.
Consider the integral sought in Example 5.1, Equation (5.7), for the parameter values given there. Find a simple rejection sampling envelope that will produce extremely few rejections when used to generate draws from the density proportional to that integrand.
6.2.
Consider the piecewise exponential envelopes for adaptive rejection sampling of the standard normal density, which is log-concave. For the tangent-based envelope, suppose you are limited to an even number of nodes at ±c1, . . . ,±cn. For the envelope that does not require tangent information, suppose you are limited to an odd number of nodes at 0,±d1, . . . ,±dn. The problems below will require optimization using strategies like those in Chapter 2.
a.
For
n
= 1,
2,
3,
4,
5, find the optimal placement of nodes for the tangent-based envelope.
b.
For
n
= 1,
2,
3,
4,
5, find the optimal placement of nodes for the tangent-free envelope.
c.
Plot these collections of envelopes; also plot rejection sampling waste against number of nodes for both envelopes. Comment on your results.