Estimating the Lambda measure in multiple-merger coalescents

Multiple-merger coalescents, also known as $\Lambda$-coalescents, have been used to describe the genealogy of populations that have a skewed offspring distribution or that undergo strong selection. Inferring the characteristic measure $\Lambda$, which describes the rates of the multiple-merger events, is key to understand these processes. So far, most inference methods only work for some particular families of $\Lambda$-coalescents that are described by only one parameter, but not for more general models. This article is devoted to the construction of a non-parametric estimator of the density of $\Lambda$ that is based on the observation at a single time of the so-called Site Frequency Spectrum (SFS), which describes the allelic frequencies in a present population sample. First, we produce estimates of the multiple-merger rates by solving a linear system, whose coefficients are obtained by appropriately subsampling the SFS. Then, we use a technique that aggregates the information extracted from the previous step through a kernel type of re-construction to give a non-parametric estimation of the measure $\Lambda$. We give a consistency result of this estimator under mild conditions on the behavior of $\Lambda$ around 0. We also show some numerical examples of how our method performs.