Let U1,..., Un be i.i.d. with c.d.f. G(u) = u and let G n denote the empirical

Let U1,..., Un be i.i.d. with c.d.f. G(u) = u and let Gˆ n denote the empirical c.d.f. of U1,..., Un. Define Bn(u) = n1/2

[Gˆ n(u) − u] .

(Note that Bn(·) is a random function, called the uniform empirical process).

(i) Show that the distribution of the Kolmogorov–Smirnov test statistic n1/2dK (Gˆ n, G) under G is that of supu |Bn(u)|.

(ii) Suppose X1,..., Xn are i.i.d. F (not necessarily continuous), and let Fˆ

n denote the empirical c.d.f. of X1,..., Xn. Show that the distribution of the Kolmogorov–

Smirnov test statistic n1/2dK (Fˆ

n, F) under F is that of supt |Bn(F(t))|, where Bn is defined in (i). Deduce that this distribution does not depend on F when F is continuous.