2.27 Suppose that a random sample of size m, X1, X2, . . . , Xm, is available from a continuous cdf FX and a second independent random sample of size n, Y1, Y2, . . . , Yn, is available from a continuous cdf FY. Let Sj be the random variable representing the number of Y blocks I1, I2, . . . , Inþ1 (defined in Section 2.11.2) that contain exactly j observations from the X sample, j¼0, 1, . . . , m.

(a) Verify that S0þS1 þ    þ Sm¼nþ1 and S1þ2S2 þ    þ mSm¼m.

(b) If FX¼FY, show that the joint distribution of S0, S1, . . . , Sm is given by (n þ 1)!
s0!s1!    sm!
m þ n n  
1 .

(c) In particular show that, if FX¼FY, the marginal distribution of S0 is given by n þ 1 s0  
m þ 1 n  s0  
m þ n n  
for s0¼nmþ1, n mþ2, . . . , n. A simple distribution-free test for the equality of FX and FY can be based on S0, the number of blocks that do not contain any X observation. This is the ‘‘empty block’’ test (Wilks, 1962, pp. 446–452).