109年 - 109 國立臺北大學_碩士班一般入學考試試題_統計學系：數理統計#146988

(a) (5%) What is the mean of \( X_{i} \)? Justify your answer.

(b)(10%) DERIVE the cumulative distribution function of \( Y_n \), which is defined as the minimum of \( X_1, X_2, ..., X_n \). (Do not use the formula for the CDF of order statistics.)

(c) (10%) Does \( Y_{n} \) converge in probability? If yes, what random variable does it converge to? Prove the convergence or divergence.

(d) (10%) Does \( Y_{n} \) converge in distribution? If yes, what distribution does it converge to? Prove the convergence or divergence.

(e) (5%) Why do you think statisticians are interested in the convergence of random variables?

(f) (10%) How does (a) to (e) give you any thought on the estimation for \( \theta \)? Briefly describe your thoughts, and justify your thoughts.

(a) (10%) Find the maximum likelihood estimators \( \hat{\mu},\hat{\theta} \) of \( \mu ,\theta \)

(b) (10%) Find the Rao-Cramer lower bound of \( \hat{\mu} \) and \( \hat{\theta} \), respectively.

(c) (10%) Find the asymptotic distribution of \( \hat{\mu},\hat{\theta} \)

(d) (10%) Find the asymptotic distribution of \( \Phi(x; \hat{\mu}, \hat{\theta}) \). [Hint: Use Delta method and the derivatives with respect to \( \mu \) and \( \theta \) do not need to be expanded.]

(e) (10%) Find the Likelihood ratio test (LR test) for testing \( H0: \theta = \theta0 \) versus \( H0: \theta \neq \theta0 \), where \( \theta_0 \) is a pre-specified value and provide the corresponding critical value for the LR test.