26. Stein's two-stage procedure. (i) If mS2/(12 has a X2 = distribution with m degrees of freedom, and if the conditional distribution of Y given S = s is N(O, (12/S2), then Y has Student's r-distribution with m degrees of freedom . (ii) Let XI' X2 , . .. be independently distributed as N(t (12). Let Xo = E:'~IX;lno, S2 = E7~1(X; - Xo)2/ ( no - 1), and let a l = . . . = all o =

a, a"o+1 = . . . = all =

b, and n no be measurable functions of S. Then n S2 La; i-I n L a;(X; - n ;-1 Y= , has Student's distribution with no - 1 degrees of freedom .

(iii) Consider a two-stage sampling scheme ill' in which S2 is computed from an initial sample of size no, and then n - no additional observations are taken. The size of the second sample is such that n = max{ no + 1,[:2] + I} where c is any given constant and where [y] denotes the largest integer y . There then exist numbers a l, ... , an such that al = . . . = a"o' a"o+1 = .. . = an' 1:7_laj = 1, 1:7_la; = c1S 2 • It follows from (ii) that 1:7-1 a,(X; - ~)/ IC has Student's r-distribution with no - 1 degrees of freedom. (iv) The following sampling scheme il2, which does not require that the second sample contain at least one observation, is slightly more efficient than ill for the applications to be made in Problems 27 and 28. Let no, S2, and c be defined as before ; let n = max{ no,[ :2] + I}, a; = lin (i = 1, .. . , n), and X= 1:7_lajX; . Then In(X -~)/S has again the r-distribution with no - 1 degrees of freedom. [(ii): Given S = s, the quantities

a, b, and n are constants, 1:7~laj(X; - ~) = noa( Xo -~) is distributed as N(O, noa 2 0 2 ) , and the numerator of Y is therefore normally distributed with zero mean and variance 0 2 1:;'_1a], The result now follows from (i).