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Rigorously characterizing the statistical properties of cyber attacks is an important problem. In this paper, we propose the first statistical framework for rigorously analyzing honeypot-captured cyber attack data. The framework is built on the novel concept of stochastic cyber attack process, a new kind of mathematical objects for describing cyber attacks. To demonstrate use of the framework, we apply it to analyze a low-interaction honeypot dataset, while noting that the framework can be equally applied to analyze high-interaction honeypot data that contains richer information about the attacks. The case study finds, for the first time, that long-range dependence (LRD) is exhibited by honeypot-captured cyber attacks. The case study confirms that by exploiting the statistical properties (LRD in this case), it is feasible to predict cyber attacks (at least in terms of attack rate) with good accuracy. This kind of prediction capability would provide sufficient early-warning time for defenders to adjust their defense configurations or resource allocations. The idea of “gray-box” (rather than “black-box”) prediction is central to the utility of the statistical framework, and represents a significant step towards ultimately understanding (the degree of) the predictability of cyber attacks. [ABSTRACT FROM PUBLISHER]

This paper investigates the stochastic comparison of spacings of record values. The likelihood ratio order, the hazard rate order and the mean residual life order between spacings of record values from one sample sequence are developed. Furthermore, the increasing convex order and the mean residual life order between simple spacings of record values from two sample sequences are built as well. [ABSTRACT FROM AUTHOR]

Let X1, ? , Xnbe independent random variables with Xihaving survival function F?i, i= 1, ? , n, and let Y1, ? ,Ynbe a random sample with common population survival distribution F, where c= ?i=1n?i/n. Let Xn:nand Yn:ndenote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:nis greater than Yn:nin terms of likelihood ratio order. It is also proved that the sample range Xn:n? X1:nis larger than Yn:n? Y1:naccording to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively. [ABSTRACT FROM AUTHOR]

First of all, we express our sincere thanks to all the discussants for their valuable comments and suggestions as well as for their own significant contributions to the area of order statistics in general, and to the topic of stochastic comparison in particular. We shall now provide our response to the comments and suggestions of all the discussants. [ABSTRACT FROM AUTHOR]

In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order. [ABSTRACT FROM AUTHOR]