4.2 This is regarding Example 4.3.

(a) Show that l(0) = −

n i=1

{Xi + 2 log(1 + e

−Xi )}, l



(0) =

n i=1 1 − e

−Xi 1 + e−Xi

,

l 
(θ) = −2 n i=1 eθ−Xi (1 + eθ−Xi )2 .

(b) Show that n −1l(0)
−→P a as n→∞, where a is a positive constant.

(c) Show that n −1/2l 
(0)
−→d N(0, σ2) as n→∞, and determine σ2.

(d) Show that there is a sequence of positive random variables ξn and a constant c > 0 such that ξn −→P

b, where b is a positive constant, and ξnn ≤ sup θ
|l 
(θ)| ≤ cn.