Stat 5132 Second Midterm Exam, March 2, 1998

Problem 1

Let X₁, $\ldots$, X_n be i. i. d. $\mathcal{N}(\mu, \lambda^{-1})$.Then Example 8.4.1 in the notes shows that if we use the improper prior $g(\mu, \lambda) \propto \lambda^{-1}$, then the posterior distribution of $\mu$ given the data is the distribution of

$$\bar{x} + T \frac{s}{\sqrt{n}}$$

where $\bar{x}$ and s are the sample mean and standard deviation, respectively, and $T \sim t(n - 1)$.

1. Find the posterior mean of $\mu$, assuming the sample size is large enough so the posterior mean exists.
2. Find the posterior median of $\mu$ when it exists.

Problem 2

Suppose we observe $X \sim \text{Poi}(\mu)$ and we want to do a Bayesian analysis with prior distribution $\text{Gam}(\alpha, \beta)$ for $\mu$,where $\alpha$ and $\beta$ are known numbers expressing our prior opinion about probable values of $\mu$.

1. Find the posterior distribution of $\mu$.
2. Find the posterior mean of $\mu$.

Problem 3

Let X₁, X₂, $\ldots$, X_n be an i. i. d. sample from a model having densities

$$f_\theta(x) = \frac{\theta}{x^{\theta + 1}}, \qquad 1 < x < \infty,$$

where $\theta \gt 0$ is an unknown parameter.

1. Find the MLE of $\theta$. You do not have to prove that your solution is the global maximizer of the likelihood. It is enough to find it.
2. Find the expected Fisher information for $\theta$.
3. Give an asymptotic 95% confidence interval for $\theta$.

Problem 4

Let X₁, X₂, $\ldots$, X_n be an i. i. d. sample from a $\mathcal{N}(\mu, \sigma^2)$ model, where $\mu$ and $\sigma^2$ are unknown parameters, and let S_n² denote the sample variance (defined, as usual, with n - 1 in the definition as on p. 204 in Lindgren). Suppose n = 5 and S_n² = 53.3. Give an exact (not asymptotic) 95% confidence interval for $\sigma^2$.

Problem 5

Let X₁, X₂, $\ldots$, X_n be an i. i. d. sample from a $\text{Beta}(\theta, \theta)$ model, where $\theta$ is an unknown parameter. Find a consistent estimator of $\theta$.(Hint: method of moments estimators are always consistent.)