Minimax optimization of the numerical-statistical similar trajectory method
Importance sampling, weighted Monte-Carlo and similar trajectory methods need the estimation of some density \(p\) from a family \(P\) of densities such that the variance of \(g / p\) with respect to the density \(p\) is minimal. It is well-known that in the case of positive densities, the minimum is given for \(p=g\) whenever \(g\) itself is a positive density. The authors solve for given positive densities \(g_1,g_2\) the minimax problem \(\min_{p\in P}\max_{i=1,2}\{\int_{0}^{\infty}g_i^2(x)/p(x)dx\}\) for a couple of simple families \(P\) explicitly and extend the result to some \(g_1\) and \(g_2\) connecting one parametric families \(g(t), t\in \Sigma\). Finally, they shortly discuss in an example the application to integral equations with a similar trajectory method.
scientific article; zbMATH DE number 6124988