Statistics Prelim Exam University of Utah Department of Mathematics 0.8 August 2017

Let \(X_1,X_2,\ldots ,X_n\) be iid random variables with probability mass function \[ f(t;\theta )=\theta (1-\theta )^{t-1}, \qquad t=1,2,3,\ldots \] Find the uniformly minimum variance unbiased estimator of \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be iid random variables with density function \[ h(t;\eta )=e^{-(t-\eta )}I\{t\geq \eta \}, \] where \(I\) indicates the indicator function. We want to test \(H_0:\eta =\eta _0\) against \(H_A:\eta \ne \eta _0\). Find a test using the likelihood method and compute the power function.

Let \(X_1,X_2,\ldots ,X_n\) be iid random variables with density \[ h(t;\theta )=\theta t^{\theta -1}I\{0\leq t\leq 1\}, \qquad \theta >0. \] Find the maximum likelihood estimator of \(\theta \) and compute the asymptotic variance of the maximum likelihood estimator.

Let \(X_1,X_2,\ldots ,X_n\) be iid with density \[ h(t;\theta )=\frac {1}{2}I\{\theta -1\leq t\leq \theta +1\}. \] Find all maximum likelihood estimators of \(\theta \). Show that if \(\hat \theta _n\) is a maximum likelihood estimator then \(\mathbb {E}[\hat \theta _n]\to \theta \) as \(n\to \infty \).

Let \(X_1,X_2,\ldots ,X_n\) be iid \(N(\mu _1,\sigma _1^2)\) and \(Y_1,Y_2,\ldots ,Y_m\) be iid \(N(\mu _2,\sigma _2^2)\). The \(X\) and the \(Y\) are also independent of each other. Assume that \(\sigma _1^2,\sigma _2^2\) are known while \(\mu _1,\mu _2\) are unknown. We wish to test \(H_0:\mu _1=\mu _2\) against \(H_A:\mu _1\ne \mu _2\). Find a test using the likelihood method. Compute the distribution of the likelihood ratio test.

Let \(X_1\) be \(N(\mu _1,\sigma _1^2)\) and \(X_2\) be \(N(\mu _2,\sigma _2^2)\) and assume \(X_1\) and \(X_2\) are independent. Show that \(Y_1=X_1+X_2\) and \(Y_2=X_1-X_2\) are independent if and only if \(\sigma _1=\sigma _2\).

Let \(X_1,X_2,\ldots ,X_n\) be iid with density \[ f(x)=2x/\theta ^2, \qquad 0<x<\theta . \] Let \(Y=X_{(n)}=X_{n:n}\) be the largest order statistic of the \(X_i\). Show that \(Y/\theta \) is a pivotal quantity for \(\theta \) and use it to construct a 99% confidence interval for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be iid with a Poisson\((\theta )\) distribution, so that their common probability mass function is \[ \mathbb {P}(X_i=k)=e^{-\theta }\frac {\theta ^k}{k!}, \qquad k=0,1,2,\ldots ,\ \theta >0. \] Find a sufficient statistic for \(\theta \).

Assume that the lifetimes of 100 lightbulbs are iid random variables with an Exponential(5) distribution, meaning that their common pdf is \[ f(x;\theta )=\frac {1}{5}e^{-x/5}, \qquad x>0. \] Use normal approximation to estimate the probability that in total the 100 lightbulbs last longer than 550 hours. You can express your answer as an integral.

Let \(X_1,X_2,\ldots ,X_n\) be iid with common pdf \[ f(x;\theta )=\frac {\theta }{(1+x)^{\theta +1}}, \qquad 0<x<\infty ,\ \theta >0. \] What is a lower bound for the variance of an unbiased estimator of \(\theta \)?