Homework 7 in STAT400: Computational Statistics @ CSU
Be sure to set.seed(400) at the beginning of your homework.
Compute a Monte Carlo estimate \(\hat{\theta}_1\) of \[ \theta = \int\limits_0^{0.5} e^{-x} dx \] by sampling from the Uniform(\(0, 0.5\)) and estimate the variance of \(\hat{\theta}_1\). Find another Monte Carlo estimator \(\hat{\theta}_2\) by sampling from the Exponential(1) distribution and estimating its variance.
Which of the variances (of \(\hat{\theta}_1\) or \(\hat{\theta}_2\)) is smaller?
Find two importance functions \(\phi_1\) and \(\phi_2\) that are supported on \((1, \infty)\) and are “close” to \[ h(x) = \frac{x^2}{\sqrt{2\pi}} e^{-x^2/2}, \qquad x > 1. \]
Which of your two importance functions should produce the smallest variance in estimating \[ \int\limits_1^\infty \frac{x^2}{\sqrt{2\pi}} e^{-x^2/2} dx \] by importance sampling? Explain.
Hint: You will need to create plots of \(\phi_1\), \(\phi_2\), and \(g(x)f(x)\) as well as \(g(x)f(x)/\phi(x)\) to answer this question.
Obtain a Monte Carlo estimate of
\[
\int\limits_1^\infty \frac{x^2}{\sqrt{2\pi}} e^{-x^2/2} dx
\] using importance sampling with the two importance sampling functions you chose in Problem 2 (\(\phi_1\) and \(\phi_2\)). Obtain an estimate of the variance for each and compare.
Turn in in a pdf of your homework to canvas. Your .Rmd file on rstudio.cloud will also be used in grading, so be sure they are identical and reproducible.