Let X U[0,b] and Mn maxin Xi . Derive the asymptotic distribution using the following the

Let X »U[0,b] and Mn Æ maxi·n Xi . Derive the asymptotic distribution using the following the steps.

(a) Calculate the distribution F(x) ofU[0,b].

(b) Show Zn Æ n (Mn ¡b) Æ n

µ

max 1·i·n Xi ¡b

¶

Æ max 1·i·n n (Xi ¡b) .

(c) Show that the CDF of Zn is Gn(x) Æ P[Zn · x] Æ

³

F

³

b Å

x n

´´n

.

(d) Derive the limit of Gn(x) as n!1for x Ç 0.
Hint: Use limn!1 ³
1Å
x n ´n Æ exp(x).

(e) Derive the limit of Gn(x) as n!1for x ¸ 0.
(f ) Find the asymptotic distribution of Zn as n!1.