STATISTICS 0.6 PhD Written Qualifying Examination -- 15 Aug. 2000 9:00--12:00

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential (1) random variables with common density function \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ e^{-t}, & \text {if } 0\leq t<\infty . \end {cases} \] (a) Show that \(\left (\sum _{1\leq i\leq n}X_i\right )^{1/2}-n^{1/2}\) converges in distribution to a normal random variable.

(b) Compute the mean and the variance of the limiting normal random variable.

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed Poisson random variables with common probability mass function \[ p(k,\theta )=\frac {\theta ^k}{k!}e^{-\theta },\qquad k=0,1,\ldots \] Let \[ \tau =\theta e^{-\theta }. \] Find the uniformly minimum variance unbiased estimator for \(\tau \).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed uniform \((0,\theta )\) random variables with common density function \[ f(t)= \begin {cases} 0, & \text {if } t\notin [0,\theta ],\\ 1/\theta , & \text {if } t\in [0,\theta ]. \end {cases} \] (a) Find the maximum likelihood estimator for \(\theta \).

(b) Find a \(1-\alpha \) equal-tail confidence interval for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be independent exponential \((\theta _i)\) random variables, i.e. the density of \(X_i\) is \[ f(t,\theta _i)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ \dfrac {1}{\theta _i}e^{-t/\theta _i}, & \text {if } 0\leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta _1=\theta _2=\cdots =\theta _n\) against the alternative that \(H_0\) is not true.

(a) Find the likelihood ratio test.

(b) Show that the likelihood ratio test is distribution free under \(H_0\).

(c) Show that the likelihood ratio test is equivalent with the \(\chi ^2\) test under \(H_0\).

(d) Approximate the critical values of the likelihood ratio test under \(H_0\).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with common density function \(f\). Let \[ \hat {f}_n(t)=\frac {1}{nh}\sum _{1\leq i\leq n}K\left (\frac {t-X_i}{h}\right ), \] where \(K\) is a symmetric density function and \(K(t)=0\) if \(|t|>1\). We assume that \(h=h(n)\to 0\) as \(n\to \infty \). We assume that the second derivative of \(f\) exists and continuous in a neighborhood of \(t\). We use \(\hat {f}\) to estimate \(f\). Compute the bias of \(\hat {f}_n(t)\).

\(X\) and \(Y\) are two independent identically distributed normal \(N(0,\sigma ^2)\) random variables. Let \[ Z=X^2+Y^2 \] and \[ W=\frac {X}{(X^2+Y^2)^{1/2}}. \] (a) Show that \(Z\) and \(W\) are independent.

(b) Compute the marginal distributions of \(Z\) and \(W\).

\(Z_1\) and \(Z_2\) are two independent identically distributed exponential \((\lambda )\) random variables, i.e. the common density is \[ f(t)= \begin {cases} 0, & \text {if } -\infty <t<0,\\ \dfrac {1}{\lambda }e^{-t/\lambda }, & \text {if } 0\leq t<\infty . \end {cases} \] Let \(X=Z_1\) and \(Y=Z_1+Z_1Z_2\). Compute

(a) \(E(Y\mid X=x)\)

(b) \(\operatorname {var}(E(Y\mid X))\).

Let \(y_i=\alpha x_i+\epsilon _i\), where \(\epsilon _i\), \(1\leq i\leq n\), are independent, identically distributed normal \(N(0,\sigma ^2)\) random variables.

(a) Find the maximum likelihood estimators for \(\alpha \) and \(\sigma ^2\).

(b) Find the distributions of the maximum likelihood estimators.

(c) Compute the least-squares estimator for \(\alpha \).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed Poisson random variables with common probability mass function \[ p(k,\theta )=\frac {\theta ^k}{k!}e^{-\theta },\qquad k=0,1,\ldots \] Using the Cramer-Rao inequality show that the sample mean is a uniformly minimum variance unbiased estimator for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential \((\lambda )\) random variables. Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential \((\mu )\) random variables. The two samples are independent. We wish to test \(H_0:\theta =\mu \) against the alternative that \(H_0\) is not true.

(a) Find the likelihood ratio test.

(b) Find the asymptotic distribution of the likelihood ratio under \(H_0\).

(c) Find an equal-tail confidence interval for \(\tau =\lambda /\mu \).