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.

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Background Obstructive sleep apnea (OSA) is a chronic disease characterized by recurrent pharyngeal collapses during sleep. In most severe cases, continuous positive airway pressure (CPAP) consists in keeping the airways open by administering mild air pressure. This treatment faces adherence issues. Objectives Eight hundred and forty‐eight subjects were equipped with CPAP prescribed at the Grenoble University Hospital between 2016 and 2018. Their daily CPAP uses have been recorded during the first 3 months. Our aim is to cluster these adherence time series. With hierarchical agglomerative clustering, we focused on the choices of the dissimilarity measure and the internal cluster validation index (CVI). Methods The Euclidean distance, the dynamic time warping (DTW) and the generalized summed discrete Fréchet dissimilarity were implemented with three linkage strategies (“average,” “complete,” and “Ward”). The performances of each method (dissimilarity and linkage) were evaluated on a simulation study through the adjusted Rand index (ARI). The Ward linkage with DTW dissimilarity provided the best ARI. Then six different internal CVIs (Silhouette, Calinski Harabasz, Davies Bouldin, Modified Davies Bouldin, Dunn, and COP) were compared on their ability to choose the best number of clusters. The Dunn index beat the others. Results CPAP data were clustered with the Ward linkage, the DTW dissimilarity and the Dunn index. It identified six clusters, from a cluster of patients (N = 29 subjects) whose stopped the therapy early on to a cluster (N = 105) with increasing adherence over time. Other clusters were extremely good users (N = 151), good users (N = 150), moderate users (N = 235), and poor adherers (N = 178).

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