The Leader Property in Quasi Unidimensional Cases

Abstract

The following problem was studied: let Zjj≥1 be a sequence of i.i.d. d-dimensional random vectors. Let F be their probability distribution and for every n≥1 consider the sample Sn={Z1,Z2,…,Zn}. Then Zj was called a “leader” in the sample Sn if Zj≥Zk,∀k∈{1,2,…,n} and Zj was an “anti-leader” if Zj≤Zk,∀k∈{1,2,…,n}. The comparison of two vectors was the usual one: if Zj=Zj1,Zj2,…,Zjd,j≥1, then Zj≥Zk means Zji≥Zki, while Zj≤Zk means Zji≤Zki,∀1≤i≤d,∀j,k≥1. Let an be the probability that Sn has a leader, bn be the probability that Sn has an anti-leader and cn be the probability that Sn has both a leader and an anti-leader. Sometimes these probabilities can be computed or estimated, for instance in the case when F is discrete or absolutely continuous. The limits a=liminfan,b=liminfbn,c=liminfcn were considered. If a>0 it was said that F has the leader property, if b>0 they said that F has the anti-leader property and if c>0 then F has the order property. In this paper we study an in-between case: here the vector Z has the form Z=fX where f=f1,…,fd:0,1→Rd and X is a random variable. The aim is to find conditions for f in order that a>0,b>0 or c>0. The most examples will focus on a more particular case Z=X,f2X,…,fdX with X uniformly distributed on the interval [0,1].

Keywords

• stochastic order
• random vector
• multivariate distributions