hw-7

Homework 7 in STAT400: Computational Statistics @ CSU

Assignment

Be sure to set.seed(400) at the beginning of your homework.

1. Estimating the cdf of a normal distribution. Use \(m = 1000\).

   1. Implement all 3 methods that we discussed in class (Example 1.7, Page 9-10 of Ch. 6 Notes) to estimate the cdf of a normal distribution \(\Phi(x)\). Note that you will need to show some derivations for method 2.
   2. Compare your estimates with the output from the R function pnorm() for \(x = 0.5, 1, 2, 3\). Summarise your findings comparing the performance of the methods.
   3. For each method, compute an estimate of the variance of your Monte Carlo estimate of \(\Phi(2)\). Summarise your findings.
   4. For each method, compute a \(95\%\) confidence interval for \(\Phi(2)\). Summarise your findings. Which CI is narrower and why does that matter?

2. 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.

3. 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 4 (\(\phi_1\) 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 the server will also be used in grading, so be sure they are identical and reproducible.