(Cramr-Rao) Let ${U_1, dots, U_n}$ be a random sample from the distribution with c.d.f. $F(u; theta)$, $theta in mathbb{R}^k$. Suppose that $F(u; theta)$ satisfies conditions

(Cramér-Rao) Let $\{U_1, \dots, U_n\}$ be a random sample from the distribution with c.d.f. $F(u; \theta)$, $\theta \in \mathbb{R}^k$. Suppose that $F(u; \theta)$ satisfies conditions that admit the existence of the information matrix. Suppose also that there is an unbiased estimator $\tilde{\theta} = \tilde{\theta}(U_1, \dots, U_n)$ for $\theta_0$ whose variance matrix attains the Cramér-Rao lower bound.

(a) Show that $E[\tilde{\theta} - \theta^*] = 0$ and $Var[\tilde{\theta} - \theta^*]$ is a matrix of zeros, where $\theta^*$ is the Cramér-Rao estimator defined in (14.31).

(b) Show that $\tilde{\theta}$ exists if and only if the average score can be expressed as

except perhaps for a set of outcomes of probability zero.

(c) Give two examples in which such a $\tilde{\theta}$ exists and confirm that the average score satisfies the restriction above.

Ex[L(00)] = 3(00) (400)