University of Minnesota, Twin Cities School of Statistics Stat 8701 Rweb
The policy is the same as the policy for Homework No. 2.
This assignment is due (initially) Friday, March 1, 2002.
A problem about control variates. Estimate the probability that a bivariate standard normal random vector (mean zero and variance matrix the identity) falls in a regular hexagon inscribed in the unit circle (by the circular symmetry of the normal distribution it doesn't matter which inscribed regular hexagon you use).
Consider using as a control variate the empirical probability that
the random vector falls in the unit circle, the theoretical probability
of which is easily determined (explain how). Use a preliminary sample
to estimate the optimal loading
(Robert and Casella, display at
the bottom of p. 113).
Using a second, independent sample to estimate the probability of falling in the hexagon with and without control variate and provide Monte Carlo standard errors for both. What is the asymptotic relative efficiency of the two methods of estimation?
The problem in Chapter 1 of the handout about Monte Carlo maximum likelihood.