Statistics Prelim Exam University of Utah Department of Mathematics 0.8 August 2016

Let \(X_1,X_2,\ldots ,X_n\) be iid with common pdf \(f(x)=2x\), \(0<x<1\) and zero elsewhere. Find the limiting distribution of \(n(1-X_{(n)})\), where \(X_{(n)}=\max \{X_1,\ldots ,X_n\}\).

Let \(X_1\sim N(0,4\theta )\), \(X_2\sim N(0,9\theta )\), \(X_3\sim N(0,4\theta ^2)\), \(X_4\sim N(0,12\theta ^2)\), and assume that all \(X\)s are independent of each other. Using all four \(X\) variables and the \(t\) distribution, compute a \(100(1-\alpha )\%\) confidence interval for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be iid with common pdf \[ f(x;\theta )=\frac {1}{2\theta }e^{-|x|/\theta }, \qquad -\infty <x<\infty , \] with \(\theta \) unknown but positive. Compute the Cramer-Rao lower bound for the variance of an unbiased estimator of \(\theta \), and extend your calculation to finding the UMVUE by finding a linear combination of \[ \sum _{i=1}^n \frac {\partial }{\partial \theta }\log f(X_i;\theta ) \] that is unbiased.

If a random sample of size \(n\) is taken from a distribution having pdf \(f(x;\theta )=2x/\theta ^2\), \(0<x\leq \theta \) and zero elsewhere, find:

(a) the MLE \(\hat \theta \) for \(\theta \),

(b) the constant \(c\) so that \(\mathbb {E}[c\hat \theta ]=\theta \),

(c) the MLE for the median of the distribution.

If \(X_1,X_2,\ldots ,X_n\) is a random sample from a distribution having pdf of the form \(f(x;\theta )=\theta x^{\theta -1}\), \(0<x<1\) and zero elsewhere, show that a best critical region for testing \(H_0:\theta =1\) against \(H_1:\theta =2\) is \[ C=\left \{(x_1,x_2,\ldots ,x_n): c\leq \prod _{i=1}^n x_i\right \}. \]

Let \(X_1,X_2,\ldots ,X_{25}\) denote a random sample of size 25 from a normal distribution with mean \(\theta \) and variance 100. Find a uniformly most powerful critical region of size \(\alpha =0.10\) for testing \(H_0:\theta =75\) against \(H_1:\theta >75\).

Let \(X_1\) and \(X_2\) be two independent random variables. Suppose that \(X_1\) is \(\chi ^2(r_1)\) and \(X_1+X_2\) is \(\chi ^2(r)\), where \(r_1,r\) are the respective degrees of freedom and \(r_1<r\). Show that \(X_2\) is \(\chi ^2(r-r_1)\).

Let \(Y_1,Y_2,\ldots ,Y_n\) be independent random variables, each with the \(N(\beta x_i,\gamma ^2x_i^2)\) distribution, where the numbers \(x_1,x_2,\ldots ,x_n\) are known, not all are equal, and none of them is zero. Suppose \(\beta \) and \(\gamma \) are unknown. Find the maximum likelihood estimators of \(\beta \) and \(\gamma ^2\).

Let \(X_1,X_2,\ldots ,X_n\) be a random sample from a Gamma\((\alpha ,\theta )\) distribution, with \(\alpha \) known and \(\theta \) unknown. Find the Fisher information of this distribution and show that the MLE of \(\theta \) is an efficient estimator of \(\theta \).

Suppose that an engineer wishes to compare the number of complaints per week filed by union stewards for two different shifts at a manufacturing plant. It is known that the number of complaints per week on the \(i\)th shift has a Poisson distribution with mean \(\theta _i\) for \(i=1,2\). One hundred independent observations on the number of complaints gave means \(\bar x=20\) for shift 1 and \(\bar y=22\) for shift 2. Using this data test \(H_0:\theta _1=\theta _2\) versus \(H_1:\theta _1\ne \theta _2\) by the likelihood ratio method, with \(\alpha =0.01\). (Hint: First find the likelihood ratio test statistic for \(\lambda \) and then use the distribution of \(-2\log \lambda \).)