Let Xi = + Ui, and suppose that the joint density f of the Us is

Let Xi = ξ + Ui, and suppose that the joint density f of the U’s is spherically symmetric, that is, a function of U2 i only, f(u1,...,un) = q(

u2 i ) .

Show that the null distribution of the one-sample t-statistic is independent of q and hence is the same as in the normal case, namely Student’s t with n − 1 degrees of freedom. Hint: Write tn as n1/2X¯n/
,X2 j ,(Xi − X¯n)2/(n − 1)X2 j , and use the fact that when ξ = 0, the density of X1,...,Xn is constant over the spheres x2 j = c and hence the conditional distribution of the variables Xi/
,X2 j given X2 j = c is uniform over the conditioning sphere and hence independent of q. Note. This model represents one departure from the normaltheory assumption, which does not affect the level of the test. The effect of a much weaker symmetry condition more likely to arise in practice is investigated by Efron (1969).
Section 5.3