4.35. (Sec. 4.4) Prove that conditional on ZI" = ZI,,' a = 1, ... , n, RZ /0 - RZ) is distributed like T 2/(N* - 1), where T Z = N* i' S-I i based on N* = n observations on a vector X with p* = p - 1 components, with mean vector (c / 0"11)0"(1)

(nc z = EZT,,) and covariance matrix l:ZZ'1 = l:zz -1l/0"1l)0"(1)0"(1)' [Hint: The conditional distribution of Z~Z) given ZI" =ZI" is N[O/O"ll)O"(I)ZI,,' l:22.d.

There is an n X n orthogonal matrix B which carries (z II" .. , Z In) into

(c, ... , c)

and (Zi!"'" Zili) into CY;I"'" l'ill' i = 2, ... , p. Let the new X~ be

(YZ" , ••• , Yp,,}.l