The selection of the number of terms in an orthogonal series cumulative function estimator

International audienceLet h(.) be a continuous, strictly positive probability density function over an interval [a, b] and H(.) its associated cumulative distribution function (cdf). Given a sample set X1,…,Xn of independent identically distributed variables, we want to estimate H(.) from this sample set. The present work has two goals. The first one is to propose an estimator of a cdf based on an orthogonal trigonometric series and to give its statistical and asymptotic proprieties (bias, variance, mean square error, mean integrated squared error, convergence of the bias, convergence of variance, convergence of the mean squared error, convergence of the mean integrated squared error, uniform convergence in probability and the rate of convergence of the mean integrated squared error). The second is to introduce a new method for the selection of a “smoothing parameter”. The comparison by simulation between this method and Kronmal–Tarter’s method, shows that the new method is more performant in the sense of the mean integrated square error

Application des lois non paramétriques dans les systèmes d'attente et la théorie de renouvellement

Les distributions non paramétriques de survie trouvent, de plus en plus, des applications dans des domaines très variés, à savoir: théorie de fiabilité et analyse de survie, files d'attente, maintenance, gestion de stock, théorie de l'économie, ... L'objet de ce travail est d'utiliser les bornes inférieures et supérieures (en terme de la moyenne) des fonctions de fiabilité appartenant aux classes de distribution de type IFR, DFR, NBU et NWU, présentées par Sengupta (1994), pour l'évaluation de certaines caractéristiques. Nous utilisons certaines de ces lois pour l'évaluation des bornes du temps moyen d'attente dans la file d'un système d'attente de type GI/GI/1, en actualisant celles trouvées par Stoyan (1983). Comme application à la théorie de renouvellement et de fiabilité, nous utilisons les propriétés qualitatives des temps de réparation pour borner le temps moyen de vie d'un système à deux éléments réparables

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

International audienceThis study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estima-tors. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivari-ate regression model used as start regression function belongs to a neighbourhood of the true regression model

A multivariate non-parametric kernel estimator for global sensitivity analysis

International audienceTo estimate how a model output is influenced by the variations of inputs has become an important problematic in reliability and sensitivity analyses. This paper is interested in estimating sensitivity indices useful to quantify the contribution of inputs to the variance of model output. A multivariate mixed kernel estimator is investigated since, until now, discrete and continuous inputs have been separately considered in kernel estimation of sensitivity indices. To illustrate the differences between the influence of mixed, discrete and continuous inputs, analytical expressions of Sobol sensitivity indices are expressed in these three cases for the Ishigami test function. Besides, the performance of mixed kernel estimator is illustrated through simulations in which the Bayesian procedure is applied for bandwidth parameter choice. An application is also realized on a real example. Finally, to use an appropriate kernel estimator according to the type of inputs is found to be influential on the accuracy of sensitivity indice estimates

Sur l'équivalence entre la théorie de risque et la théorie de files d'attente

International audienceDans ce travail, nous nous intéressons à l'étude de l'interaction qui existe entre la théorie des files d'attente et la théorie de risque. En particulier, nous détaillons la dualité qui existe entre le système de files d'attente M/M/1 et le modèle de risque classique. Par la suite, nous développons une approche de simulation pour calculer la probabilité de ruine dans un modèle de risque et le temps d'attente dans un système de files d'attente afin d'illustrer numériquement cette dualité.</p

Influence of the density pole on the performances of its gamma-kernel estimator

In this paper, we aim at highlighting the in uence of the density pole on the performances of its gamma-kernel estimator. To do this, we performed a comparative study for the performances of the gamma-kernel estimators with those provided by other bias effect correction techniques at the bounds, using the simulation technique. In conclusion, the results obtained conrm those provided in the literature and show that in some cases the normalization of the gamma estimators can considerably improve local and global performances of the gamma-kernel estimators