Confidence intervals for nonhomogeneous branching processes and polymerase chain reactions

Abstract

We extend in two directions our previous results about the sampling and the empirical measures of immortal branching Markov processes. Direct applications to molecular biology are rigorous estimates of the mutation rates of polymerase chain reactions from uniform samples of the population after the reaction. First, we consider nonhomogeneous processes, which are more adapted to real reactions. Second, recalling that the first moment estimator is analytically known only in the infinite population limit, we provide rigorous confidence intervals for this estimator that are valid for any finite population. Our bounds are explicit, nonasymptotic and valid for a wide class of nonhomogeneous branching Markov processes that we describe in detail. In the setting of polymerase chain reactions, our results imply that enlarging the size of the sample becomes useless for surprisingly small sizes. Establishing confidence intervals requires precise estimates of the second moment of random samples. The proof of these estimates is more involved than the proofs that allowed us, in a previous paper, to deal with the first moment. On the other hand, our method uses various, seemingly new, monotonicity properties of the harmonic moments of sums of exchangeable random variables.