Total positivity and dependence of order statistics
Enrique de Amo, José Juan Quesada-Molina, Manuel Úbeda-Flores
TL;DR
This paper extends multivariate total positivity (MTP) to directional settings to accommodate negative components and studies dependence structures among order statistics. It introduces two directional generalizations, $MTPP_2(oldsymbol{eta})$ and $MTP_2(oldsymbol{eta})$, and proves their equivalence, along with key invariance properties such as ${oldsymbol{eta}{f X}}$ being $MTP_2({f 1})$ equivalent to ${f X}$ being $MTP_2(oldsymbol{eta})$. The authors then derive dependence results for order statistics, showing PRD and PLRD properties for pairs $(X_{(i)},X_{(j)})$, CIS for the full order-statistic vector, and conditional increasing behavior for gaps and differences under DFR/DFR-related assumptions. These contributions provide a unified directional framework for positive dependence in multivariate settings and illuminate the dependence structure of order statistics under this framework.
Abstract
In this comprehensive study, we delve deeply into the concept of multivariate total positivity, defining it in accordance with a direction. We rigorously explore numerous salient properties, shedding light on the nuances that characterize this notion. Furthermore, our research extends to establishing distinct forms of dependence among the order statistics of a sample from a distribution function. Our analysis aims to provide a nuanced understanding of the interrelationships within multivariate total positivity and its implications for statistical analysis and probability theory.