Statistics Prelim Exam University of Utah Department of Mathematics 0.8 January 2020

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\theta }e^{-t/\theta }, & \text {if } t\geq 0. \end {cases} \] Let \(\alpha >0\) and \[ \tau =\frac {1}{\theta ^\alpha }. \]

(a) How big \(n\) should be (as a function of \(\alpha \)) so that \(\tau \) has uniformly minimum variance unbiased estimator?

(b) Find the uniformly minimum variance unbiased estimator for \(\tau \) (you need to justify that your estimator has the required property).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\notin [-\theta ,\theta ],\\ \dfrac {5t^4}{2\theta ^5}, & \text {if } -\theta \leq t\leq \theta . \end {cases} \] Find the maximum likelihood estimator for \(\theta \) and compute its bias.

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\notin [-\theta ,\theta ],\\ \dfrac {10t^9}{2\theta ^{10}}, & \text {if } -\theta \leq t\leq \theta . \end {cases} \] Find a moment estimator for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \[ f(t\theta )=\frac {1}{2}e^{-|t-\theta |}, \qquad -\infty <t<\infty . \] The parameter \(\theta \) could be any real number. Find all maximum likelihood estimators for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be iid with the same pdf as in question 5. Find the form of the rejection region coming from the generalized likelihood ratio test for \(H_0:\theta =\theta _0\) versus \(H_a:\theta \ne \theta _0\).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables that have the uniform distribution on \([0,1]\). Let \(X_{1,n}\leq X_{2,n}\leq \cdots \leq X_{n,n}\) be the order statistics. Let \(k\geq 1\) be a fixed integer. Show that \[ Y_n=nX_{k,n} \] converges in distribution and compute the limiting distribution.

Let \(X\) and \(Y\) be independent identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\theta }e^{-t/\theta }, & \text {if } t\geq 0. \end {cases} \] Compute the density function of \[ Z=\frac {X}{X+Y}. \]

Let \(X_1,\ldots ,X_n\sim N(\mu _1,1)\) and \(Y_1,\ldots ,Y_m\sim N(\mu _2,1)\), where all random variables are independent. Derive the likelihood ratio test for \[ H_0:\mu _1^2=\mu _2^2 \quad \text {vs.}H_a:\mu _1^2\ne \mu _2^2. \] What is the distribution of the test statistic?

Suppose a box contains 6 marbles in total. Suppose \(\theta \) of them are white and \(6-\theta \) of them are black. Test \(H_0:\theta =2\) against \(H_a:\theta \ne 2\) as follows: draw two marbles without replacement and reject \(H_0\) if both marbles are the same color, otherwise do not reject. What is the probability of Type II error for each of the alternatives \(\theta =1\) and \(\theta =3\)?

Let \(X_1,\ldots ,X_n\) be iid random variables. The density function of \(X_i\) is \[ g(t;\theta _i)=\theta _i t^{-\theta _i-1}1\{t\geq 1\}. \] Here \(\theta _i>0\) is an unknown parameter. We wish to test \[ H_0:\theta _1=\theta _2=\cdots =\theta _n \] against the alternative that \(H_0\) is not true. Find a test using the generalized likelihood ratio.