Statistics Qualifying Examination 1.2 January 7, 2009

Let \(X_1\) and \(X_2\) be independent, each with probability density function \(f(x)=1/x^2\), \(x>1\). Find the joint probability density function of \(Y_1=X_1X_2\) and \(Y_2=X_1/X_2\). It is important to state where your formula is valid. (If you have time, check that your joint pdf integrates to 1; if it doesn’t, find the error.)

Let \(X_1,\ldots ,X_n\) be independent \(\operatorname {LogNormal}(\mu ,\sigma ^2)\) (i.e., \(X_1=e^{Y_1},\ldots ,X_n=e^{Y_n}\) with \(Y_1,\ldots ,Y_n\) being independent \(N(\mu ,\sigma ^2)\)). Argue that the sample median is asymptotically normal and find the asymptotic mean and variance.

Let \(X_1,\ldots ,X_n\) be a random sample from \(N(\theta ,\theta )\), where \(\theta >0\). In particular, the population mean and the population variance are equal but unknown.

(a) Find the method-of-moments estimator of \(\theta \) based on the second sample moment.

(b) Find the maximum likelihood estimator of \(\theta \).

Let \(X_1,\ldots ,X_n\) be a random sample from a \(N(\theta ,1)\) distribution, and let the prior distribution of \(\theta \) be \(N(\mu _0,1)\), where \(\mu _0\) is known.

(a) Find the posterior distribution of \(\theta \).

(b) Find the Bayes estimator of \(\theta \) under squared error loss, and show that it is a weighted average of the prior mean \(\mu _0\) and the sample mean \(\bar X\).

Let \(X_1,\ldots ,X_n\) be a random sample from \(\operatorname {Poisson}(\theta )\).

(a) Find a complete, sufficient statistic.

(b) Noting that \(\Prob _\theta (X_1=1)=\theta e^{-\theta }\), use the Rao–Blackwell and Lehmann–Scheffé theorems to find a UMVUE of \(\tau (\theta ):=\theta e^{-\theta }\).

Assume a sample of size \(n\) from \(\operatorname {Uniform}[\eta -\theta ,\eta +\theta ]\), which is a location-scale family. Here \(\eta \) is real and \(\theta >0\).

(a) Find a \(100(1-\alpha )\%\) confidence interval for \(\theta \).

(b) Find a \(100(1-\alpha )\%\) confidence interval for \(\eta \).

It is not necessary to find the explicit distribution of the pivotal quantities; just denote the needed quantiles by \(x_p\) and \(y_p\) for an appropriate \(p\).

Let \(X_1,\ldots ,X_n\) be a random sample from \(\operatorname {Uniform}[0,\theta ]\). Find the UMP test of size \(\alpha \) of \(H_0:\theta \geq \theta _0\) vs. \(H_a:\theta <\theta _0\) by first deriving a most powerful test of simple hypotheses and then extending it to composite hypotheses.

Let \(X_1,\ldots ,X_n\) be a random sample from \(N(\mu ,\sigma _0^2)\), where \(\sigma _0^2\) is known. Find the generalized likelihood ratio test of size \(\alpha \) of \(H_0:\mu \geq \mu _0\) vs. \(H_a:\mu <\mu _0\).

According to a genetic model the proportions of individuals having the four blood types should be given by O: \(q^2\); A: \(p^2+2pq\); B: \(r^2+2qr\); AB: \(2pr\). Here \(p>0\), \(q>0\), \(r>0\), and \(p+q+r=1\), but the three parameters are unknown. We observe \(n_O\), \(n_A\), \(n_B\), \(n_{AB}\) individuals of the four blood types in a random sample of size \(n\). Describe a goodness-of-fit test of size \(\alpha \) of the stated hypothesis.

Consider the simple linear regression model \(Y_i=\beta _0+\beta _1x_i+\epsilon _i\) (\(i=1,2,\ldots ,n\)). Assume independent \(N(0,\sigma ^2)\) errors.

(a) Find the least squares estimators \(\hat \beta _0\) of \(\hat \beta _1\) of \(\beta _0\) and \(\beta _1\).

(b) Calculate the covariance matrix of \((\hat \beta _0,\hat \beta _1)\).