Statistics for the Luria-Delbr\"uck distribution

TL;DR

This paper provides a rigorous probabilistic foundation for the Luria-Delbrück distribution, introduces robust estimators based on empirical generating functions, and compares their performance to maximum likelihood methods.

Contribution

It offers a rigorous proof of the distribution's convergence, interprets it probabilistically, and proposes stable, efficient estimators for its parameters.

Findings

Robust estimators outperform maximum likelihood in numerical stability.

Estimators have comparable precision to MLE with broader calculability.

Asymptotic variance of estimators is derived.

Abstract

The Luria-Delbr\"uck distribution is a classical model of mutations in cell kinetics. It is obtained as a limit when the probability of mutation tends to zero and the number of divisions to infinity. It can be interpreted as a compound Poisson distribution (for the number of mutations) of exponential mixtures (for the developing time of mutant clones) of geometric distributions (for the number of cells produced by a mutant clone in a given time). The probabilistic interpretation, and a rigourous proof of convergence in the general case, are deduced from classical results on Bellman-Harris branching processes. The two parameters of the Luria-Delbr\"uck distribution are the expected number of mutations, which is the parameter of interest, and the relative fitness of normal cells compared to mutants, which is the heavy tail exponent. Both can be simultaneously estimated by the maximum likehood method. However, the computation becomes numerically unstable as soon as the maximal value of the sample is large, which occurs frequently due to the heavy tail property. Based on the empirical generating function, robust estimators are proposed and their asymptotic variance is given. They are comparable in precision to maximum likelihood estimators, with a much broader range of calculability, a better numerical stability, and a negligible computing time.