduling on two uniform machines with known optimum

We consider a semi-online version of the problem of scheduling a sequence of jobs of different lengths on two uniform machines with given speeds 1 and s. Jobs are revealed one by one (the assignment of a job has to be done before the next job is revealed), and the objective is to minimize the makespan. In the considered variant the optimal offline makespan is known in advance. The most studied question for this online-type problem is to determine the optimal competitive ratio, that is, the worst-case ratio of the solution given by an algorithm in comparison to the optimal offline solution. In this paper, we make a further step towards completing the answer to this question by determining the optimal competitive ratio for s between \(\frac{5 + \sqrt{241}}{12} \approx 1.7103\) and \(\sqrt{3} \approx 1.7321\), one of the intervals that were still open. Namely, we present and analyze a compound algorithm achieving the previously known lower bounds.

     

Upper bounds on stop-loss premiums in case of known moments up to the fourth order

In actuarial sciences recently a lot of results have been derived for solving the problem sup {E(X−t) +:r.υ. X >0,EX′ = μ, for i = 1, 2, …, k}, x where μ, i = 1 to k as well as t are given. The present contribution solves this problem up to k = 4 analytically.     

Quantifying the error of convex order bounds for truncated first moments

The concepts of convex order and comonotonicity have become quite popular in risk theory, essentially since Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M.J., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168] constructed bounds in the convex order sense for a sum S of random variables without imposing any dependence structure upon it. Those bounds are especially helpful, if the distribution of S cannot be calculated explicitly or is too cumbersome to work with. This will be the case for sums of lognormally distributed random variables, which frequently appear in the context of insurance and finance.In this article we quantify the maximal error in terms of truncated first moments, when S is approximated by a lower or an upper convex order bound to it. We make use of geometrical arguments; from the unknown distribution of S only its variance is involved in the computation of the error bounds. The results are illustrated by pricing an Asian option. It is shown that under certain circumstances our error bounds outperform other known error bounds, e.g. the bound proposed by Nielsen and Sandmann [Nielsen, J.A., Sandmann, K., 2003. Pricing bounds on Asian options. J. Financ. Quant. Anal. 38, 449-473].     

Algebraic bounds on standardized sample moments

Best possible bounds are given for standardized sample (or finite population) moments and absolute moments of arbitrary order, generalizing those given by Cramér (1945) and Kirby (1974) on sample skewness and kurtosis.     

具有均值和方差的HNBUE分布类中可靠度的下界

本文导出了具有均值和方差的ＨＮＢＵＥ分布类中可靠度的指数型下界，给出了ＮＢＵ和ＮＢＵＥ分布类中可靠度的新的下界，提供了估计这些分布类中可靠度最低限的方法。     

High moments of two optimal rules of detecting a change in distributions

The CUSUM rule and Shiryayev-Roberts (S-R) rule, proposed by Page and Shiryayev (or Roberts) respectively, are two asymptotically optimal rules of detecting a change in distributions. This paper is concerned with the properties of their moments. In the context of continuous times, explicit formulas of their high moments are established.     

Characterizations of geometric distributions based on moments

Restricted to certain classes of discrete life distributions, and based on moments, conditional binomial moments, order statistics, spacing or record values, characterizations of geometric distributions are given.This project is supported by the National Natural Science Foundation of China Grant No. 1880492.     

Application of the problem of moments to derive bounds on integrals with integral constraints

Recently a lot of results (for a review see Goovaerts et al. (1983)) have been obtained for bounds on stop-loss premiums in case of incomplete information on the claim distribution.As a consequence some extremal distributions (depending on the retention limit) have been characterized. The extremal distributions for the stop-loss ordering in case of fixed values of the retention limit are obtained by means of deep results from the theory of convex analysis. In the present contribution it is shown, by means of some results from the problem of moments, how bounds on integrals with integral constraints can be obtained. We assume only the knowledge of the moments μ₀, μ₁, …, μ_n.     

Remarks on properties of probability distributions determined by conditional moments

Summary In this paper we consider some properties of rotation — invariant distributions onR ⁿ , which are determined by a form of conditional moment of order >0. In particular we prove that the Gaussian distribution is determined uniquely by its conditional moments and we investigate the related question of finiteness of exponential moments. The case of general >0 appears to be more difficult to analyze than the case =2, studied previously by other authors.     

Sharp bounds for Laplacian spectral moments of digraphs with a fixed dichromatic number

The k-th Laplacian spectral moment of a digraph G is defined as ${\sum }_{i=1}^n{\lambda }_i^k$, where ${\lambda }_i$ are the eigenvalues of the Laplacian matrix of G and k is a nonnegative integer. For $k=2$, this invariant is better known as the Laplacian energy of G. We extend recently published results by characterizing the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We also determine sharp bounds for the third Laplacian spectral moment within the special subclass which we define as join digraphs. We leave the full characterization of the extremal digraphs for $k\ge 3$ as an open problem.     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

STRUCTURES OF SYSTEMS WITH EXPONENTIAL LIFE DISTRIBUTIONS

Abstract. Structures of monotone systems and cold standby systems with     

Characterizations and closure under convolution of two classes of multivariate life distributions

Let ?(η) be the class of positive random vectors $\text{T}$ for which min_1?i?nα_iT_i is IFRA (NBU) for all α_i > 0, i = 1,…,n where n is an arbitrary positive integer. Characterizations of the classes ? and η are obtained and utilized to show that η is closed under convolution and that ? is closed under convolution provided one of the two convoluted vectors has independent components.