Electronic Journal of Statistics

Statistics for the Luria-Delbrück distribution

Agnès Hamon and Bernard Ycart

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The Luria-Delbrück 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-Delbrück 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 when the maximal value of the sample is large, which occurs frequently due to the heavy tail property. Based on the empirical probability 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.

Article information

Electron. J. Statist. Volume 6 (2012), 1251-1272.

First available in Project Euclid: 26 July 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D25: Population dynamics (general)
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces

Luria-Delbrück distribution fluctuation analysis Bellman-Harris branching process probability generating function estimator


Hamon, Agnès; Ycart, Bernard. Statistics for the Luria-Delbrück distribution. Electron. J. Statist. 6 (2012), 1251--1272. doi:10.1214/12-EJS711. https://projecteuclid.org/euclid.ejs/1343310297.

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Supplemental materials

  • Supplementary material: R script. Set of R functions for statistical computation with Luria-Delbrück distributions, including random sample simulation, estimation of parameters with ML and GF methods, asymptotic variance matrices, confidence intervals and p-values for hypothesis testing.