2004 Qualifying Exam in Statistics

Let \(U_1,U_2,\ldots ,U_n\) be independent identically distributed random variables, uniformly distributed on \([0,1]\). Compute the covariance between \(U_{1,n}=\min _{1\leq i\leq n}U_i\) and \(U_{n,n}=\max _{1\leq i\leq n}U_i\).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed normal \(N(\mu ,\sigma ^2)\) random variables. We wish to test \(H_0:\mu =\mu _0\) against \(H_A:\mu \neq \mu _0\) (\(\sigma \) is unknown). Show that the t-test and the likelihood tests are equivalent.

Let \(X_1\) and \(X_2\) be two independent, identically distributed exponential (1) random variables. Compute the joint density of \(Y_1=X_1+X_2\) and \(Y_2=X_1/(X_1+X_2)\).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed uniform on \([0,\theta ]\), \(\theta >0\), random variables. We wish to test \(H_0:\theta \leq \theta _0\) against \(H_A:\theta >\theta _0\). Find the uniformly most powerful test of size \(\alpha \) and compute the power function. (You must explain why your test has this property.)

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\notin [0,\theta ],\\ \dfrac {1}{\theta }, & \text {if } t\in [0,\theta ]. \end {cases} \] We use \(\hat {\theta }=cX_{1,n}\) and \(\hat {\eta }=\alpha \bar {X}\) to estimate \(\theta \), where \[ X_{1,n}=\min _{1\leq i\leq n}X_i\quad \text {and}\quad \bar {X}=\frac {1}{n}\sum _{1\leq i\leq n}X_i. \] (a) Find \(c\) and \(\alpha \) such that the estimators are unbiased.

(b) Compute the relative efficiency.

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed Poisson \((\theta )\) random variables. Find the uniformly minimum unbiased estimator for \(\eta =\theta ^2 e^{-\theta }\). (You must explain your answer.)

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be independent, identically distributed random variables with a common continuous distribution function. Compute the expected value and the variance of \[ Z=\sum _{1\leq i\leq n}\sum _{1\leq j\leq n} I\{X_i\leq Y_j\}. \]

where \(I\{A\}\) denotes the indicator function of the event \(A\).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\leq \theta ,\\ e^{-(t-\theta )}, & \text {if } t>\theta . \end {cases} \] (a) Find two \(1-\alpha \) equal-tail confidence intervals for \(\theta \). One should be based on the smallest order statistic and the other should be based on the sample mean.

(b) Compare the expected lengths of the confidence intervals.

Let \(\xi _n\) and \(\eta _m\) be two sequences of random variables. Prove or give counterexamples to the following statements:

(a) If \(\xi _n\overset {d}{\to }\xi \) and \(\eta _m\overset {d}{\to }\eta \), then \(\xi _n+\eta _m\overset {d}{\to }\xi +\eta \).

(b) \(\xi _n\overset {P}{\to }\xi \) and \(\eta _m\overset {P}{\to }\eta \), then \(\xi _n+\eta _m\overset {P}{\to }\xi +\eta \).

(c) \((\xi _n,\eta _n)\overset {P}{\to }(\xi ,\eta )\) then \(\xi _n+\eta _n\overset {P}{\to }\xi +\eta \).

(\(\overset {d}{\to }\) and \(\overset {P}{\to }\) denote convergence in distribution and convergence in probability.)

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with \(P\{X_i=0\}=\theta \) and \(P\{X_i=1\}=1-\theta \) \((0<\theta <1)\). Find the numerical sequences \(a_n\) and \(b_n\) such that \[ Y_n=a_n\hat {X}_n(1-\hat {X}_n)-b_n \] converges in distribution to a nondegenerate random variables, where \[ \hat {X}_n=\frac {1}{n}\sum _{1\leq i\leq n}X_i. \]