Qualifying Exam in Statistics, 2005

Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta _1)= \begin {cases} 0, & \text {if } t\notin [1,\infty ),\\ \theta _1 t^{-\theta _1-1}, & \text {if } 1\leq t<\infty . \end {cases} \] Let \(Y_1,\ldots ,Y_m\) be independent, identically distributed random variables with density function \[ f(t,\theta _2)= \begin {cases} 0, & \text {if } t\notin [1,\infty ),\\ \theta _2 t^{-\theta _2-1}, & \text {if } 1\leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta _1=\theta _2\) against the alternative that \(\theta _1\neq \theta _2\). The two samples are independent.

(a) Find a test using the likelihood ratio.

(b) Provide a large sample approximation for the likelihood ratio test.

Let \(X_1,\ldots ,X_n\) be independent random variables with density functions \[ f(t,\theta _i)= \begin {cases} 0, & \text {if } t\notin [0,\infty ),\\ \dfrac {1}{\theta _i}e^{-t/\theta _i}, & \text {if } 0\leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta _1=\theta _2=\cdots =\theta _n\) against the alternative that \(H_0\) is not true.

(a) Find a test using the likelihood ratio.

(b) Provide a large sample approximation for the likelihood ratio test.

Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ g(t;\theta )= \begin {cases} 0, & \text {if } -\infty <t<\theta ,\\ e^{-(t-\theta )}, & \text {if } \theta \leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta \leq \theta _0\) against \(H_A:\theta >\theta _0\).

(a) Find the uniformly most powerful test of size \(\alpha \).

(b) Compute the power function.

Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with a continuous distribution function \(F\). Let \(x_{1/4}\) be defined by \(F(x_{1/4})=1/4\).

(a) Find a \(1-\alpha \) lower confidence bound for \(x_{1/4}\).

(b) Provide a large sample approximation.

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with distribution function \[ F(t)=\frac {1}{1+e^{-t}},\qquad -\infty <t<\infty . \] Show that \[ Y_n=X_{n,n}-\log n \] converges in distribution, where \(X_{n,n}=\max \{X_1,X_2,\ldots ,X_n\}\).

Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with probability mass function \[ f(t,\theta )=\theta (1-\theta )^{t-1}I\{t=1,2,\ldots \}\qquad (0<\theta <1). \] Find the uniformly minimum variance unbiased estimator for \(\theta \).

Let \(X\) and \(Y\) be independent random variables with densities, \[ g(t)= \begin {cases} 0, & \text {if } t\notin [-2,2],\\ 1/4, & \text {if } t\in [-2,2], \end {cases} \] and \[ f(t)= \begin {cases} 0, & \text {if } t\notin [-1,1],\\ 1/2, & \text {if } t\in [-1,1]. \end {cases} \] Compute the density of \(X+Y\).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with probability mass function \[ P\{X_i=1\}=\theta \quad \text {and}\quad P\{X_i=0\}=1-\theta . \] Let: \[ \bar {X}_n=\frac {1}{n}\sum _{1\leq i\leq n} X_i \] and \(h(t)=t^3-t\). Find the sequences \(a_n\) and \(b_n\) such that \(a_n\bigl (h(\bar {X}_n)-h(\theta )-b_n\bigr )\) has a non-degenerate limit distribution.

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with a continuous probability distribution function \(F\) and \(Y_1,Y_2,\ldots ,Y_m\) be independent, identically distributed random variables with a continuous probability distribution function \(H\). The two samples are independent and \(F=H\). Let \[ T=\sum _{1\leq i\leq n}\sum _{1\leq j\leq m} I\{X_i\leq Y_j\}. \] Compute \(E T\) and \(\operatorname {var}T\).

Let \(\epsilon _i\), \(1\leq i\leq n\), be independent identically distributed normal \(N(0,\sigma ^2)\) random variables and consider the model \[ y_i=i\alpha +\gamma i^2+\epsilon _i,\qquad 1\leq i\leq n. \] We observe \(y_i\), \(1\leq i\leq n\).

(a) Find the maximum likelihood estimators for \(\alpha \), \(\gamma \), and \(\sigma ^2\).

(b) Compute the joint distribution of the maximum likelihood estimators for \((\alpha ,\gamma )\).