Stochastic Comparisons of Order Statistics and Spacings: A Review

We review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as nonidentically distributed. But most of the time we will be assuming that the observations are independent. The case of independent exponentials with unequal scale parameters as well as the proportional hazard rate model is discussed in detail.

Sample-Spacings-Based Density and Entropy Estimators for Spherically Invariant Multidimensional Data

While the sample-spacings-based density estimation method is simple and efficient, its applicability has been restricted to one-dimensional data. In this letter, the method is generalized such that it can be extended to multiple dimensions in certain circumstances. As a consequence, a multidimensional entropy estimator of spherically invariant continuous random variables is derived. Partial bias of the estimator is analyzed, and the estimator is further used to derive a nonparametric objective function for frequency-domain independent component analysis. The robustness and the effectiveness of the objective function are demonstrated with simulation results.

TP2 DEPENDENCE OF SAMPLE SPACINGS WITH APPLICATIONS

This article investigates TP2 dependence of sample spacings. It is proved that TP2 (RR2) dependence between a general spacing and a nonadjacent order statistic might be characterized by the DLR (ILR) property of the parent distribution, and TP2 dependence between any pair of consecutive spacings might be characterized by the DLR aging property of the population. Furthermore, TP2 dependence between any two consecutive spacings in multiple outliers exponential models is also derived. In addition, some applications in reliability and business auction are presented as well.