  Multiple importance sampling (MIS) is an increasingly used methodology where
several proposal densities are used to approximate integrals, generally
involving target probability density functions. The use of several proposals
allows for a large variety of sampling and weighting schemes. Then, the
practitioner must choose a given scheme, i.e., sampling mechanism and weighting
function. A variance analysis has been proposed in Elvira et al (2019,
Statistical Science 34, 129-155), showing the superiority of the balanced
heuristic estimator with respect to other competing schemes in some scenarios.
However, some of their results are valid only for two proposals. In this paper,
we extend and generalize these results, providing novel proofs that allow to
determine the variance relations among MIS schemes.
