Statistics Prelim Exam University of Utah Department of Mathematics 0.8 January 2018

Let \(X_1,X_2\) be iid with density \[ f(x;\theta )= \begin {cases} 3x^2/\theta ^3, & 0<x<\theta ,\\ 0, & \text {otherwise.} \end {cases} \] Find a complete, sufficient statistic for \(\theta \in [0,\infty )\) based on \(X_1\) and \(X_2\).

Let \(X_1,X_2,\ldots ,X_n\) be iid random variables with a Beta\((\theta ,\theta )\) distribution, i.e. their density is \[ f(x;\theta )=\frac {\Gamma (2\theta )}{\Gamma (\theta )^2} x^{\theta -1}(1-x)^{\theta -1}, \qquad 0<x<1. \] Show that a best critical region for testing \(H_0:\theta =1\) versus \(H_A:\theta =2\) is \[ \left \{(x_1,x_2,\ldots ,x_n): c\leq \prod _{i=1}^n x_i\right \}. \]

A particular gene occurs as one of two alleles (\(A\) and \(a\)), with allele \(A\) having frequency \(\theta \) in the population. In other words, a random copy of the gene is \(A\) with probability \(\theta \) and \(a\) with probability \(1-\theta \). A diploid genotype is just two genes put together (in any particular order) so the probability of each diploid genotype is \(\theta ^2\) for \(AA\), \(2\theta (1-\theta )\) for \(Aa\), and \((1-\theta )^2\) for \(aa\). Suppose we test a random sample of people and find that \(k_1\) of them are \(AA\), \(k_2\) of them are \(Aa\), and \(k_3\) of them are \(aa\). Find the MLE of \(\theta \).

Let \(X_1,X_2,X_3\) be iid and distributed according to \(f(x)=2x\), \(0<x<1\). Let \(X_{(1)}<X_{(2)}<X_{(3)}\) be the order statistics. Compute the probability that the smallest of the \(X_i\) exceeds the median of the distribution.

Let \(X\) be discrete taking values in \(\{0,1\}\) and have the pdf \(f(x;\theta )=\theta ^x(1-\theta )^{1-x}\) for \(x\in \{0,1\}\). We test the simple hypothesis \(H_0:\theta =1/4\) agains the alternative composite hypothesis \(H_A:\theta <1/4\) by taking a random sample of size 10 and rejecting \(H_0\) if and only if the observed values \(x_1,\ldots ,x_{10}\) satisfy \[ \sum _{i=1}^{10}x_i\leq 1. \] Find the power function \(K(\theta )\) of this test for \(0\leq \theta \leq 1/4\).

Consider the density function \[ p(x;\alpha ,\lambda )=(2\pi x^3)^{-3/2} \exp \left \{(\alpha \lambda )^{1/2}-\frac {1}{2}\log \lambda -\frac {1}{2}\alpha x-\frac {\lambda }{2}x^{-1}\right \}, \qquad 0<x<\infty . \] Given \(X_1,X_2,\ldots ,X_n\) iid from this density find a sufficient statistic (that involves all of the \(X_i\)) for the parameters \((\alpha /2,\lambda /2)\).

Suppose that \(X_1,\ldots ,X_n\) form a random sample from a distribution for which the pdf \(f(x|\theta _1,\theta _2)\) is as follows: \[ f(x|\theta _1,\theta _2)= \begin {cases} \dfrac {1}{\theta _2-\theta _1}, & \theta _1<x<\theta _2,\\ 0, & \text {otherwise.} \end {cases} \] The values of \(\theta _1\) and \(\theta _2\) are unknown, other than \(-\infty <\theta _1<\theta _2<\infty \). Find their MLEs.

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 the UMVUE of \(e^{-\theta }\).

A food processing company packages honey in glass jars. Each jar is supposed to contain 10 fluid ounces of honey. Previous experience suggests that the volume of a randomly selected jar of honey is normally distributed with a known variance of 2 ounces. At a significance level of \(\alpha =.05\) derive the likelihood ratio test for testing the null hypothesis \(H_0:\mu =10\) versus \(H_A:\mu \ne 10\).

Let \(X_1,X_2,\ldots ,X_{25}\) denote a random sample of size 25 from a \(N(\theta ,100)\) distribution. Find a uniformly most powerful critical region of size \(\alpha =.10\) for testing \(H_0:\theta =75\) against \(H_A:\theta >75\).