A sub-critical branching process model for application to analysing Y haplotype DNA mixtures.

The treatment of short-tandem-repeat (STR) loci on the Y chromosome presents special problems in the forensic analysis of DNA mixtures, chiefly but not exclusively relating to the linkage of Y-STR loci which precludes the use of the `product rule' for estimating Y-haplotype match probabilities. In recent paper, Andersen and Balding(2017) estimated, via a population simulation model, the distribution of the number of haplotypes sharing a common profile over a set of Y-STR loci, and argued for its use as an alternative to estimating Y-haplotype match probabilities. In this paper we present a sub-critical branching process model that approximates their population model, and show how to estimate the haplotype number distribution numerically using multivariate probability generating functions. It is shown that the approximation provides a good fit to their simulations. The model is extended to propose a new framework for evaluating the weight-of-evidence of Y-STR haplotype mixtures, and it is illustrated with publicly available data of a three person DNA mixture.