Preliminary Examination Probability \& Statistics 1994

Let \(X_1,X_2,\cdots \) be independent, identically distributed random variables, uniformly distributed on \([0,1]\). Show that \[ Y_n=\frac {4\sum _{1\le k\le n}kX_k-n^2}{n^{3/2}}, \] converges in distribution to a normal random variable.

Let \((\Omega ,\mathcal A,P)\) be a probability space and \(\{B(t),0\le t\le 1\}\) be a Brownian motion on it. Since almost all sample paths of \(B\) are continuous, \(\int _0^t B(t)dt\) can be defined as a usual Rieman integral. Compute the distribution of \(\int _0^1 B(t)dt\).

Let \(\{S_n,\mathcal F_n,\,n\ge 1\}\) be a nonnegative martingale with \(\mathbb {E} S_n=1\). Show that for all \(\lambda >0\), \[ \mathbb {P}\{S_n\ge \lambda ,\text { for some } n\ge 1\}\le \frac {1}{\lambda }. \]

Let \(X_1,X_2,\cdots \) be independent, identically distributed random variables. Show that the following statements are equivalent:

(a) \(\mathbb {E}|X_1|^\nu <\infty \);

(b) \(X_n/n^{1/\nu }\to 0\), almost surely;

(c) \(\max _{1\le i\le n}|X_i|/n^{1/\nu }\to 0\), almost surely.

Let \(X_1,X_2,\cdots \) be independent, identically distributed normal random variables. Find two numerical sequences, \(a_n\) and \(b_n\), such that \[ \frac {\max _{1\le i\le n}X_i-a_n}{b_n} \] converges in distribution to a non-degenerate random variable.

Let \(0\le X_n\le 1\) be adapted to \(\mathcal F_n\). Let \(\alpha ,\beta >0\) with \(\alpha +\beta =1\) and suppose \[ \mathbb {P}(X_{n+1}=\alpha +\beta X_n\mid \mathcal F_n)=X_n \] \[ \mathbb {P}(X_{n+1}=\beta X_n\mid \mathcal F_n)=1-X_n. \]

(a) Show that \(\mathbb {P}\{\lim _{n\to \infty }X_n=0\text { or }1\}=1\).

(b) Show that if \(X_0=\theta \), then \(\mathbb {P}\{\lim _{n\to \infty }X_n=1\}=\theta \).

Suppose \(X_1,\cdots ,X_m\) are independent with \(X_j\sim \operatorname {Binomial}(n_j,p_j)\).

(a) Find the UMVUE’s of \(p_1,\cdots ,p_m\).

(b) Suppose you know that \(p_1=p_2=\cdots =p_m\). Let \(p\) denote this common (but unknown) value. Find the UMVUE of \(p\).

(c) Find the likelihood ratio statistic, \(\lambda \), for \(H_0:p_1=\cdots =p_m\) versus \(H_1:p_i\ne p_j\), for some \(i\) and \(j\).

(d) It can be shown that \(2\log \lambda \) is approximately the same as the usual \(\chi ^2\) statistics. Using this fact, find an approximate test for \(H_0\) vs \(H_1\) above.

Consider the linear model: \[ Y_{ij}=\beta _i+\epsilon _{ij},\qquad 1\le i\le 2,\qquad 1\le j\le J. \] Suppose \(\epsilon _{ij}\)’s are independent and for some (known) \(a_1\) and \(a_2\), \(\epsilon _{ij}\sim N(0,a_i^2\sigma ^2)\).

(a) Find the U.M.V.U.E.’s of \(\beta _1\) and \(\beta _2\).

(b) Suppose you know that for some unknown \(\beta \), \(\beta _1=\beta _2\). Find the U.M.V.U.E. of \(\beta \).

Suppose \(X\sim \operatorname {Poisson}(\theta )\). Put a \(\operatorname {Gamma}(\alpha ,\beta )\) prior on \(\theta \) and suppose we have the following loss function: \(\ell (\theta ,a)=(a-\theta )^2/\theta \). Find the Bayes’ estimator of \(\theta \).

Suppose \(\theta \in \{\theta _0,\theta _1\}\) is an unknown (\(\theta _0\) and \(\theta _1\) are, however, known.) Put some prior, \(\pi \), on \(\theta \). We are to test \(H_0:\theta =\theta _0\) versus \(H_1:\theta =\theta _1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes’ procedure for doing this test, if the loss function, \(\ell (\theta ,a)\), is the 0-1 loss given by: \[ \begin {alignedat}{2} \ell (0,1)&=0,\qquad & \ell (0,0)&=1,\\ \ell (1,0)&=0,\qquad & \ell (1,1)&=1. \end {alignedat} \] Is this procedure minimax?

Construct a \((1-\alpha )\) two-sided confidence interval for the correlation coefficient of a bivariate normal distribution. (HINT. This is an exponential family.)