Ph.D. Qualification Examination in Statistics 0.35 September 2006

Let \(X_1, X_2, \ldots \) be an i.i.d. sample from the uniform distribution on \([\theta , 1+\theta ]\), where \(\theta > 0\) is unknown. Is there an unique maximum likelihood estimator of \(\theta \) that is based on \(\{X_i\}_{i=1}^{\infty }\)? Prove or disprove.

Suppose \(X_1, \ldots , X_n\) is an i.i.d. sample from the exponential density with mean \(\theta \), where \(\theta > 0\) is unknown. Find an estimator of \(\theta \) based on \(X_1, \ldots , X_n\). Is your estimator complete? Is it sufficient? Justify your reasoning.

Let \(X\) have the binomial distribution with parameters \(n=10\) and \(p=\theta \) unknown. Suppose we impose a uniform-\((0,1)\) prior on \(\theta \). Compute the posterior mean of \(\theta \). Is it consistent?

Let \(X_1, X_2, \ldots \) denote an i.i.d. sample with common distribution function \(F\). Define \[ F_n(x) = n^{-1}\sum _{j=1}^{n} I\{X_j \leq x\} \] for all \(x \in (-\infty , \infty )\); i.e., \(F_n\) is the empirical distribution function based on the \(X_i\)’s. Prove that for all \(x \in (-\infty , \infty )\) fixed, \[ \operatorname {Var}\bigl (F_n(x)-F(x)\bigr ) \leq \frac {1}{4n}. \] Use this to conclude that \(F_n(x)\) is a consistent estimator for \(F(x)\).

From here on let \(X_1, X_2, \ldots \) denote an i.i.d. sample from an exponential density with mean \(1/\theta \).

Is the MLE a consistent estimator of \(\theta \)? Prove or disprove.

Find a confidence interval for \(\theta \) that has asymptotic level \((1-\alpha )\%\).

Develop carefully the likelihood ratio test for \(H_0:\theta =1\) versus \(H_1:\theta =2\). Discuss the various optimality properties—if any—of your test.

Compute the distributions of: \[ \begin {array}{ll} \text {(a)} & \max _{1\leq i\leq n} X_i,\\ \text {(b)} & \min _{1\leq i\leq n} X_i,\\ \text {(c)} & X_1+\cdots +X_n. \end {array} \]