Given i.i.d. random variables X 1 ,...,X n following each of the distributions considered in Examples 2.3.1–2.3.7, obtain the MLE of the corresponding parameters. Show that Un in XXXXXXXXXXis a...

Given i.i.d. random variables X₁,...,X_n
following each of the distributions considered in Examples 2.3.1–2.3.7, obtain the MLE of the corresponding parameters.

Show that Un in (1.5.108) is a symmetric function of X₁,...,X_n
(i.e., it remains invariant under any permutation of (X₁,...,X_n); hence, show that U(X₁,...,X_n) = U(Xn:₁,...,X_n:n).

Show that for a distribution with median
  the equality sign holds when θ ≡ a.

Chapter 3