arXiv subjects
Publications and source records attributed to Zhengjun Zhang.
Discussion of "Estimating the historical and future probabilities of large terrorist events" by Aaron Clauset and Ryan Woodard [arXiv:1209.0089].
For a non-stationary or non-ergodic marked point process (MPP) on $\R^d$, the definition of averages becomes ambiguous as the process might have a different stochastic behavior in different realizations (non-ergodicity) or in different areas of the observation window (non-stationarity). We investigate different definitions for the moments, including a new hierarchical definition for non-ergodic MPPs, and embed them into a family of weighted mean marks. We point out examples of application in which different weighted mean marks all have a sensible meaning. Further, asymptotic properties of the corresponding estimators are investigated as well as optimal weighting procedures.
The quotient correlation is defined here as an alternative to Pearson's correlation that is more intuitive and flexible in cases where the tail behavior of data is important. It measures nonlinear dependence where the regular correlation coefficient is generally not applicable. One of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases where the Fisher's $Z$-transformation test may fail to reach a right conclusion. Unlike most asymptotic test statistics, which are either normal or $χ^2$, this test statistic has a limiting gamma distribution (henceforth, the gamma test statistic). More than the common usages of correlation, the quotient correlation can easily and intuitively be adjusted to values at tails. This adjustment generates two new concepts--the tail quotient correlation and the tail independence test statistics, which are also gamma statistics. Due to the fact that there is no analogue of the correlation coefficient in extreme value theory, and there does not exist an efficient tail independence test statistic, these two new concepts may open up a new field of study. In addition, an alternative to Spearman's rank correlation, a rank based quotient correlation, is also defined. The advantages of using these new concepts are illustrated with simulated data and a real data analysis of internet traffic.