Statistics Prelim Exam 0.8 January 2016

Let \(X_1,\ldots ,X_n\) be a random sample with density function \(f(x)=2e^{-2(x-\eta )}\) for \(x>\eta \), and zero otherwise. We wish to test \(H_0:\eta =4\) versus the alternative \(H_A:\eta >4\). We reject \(H_0\) if \(X_{(1)}=\min _{1\leq i\leq n}X_i\geq 4+2/n\).

(a) Compute the significance level of the test.

(b) Compute the power function.

Let \(X_1,\ldots ,X_n\) be iid exponential(1) random variables. Find the joint pdf of the pair \((X_{(1)},X_{(3)})\), where \(X_{(i)}\) denotes the \(i\)th order statistic, i.e. \[ X_{(1)}\leq X_{(2)}\leq \cdots \leq X_{(n)}. \]

Let \(X_1\) and \(X_2\) be independent with joint pdf \(f(x)=x^{-2}\) for \(1\leq x<\infty \) and 0 otherwise.

(a) Find the joint pdf of \(U=X_1X_2\) and \(V=X_2\).

(b) Find the marginal pdf of \(U\).

Assume that \(X_1,\ldots ,X_n\) are a random sample with common pdf \(f(x;\theta )=(\theta +1)x^{-\theta -2}\) for \(x>1\), and zero otherwise. Is the method of moments estimator for \(\theta \) the same as the maximum likelihood estimator for \(\theta \)?

Let \(X_1,\ldots ,X_n\) be iid with the Gamma\((\theta ,\kappa )\) distribution, meaning that their common pdf is \[ f(x;\theta ,\kappa )=\frac {1}{\theta ^\kappa \Gamma (\kappa )} x^{\kappa -1}e^{-x/\theta }, \qquad x>0, \] with \(\kappa \) known but \(\theta \) unknown. Derive a \(100(1-\alpha )\%\) equal-tailed confidence interval for \(\theta \) based on the sufficient statistic.

Let \(X_1,X_2,\ldots ,X_n\) be iid exponential(1) random variables and \[ \bar X_n=\frac {1}{n}\sum _{i=1}^n X_i \] be the sample mean. Find two sequences of numbers \(a_n\) and \(b_n\) such that \[ \frac {\log \bar X_n-a_n}{b_n} \] converges in distribution to a standard normal random variable.

Suppose that \(X_1,\ldots ,X_n\) are iid from a \(\BETA (\theta _1,\theta _2)\) distribution, meaning that the pdf is \[ f(x;\theta _1,\theta _2)= \frac {\Gamma (\theta _1+\theta _2)}{\Gamma (\theta _1)\Gamma (\theta _2)} x^{\theta _1-1}(1-x)^{\theta _2-1}, \qquad 0<x<1. \] Find jointly sufficient statistics for \(\theta _1\) and \(\theta _2\), and use them to find a UMVUEs for \(\theta _1\) and \(\theta _2\).

Let \(Y_1<Y_2<\cdots <Y_5\) be the order statistics of a random sample of size \(n=5\) from a distribution with pdf \(f(x;\theta )=e^{-|x-\theta |}/2\), \(-\infty <x<\infty \), with \(\theta \) unknown. Find the likelihood ratio test \(\lambda \) for testing \(H_0:\theta =\theta _0\) versus \(H_1:\theta \ne \theta _0\).

Let \(X_1,X_2,\ldots ,X_n\) be iid from a distribution with CDF \(F(x)=1-1/x\) for \(1\leq x<\infty \), and zero otherwise.

(a) Derive the CDF of the smallest order statistic \(X_{(1)}\).

(b) Find the limiting distribution of \(X_{(1)}\).

(c) Find the limiting distribution of \(X_{(1)}^n\).

Let \(X_1,\ldots ,X_n\) be iid Uniform\((0,\theta )\) with \(\theta >0\), and \(X_{(n)}\) be the largest order statistic.

(a) Find the probability that the random interval \((X_{(n)},2X_{(n)})\) contains \(\theta \).

(b) Find the constant \(c\) such that \((X_{(n)},cX_{(n)})\) is a \(100(1-\alpha )\%\) confidence interval for \(\theta \).

Let \(X_1,\ldots ,X_n\) be a random sample from the \(N(\theta ,1)\) distribution, \(\theta \) unknown. We wish to test \(H_0:\theta =\theta _0\) versus \(H_A:\theta <\theta _0\). Find a test using the generalized likelihood ratio \(\lambda \).

The distance in feet by which a parachutist misses a target is \(D=\sqrt {X_1^2+X_2^2}\), where \(X_1\) and \(X_2\) are independent with \(X_i\sim N(0,25)\). Find \(\Prob (D\leq 12.25)\) (you can write down an integral for it, you don’t need to evaluate it).