Seminaire Pierre Lelong (Analyse) Annee 1975-76 by Pierre Lelong

By Pierre Lelong

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N . 7) we find that Jacobian is equal to J(Yn,UJ = det[ I" OIl>/;p,l=l, ... ,n Cl D2]-1 D2 where 0IlXII(II-1)12 is a matrix of the dimension n x n(n '-1) /2 whose entries are equal to zero. Thus from this expression we have J(Yn,Un ) = IIIYl - Yplcp(u;;i l>p = 1, ... ,1) where cp(uj;i = 1, ... ,/) is some Borel function. Therefore 'V(Y,,) =c I1IYt - Ypl· l>p Let us calculate constant c. To do this, we set j= 1, where m chapter).

4) 1=1 J=1 then Jex - y2 /2dy (21t)1/2 .

Generalized variance is an important measure of spread in multidimensional statistical analysis. In this chapter we develop a general analysis for it, when observed random vectors Xl, ... ,X" do not have density and its components have an arbitrary dependence structure. It is natural that in this case we must use asymptotic methods, when dimension of observed vectors mayaiso increase together with the growing number of observations. Under certain conditions the central limit theorem for empirical generalized variance is proven.

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