prove statistic about MLE

Question

max IxiI = (-x(1), x(n)) prove it

this is the result for MLE for U(-theta, theta) by tow method and i need to prove

Answer 1

Calculate the likelihood function.

Show that L(θ) = ∏ (1/ (2θ) I _Xi (-θ,θ) = (2θ)^-n Π I _Xi (-θ,θ)

Note that Π I _Xi (-θ,θ) is either 1 or 0. To maximize L(θ) we need this to be = 1.

Π I _Xi (-θ,θ) = P( -θ <= X1 <= θ) * P( -θ <= X2 <= θ) *** P( -θ <= Xn <= θ)

= P( -θ <= X(1) < = X(2) <=...<= X(n) <= θ using order statistics

=P(X(1) >=-θ and X(n) <= θ)

= P-(X(1)<=θ and X(n) <=θ).

To satisfy both inequalities, Take θ_MLE=max(-X(1), X(n))