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

Let \(U_1,U_2,\ldots ,U_n\) be independent and identically distributed random variables, uniform on \([0,1]\). Let \(U_{1,n}\leq U_{2,n}\leq \cdots \leq U_{n,n}\) be the order statistics. Show that \(nU_{3,n}\) converges in distribution and compute the limit.

Let \(X,Y\) be independent random variables with respective densities \[ f(t)= \begin {cases} 0, & \text {if } t\notin [-2,3],\\ \dfrac {1}{5}, & \text {if } t\in [-2,3], \end {cases} \] and \[ g(t)= \begin {cases} 0, & \text {if } t\notin [-1,5],\\ \dfrac {1}{6}, & \text {if } t\in [-1,5]. \end {cases} \] Compute the density of \(X+Y\).

Let \(X\) and \(Y\) be independent normally distributed random variables with \(\mathbb {E} X=\mu _1\), \(\operatorname {Var} X=\sigma _1^2\), \(\mathbb {E} Y=\mu _2\) and \(\operatorname {Var} Y=\sigma _2^2\). Show that \(X+Y\) and \(X-Y\) are independent if and only if \(\sigma _1^2=\sigma _2^2\).

Let \(X_1,X_2,\ldots ,X_n\) be independent and identically distributed exponential\((\lambda )\) random variables, meaning that their pdf is \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ \dfrac {1}{\lambda }e^{-t/\lambda }, & \text {if } 0\leq t<\infty . \end {cases} \] Find the uniformly minimum variance unbiased estimator for \(\tau =\lambda ^5\).

Let \(X_1,X_2,\ldots ,X_n\) be independent and identically distributed random variables with density function \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<\theta ,\\ \dfrac {2\theta }{(\theta +t)^2}, & \text {if } \theta \leq t<\infty , \end {cases} \] for \(\theta >0\) a parameter. Find the maximum likelihood estimator for \(\theta \).

Let \(X_1,\ldots ,X_n\) be independent and identically distributed random variables with density function \[ h(t;\theta )=\frac {3t^2}{\theta ^3}1\{0\leq t\leq \theta \}, \] where \(\theta >0\) is a parameter. Find a \(100(1-\alpha )\%\), two-sided, equal-tailed confidence interval for \(\theta \).

Let \(X_1,\ldots ,X_n\) be independent and identically distributed \(N(\mu ,\sigma ^2)\), where \(\mu ,\sigma ^2\) are both unknown. We wish to test \(H_0:\sigma >\sigma _0\) against \(H_a:\sigma \leq \sigma _0\). We reject \(H_0\) if \(S^2<c\), where \[ S^2=\frac {1}{n-1}\sum _{i=1}^n (X_i-\bar X)^2, \qquad \bar X=\frac {1}{n}\sum _{i=1}^n X_i. \] Find \(c\) such that the size of the test is \(\alpha \).

Let \(X_1,\ldots ,X_n\) be independent and identically distributed with density function \[ h(t;\theta )=\theta t^{\theta -1}1\{0\leq t\leq 1\} \] for \(\theta >0\) unknown. Use the Neyman-Pearson Lemma to find the uniformly most powerful test of size \(\alpha \) for \(H_0:\theta =\theta _0\) versus \(H_a:\theta =\theta _1\), where \(\theta _1>\theta _0\).

Let \(X_1,\ldots ,X_n\) be independent and identically distributed 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.

Let \(X_1,\ldots ,X_n\) be independent and identically distributed random variables with (unknown) density function \(F\). Let \[ F_0(t)=(1-e^{-t})1\{t>0\}. \] We wish to test \(F(t)=F_0(t)\) for all \(t\). Explain the application of the \(\chi ^2\) goodness-of-fit test.