Four general multivariate stationary extremal Markovian processes

Two general multivariate stationary Markovian process with maximization structure (denoted by Max-AR(1) and MaxI-AR(1)) are developed respectively. Max-AR(1) is a subclass of MaxI-AR(1). The characterization of the Max-AR(1) and MaxI-AR(1) to be stationary is studied. Some properties of the two maximization processes are derived. Two more related general multivariate stochastic Markovian process with minification structure are analogously constructed (denoted by Min-AR(1) and MinI-AR(1)). Some well known maximization and minification processes are special cases of these four extermal Markovian processes. Two of them are simulated and some point estimations are provided as an illustration of the wide application of these four processes.     

Small-sample estimators of the quantiles of the normal,log-normal and Pareto distributions

Estimators of the quantiles of the normal and log-normal distributions are derived. They are more efficient than the established estimators by a wide margin for small samples and high quantiles of the log-normal distribution. Although their evaluation is iterative, it requires only moderate amount of computing, which is not related to the sample size. The method is also applied to the quantiles of the Pareto distribution, but the resulting estimator is more efficient only in some settings. An application to financial statistics, estimating the return on a unit investment in equity markets over a long term, is presented.     

Modeling and predicting extreme cyber attack rates via marked point processes

Cyber attacks have become a problem that is threatening the economy, human privacy, and even national security. Before we can adequately address the problem, we need to have a crystal clear understanding about cyber attacks from various perspectives. This is a challenge because the Internet is a large-scale complex system with humans in the loop. In this paper, we investigate a particular perspective of the problem, namely the extreme value phenomenon that is exhibited by cyber attack rates, which are the numbers of attacks against a system of interest per time unit. It is important to explore this perspective because understanding the statistical properties of extreme cyber attack rates will pave the way for cost-effective, if not optimal, allocation of resources in real-life cyber defense operations. Specifically, we propose modeling and predicting extreme cyber attack rates via marked point processes, while using the Value-at-Risk as a natural measure of intense cyber attacks. The point processes are then applied to analyze some real data sets. Our analysis shows that the point processes can describe and predict extreme cyber attack rates at a very satisfactory accuracy.     

Universal evolution patterns of degree assortativity in social networks

Degree assortativity characterizes the propensity for large-degree nodes to connect to other large-degree nodes and low-degree to low-degree. It is important to describe the forces forming the network and to predict the behavior of dynamic systems on the network. To understand the evolutionary dynamics of degree assortativity, we collect a variety of empirical temporal social networks, and find that there is a universal pattern that the degree assortativity increases at the beginning of evolution and then decreases to a long-lasting stable level. We develop a bidirectional selection model to re-construct the evolution dynamic. In our model, we assume each individual has a social status that—in analogy to Pareto’s wealth distribution —follows a power-law distribution. We assume the social status determines the probability of an interaction between two actors. By varying the ratio of link establishment from within the same status level to across different status levels, the simulated network can be tuned to be assortative or disassortative. This suggests that the rise-and-fall pattern of degree assortativity is a consequence of the different network-forming forces active at different mixing of status. Our simulations indicate that Pareto social status distribution in the population may drive the social evolution in a way of self-optimization to promote the social interaction among individuals and the status gap plays an important role for the assortativity of the social network.     

Optimal estimation for the Pareto distribution

This paper proposes an optimal estimation method for the shape parameter, probability density function and upper tail probability of the Pareto distribution. The new method is based on a weighted empirical distribution function. The exact efficiency functions of the estimators relative to the existing estimators are derived. The paper gives L ₁-optimal and L ₂-optimal weights for the new weighted estimator. Monte Carlo simulation results confirm the theoretical conclusions. Both theoretical and simulation results show that the new estimation method is more efficient relative to several existing methods in many situations.     

Current records and record range with some applications

In a sequence of independent and identically distributed (iid) random variables, the upper (lower) current records and record range are studied. We derive general recurrence relations between the single and product moments for the upper and lower current records based on Weibull and positive Weibull distributions, as well as Pareto and negative Pareto distributions, respectively. Moreover, some asymptotic results for general current records are established. In addition, a recurrence relation and an explicit formula for the moments of record range based on the exponential distribution are given. Finally, numerical examples are presented to illustrate and corroborate theoretical results.     

An Extension of Sampford's Method for Unequal Probability Sampling

Abstract. Sampford's unequal probability sampling method is extended to the case that the inclusion probabilities do not sum to an integer. In this case, the sampling outcome is left open for exactly one randomly chosen unit and that unit gets a new inclusion probability. Three applications are presented. Two of them challenge traditional sampling routines. The simple Pareto sampling design, which was introduced by Rosén in 1997, is also extended. The extended Pareto design is shown to be close to the extended Sampford design.     

Exploring the Equivalence of Two Common Mixture Models for Duration Data

The beta-geometric (BG) distribution and the Pareto distribution of the second kind (P(II)) are two basic models for duration-time data that share some underlying characteristics (i.e., continuous mixtures of memoryless distributions), but differ in two important respects: first, the BG is the natural model to use when the event of interest occurs in discrete time, while the P(II) is the right choice for a continuous-time setting. Second, the underlying mixing distributions (the beta and gamma for the BG and P(II), respectively), are very different—and often believed to be noncomparable with each other. Despite these and other key differences, the two models are strikingly similar in terms of their fit and predictive performance as well as their parameter estimates. We explore this equivalence, both empirically and analytically, and discuss the implications from both a substantive and methodological standpoint.     

The Application of Pareto＇s 80/20 Rule to Law： A Quantitative Analysis Based upon Trials Involving a Plea of Guilty

在法律领域，特别是在诉讼等争端解决领域，司法机关可以通过简化简单多数和优化复杂少数实现最优司法资源的配置。司法机关简化简单多数，是要在保证法律主体基本权利的前提下，本着简化程序而不简化权利的原则降低司法活动的边际成本，提高司法效率并促进司法公正。基于被告人认罪案件审理的定量分析结果，中国刑事司法领域通过简化简单多数来优化复杂少数的资源配置机制还有较大空间。2012年《刑事诉讼法》确认的被告人认罪案件简易程序的实施，需要充分保障被告人获得律师帮助的权利，并提高侦查、起诉和审判等环节的司法效率，才能在整体上科学配置刑事司法资源并有效促进司法公正。