Question

Cauchy distribution and rejection sampling We have seen in the lectures that we can not sample Cauchy random variables using a normal density as enveloppe function or proposal distribution. However, it is possible to generate standard normal random variables from Cauchy.

(a) Justify why the Cauchy distribution is an appropriate proposal distribution, and find the optimal value of M.

Answer 1

The Cauchy distribution is appropriate for rejection sampling and the optimal value of M depends on the problem. M should be chosen to closely envelope the target distribution.

Step-by-step explanation:

The Cauchy distribution is an appropriate proposal distribution because it has heavy tails and allows for extreme values, which can be useful in capturing the characteristics of the Cauchy random variables being sampled. When using rejection sampling, the optimal value of M (the upper bound on the proposal distribution's density) depends on the specific problem at hand. In general, M should be chosen such that the proposal distribution envelops the target distribution as closely as possible without being too inefficient.