The authors consider the problem of estimating the density g of independent and identically distributed variables X i , from a sample 21,. . . , 2, such that 2; = Xi + QE; for i = 1,. . . , n, and E is noise independent of X, with QE having a known distribution. They present a model selection procedure allowing one to construct an adaptive estimator of g and to find nonasymptotic risk bounds. The estimator achieves the minimax rate of convergence, in most cases where lower bounds are available. A simulation study gives an illustration of the good practical performance of the method. Deconvolution adaptative de densite par contraste penaliseRbumt! : Les auteurs considkrent le probltme de dkonvolution, c'est-&-dire de I'estimation de la densitt de variables altatoires identiquement distributes X,, ii partir de I'observation de Zi, oh 2; = X , + QE, pour i = 1,. . . , n, et oh les erreurs QE; sont de densit6 connue. Par une proc6dure de selection de modtles qui permet d'obtenir des bornes de risque non asymptotiques, ils construisent un estimateur adaptatif de la densitt des X ; . L'estimateur atteint automatiquement la vitesse minimax dans la plupart des cas, que les erreurs ou la densit6 & estimer soient peu ou trhs dgulitres. Une ttude par simulation illustre les bonnes performances pratiques de la mtthode.

Given an n-sample from some unknown density f on [0, 1], it is easy to construct an histogram of the data based on some given partition of [0, 1], but not so much is known about an optimal choice of the partition, especially when the set of data is not large, even if one restricts to partitions into intervals of equal length. Existing methods are either rules of thumbs or based on asymptotic considerations and often involve some smoothness properties of f . Our purpose in this paper is to give a fully automatic and simple method to choose the number of bins of the partition from the data. It is based on a nonasymptotic evaluation of the performances of penalized maximum likelihood estimators in some exponential families due to Castellan and heavy simulations which allowed us to optimize the form of the penalty function. These simulations show that the method works quite well for sample sizes as small as 25.

We consider a one-dimensional diffusion process (Xt) which is observed at n + 1 discrete times with regular sampling interval ∆. Assuming that (Xt) is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2007, Vol. 13, No. 2, 514-543. This reprint differs from the original in pagination and typographic detail.

BackgroundCirculating tumor DNA (ctDNA) is an approved noninvasive biomarker to test for the presence of EGFR mutations at diagnosis or recurrence of lung cancer. However, studies evaluating ctDNA as a noninvasive “real-time” biomarker to provide prognostic and predictive information in treatment monitoring have given inconsistent results, mainly due to methodological differences. We have recently validated a next-generation sequencing (NGS) approach to detect ctDNA. Using this new approach, we evaluated the clinical usefulness of ctDNA monitoring in a prospective observational series of patients with non-small cell lung cancer (NSCLC).Methods and FindingsWe recruited 124 patients with newly diagnosed advanced NSCLC for ctDNA monitoring. The primary objective was to analyze the prognostic value of baseline ctDNA on overall survival. ctDNA was assessed by ultra-deep targeted NGS using our dedicated variant caller algorithm. Common mutations were validated by digital PCR. Out of the 109 patients with at least one follow-up marker mutation, plasma samples were contributive at baseline (n = 105), at first evaluation (n = 85), and at tumor progression (n = 66). We found that the presence of ctDNA at baseline was an independent marker of poor prognosis, with a median overall survival of 13.6 versus 21.5 mo (adjusted hazard ratio [HR] 1.82, 95% CI 1.01–3.55, p = 0.045) and a median progression-free survival of 4.9 versus 10.4 mo (adjusted HR 2.14, 95% CI 1.30–3.67, p = 0.002). It was also related to the presence of bone and liver metastasis. At first evaluation (E1) after treatment initiation, residual ctDNA was an early predictor of treatment benefit as judged by best radiological response and progression-free survival. Finally, negative ctDNA at E1 was associated with overall survival independently of Response Evaluation Criteria in Solid Tumors (RECIST) (HR 3.27, 95% CI 1.66–6.40, p < 0.001). Study population heterogeneity, over-representation of EGFR-mutated patients, and heterogeneous treatment types might limit the conclusions of this study, which require future validation in independent populations.ConclusionsIn this study of patients with newly diagnosed NSCLC, we found that ctDNA detection using targeted NGS was associated with poor prognosis. The heterogeneity of lung cancer molecular alterations, particularly at time of progression, impairs the ability of individual gene testing to accurately detect ctDNA in unselected patients. Further investigations are needed to evaluate the clinical impact of earlier evaluation times at 1 or 2 wk. Supporting clinical decisions, such as early treatment switching based on ctDNA positivity at first evaluation, will require dedicated interventional studies.