I have two questions that have puzzled me for a while. I hope you can shed some lights. They are all about Example 3.6 of your book.
1. On page 74, there is a term x(1-x) for m(x). This is fine. But the term disappeared from (3.5) on p.75. My impression is that this is not a typo. There must be a reason for its disappearance. Can you elaborate?
I am alas afraid this is a plain typo, where I did not report the x(1-x) from one page to the next.
2. On page 75, you have the term “den=dt(normx,3)”. My impression is that you are using univariate t with 3 degrees of freedom to approximate. I thought formally you need to use a bivariatet with 3 degrees of freedom to do the importance sampling. Why would normx=sqrt(x[,1]^2+x[,2]^2) along with a univariate t work?
This is a shortcut that would require more explanation. While the two-dimensional t sample is y, a linear transform of the isotonic x, it is possible to express the density of y via the one-dimensional t density, hence the apparent confusion between univariate and bivariate t densities…