6.10

(a) Show that the distribution of the precedence statistic P(i) under the null hypothesis FX ¼ FY, given in Problem 2.28(c), can be expressed as P(P(i) ¼ jjH0) ¼

n m þ n m

j

 

n  1 i  1

 

m þ n  1 j þ i  1

 

¼

i j þ 1 m

j

 

n i

 

m þ n j þ i

  j ¼ 0, 1, . . . ,m These relationships are useful in calculating the null distribution of precedence statistics using tables of the hypergeometric distribution.

(b) Hence, show that the null distribution of the control median test statistic V, with n ¼ 2r þ 1, can be expressed as 2r þ 1 m þ 2r þ 1 m

j

 

2r r

 

m þ 2r j þ r

  j ¼ 0, 1, . . . ,m

(c) Prepare a table of the cumulative probabilities of the null distribution of V for some suitable values of m and n (odd).