4.14 Related goodness-of-fit test. The Cramrvon Mises type of statistic is defined for continuous F0(x) by v2

4.14 Related goodness-of-fit test. The Cramér–von Mises type of statistic is defined for continuous F0(x) by v2 n ¼

ð1

1

[Sn(x)  F0(x)]2fX(x)dx

(a) Prove that v2 n is distribution free.

(b) Explain how v2 n might be used for a goodness-of-fit test.

(c) Show that nv2 n ¼
1 12n þ
Xn i¼1 F0 X(i) 
2i  1 n   2 This statistic is discussed in Cramér (1928), von Mises (1931), Smirnov (1936), and Darling (1957).