Ph.D. Qualifying Examination in Statistics 0.35 August, 2008

Let \(U_1,U_2,\ldots ,U_n\) be independent random variables uniform on \([0,1]\) and let \(U_{1,n}\leq U_{2,n}\leq \cdots \leq U_{n,n}\) be the corresponding order statistics.

a. Compute the asymptotic distribution of \(nU_{2,n}\). (5 points)

b. Show that \(nU_{1,n}\) and \(n(1-U_{n,n})\) are asymptotically independent. (5 points)

Let \(y_i=\alpha x_i+\epsilon _i\), \(1\leq i\leq n\), where \(\epsilon _1,\epsilon _2,\ldots ,\epsilon _n\) are independent identically distributed normal \(N(0,\sigma ^2)\) random variables.

a. Determine the maximum likelihood estimators for \(\alpha \) and \(\sigma ^2\). (5 points)

b. Compute the distribution of the maximum likelihood estimator for \(\alpha \). (5 points)

c. Compute the distribution of the maximum likelihood estimator for \(\sigma ^2\). (5 points)

Let \(X\) and \(Y\) be two independent random variables with distribution functions \(F\) and \(G\) and density functions \(f\) and \(g\).

a. Determine the distribution function of \(XY\). (5 points)

b. Does \(XY\) always have a density function? (5 points)

c. Assuming that \(XY\) has a density function, compute it. (5 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t;\theta )= \begin {cases} 0, & \text {if } t<\theta ,\\ e^{-(t-\theta )}, & \text {if } t>\theta . \end {cases} \]

a. Find a moment estimator for \(\theta \). (5 points)

b. Find the maximum likelihood estimator for \(\theta \). (5 points)

c. Determine the asymptotic efficiency between the two estimators. (5 points)

The number of customers entering a store on a given day is a Poisson random variable with parameter \(\theta \). The money spent by a customer is uniformly distributed on \([0,\eta ]\). The number of customers and the money spent in the store are independent. We observe \(Z_1,Z_2,\ldots ,Z_n\), the total spending (the money collected by the store) on \(n\) days. We can assume that \(Z_1,Z_2,\ldots ,Z_n\) are independent.

a. Provide estimators for \(\theta \) and \(\eta \) using the method of moments. (10 points)

b. Provide estimators for \(\theta \) and \(\eta \) using the likelihood method. (10 points)

Let \(\Phi \) and \(\phi \) denote the standard normal distribution and density functions.

(a) Prove that for all \(x>0\) \[ 1-\Phi (x)\leq \frac {1}{x}\phi (x). \] (5 points)

Let \(X_{1,n}<X_{2,n}\leq \cdots \leq X_{n,n}\) denote the order statistics from a sample of \(n\) independent identically distributed standard normal random variables.

(b) Show that \(X_{n,n}\) converges in probability to infinity. (5 points)

(c) Show that for all \(\epsilon >0\) \[ \lim _{n\to \infty } \mathbb {P}\left \{X_{n,n}\leq \sqrt {(2+\epsilon )\log n}\right \}=1. \] (5 points)

Let \(X_1,\ldots ,X_n\) be independent identically distributed Poisson random variables with parameter \(\theta _1\). Let \(Y_1,\ldots ,Y_m\) be independent identically distributed Poisson random variables with parameter \(\theta _2\). We assume that the two samples are independent. We wish to test \(H_0:\theta _1=\theta _2\) against the alternative that \(H_0\) is not true.

a. Derive the likelihood ratio test. (5 points)

b. Provide a large sample approximation for the rejection region. (2 points)

Let \(X_1,\ldots ,X_n\) be independent identically distributed random variables with distribution function \(F\). We assume that \(F\) is strictly increasing and continuous on its support. Let \(x_{1/2}\) denote the median.

a. Find a \(1-\alpha \) confidence interval for \(x_{1/2}\). (5 points)

b. Provide a large sample approximation for the confidence interval. (5 points)

Let \(X_1,\ldots ,X_N\) be independent random variables. The distibution of \(X_i\) is binomial \((n_i,p_i)\). We wish to test \(H_0:p_1=p_2=\cdots =p_N\) against the alternative that \(H_0\) is not true.

a. Derive the likelihood ratio test. (5 points)

b. Provide a large sample approximation for the rejection region. (2 points)