Due 23:59 2021-06-13
The Gamma distribution with shape parameter $\alpha$ and scale parameter $1$ has density proportional to \(\tilde{f}(x) = \begin{cases} x^{\alpha-1}e^{-x}, & x > 0 \\ 0, & \text{otherwise}. \end{cases}\)
\item (30 pts) Show that, for $\alpha > 1$, \(g(x) = \frac{(\alpha-1)^{\alpha-1}e^{-(\alpha-1)}}{1 + [x - (\alpha - 1)]^2/(2\alpha - 1)}\) dominates $\tilde{f}(x)$, i.e., $g(x) \ge \tilde{f}(x)$ for all $x$. (\emph{Hint}. $\frac{d}{dx}\left(x^{\alpha-1}e^{-x}(2\alpha - 1 + [x - (\alpha - 1)]^2)\right) = -x^{\alpha-2}e^{-x}(x - \alpha)^2[x - (\alpha - 1)]$.)
Propose an algorithm that generate a $\text{Gamma}(\alpha, 1)$ random number for a given $\alpha > 1$.
Implement your algorithm and test.
Consider testing the hypotheses $H_0: \lambda=2$ versus $H_a: \lambda>2$ using 25 observations from a Poisson($\lambda$) model. Rote application of the central limit theorem would suggest rejecting $H_0$ at $\alpha=0.05$ when $Z > 1.645$, where $Z = \frac{\bar{X}-2}{\sqrt{2/25}}$.
Estimate the size of this test (i.e., the type I error rate) using three Monte Carlo approaches: standard, antithetic, (unstandardized) importance sampling. Provide a confidence interval for each estimate. Discuss the relative merits of each variance reduction technique.
For the importance sampling approach, use a Poisson envelope ($h$ in the course notes) with mean equal to the $H_0$ rejection threshold, namely $\lambda = 2.4653$.
Draw the power curve for this test for $\lambda \in [2.2,4]$, using the same three techniques. Provide pointwise confidence bands in each case. Discuss the relative merits of each technique in this setting.