Séminaire Valery B. Nevzorov

Jeudi 12 Mai 2011 (14h00)

Valery B. Nevzorov, St.Petersburg State University, St.-Petersburg, Russia.

Titre : On some regression relationships relating sample means and sucessve maxima.

Résumé : In recent years a number of papers have appeared in which characterizations of a series of distributions in terms of regression properties of order statistics were obtained.
In particular, characterizations of the Student’s t2- distribution in terms of linear regression relationships, including the sample median X2,3 and the sample mean T3, were obtained in 2002. Later, similar results were also established for samples of larger sizes. These characterizations are based, in particular on linear regression relationships relating sample means T2k+1= (X1+...+X2k+1)/(2k+1) and sample medians Xk+1,2k+1 , k=1,2,... .

In 2005 , a new characterization of t2 – distribution was obtained for samples of size 3; it was based on regression properties of successive maxima M1=X1, M2= max{X1,X2} and M3=max{X1,X2 ,X3}. Considering these results for maximal order statistics, it is of interest to examine, for k < r, the relations involving successive maxima and sample means of the form E(Mk |Mr=x)= σx + a and E(Tk|Mr=x)= σx + a.

The linear regression of this kind and families of distributions characterized by such type of relationships are discussed in the talk.