• Mathématiques/Statistiques
  
• Statistiques/Théorie
  
• Physique/Astrophysique/Cosmologie et astrophysique extra-galactique
  
• Planète et Univers/Astrophysique
  
• Planète et Univers/Océan, Atmosphère
  

A frequent issue in the study of species abundance consists in modeling empirical distributions of repeated counts by parametric probability distributions. In this setting, it is desirable that the chosen family of distributions is exible enough to take into account very diverse patterns, and that its parameters possess clear biological/ecological meanings. This is the case of the Negative Binomial distribution, chosen in this work for modeling counts of marine shes and invertebrates. This distribution depends on a vector (K, P) of parameters, and ranges from the Poisson distribution (when K → +∞) to Fisher's log-series, when K → 0. Besides, these parameters have biologi-cal/ecological interpretations detailed in the literature and reminded hereafter. We focus on the comparison of three estimators of K, P and the parameter α of Fisher's log-series, revisiting a nice paper of Rao (1971) about a three-parameter unstandardized variant of the Negative Binomial distribution. We investigate the coherency of values of the parameters resulting from these different estimators, with both real count data collected in the Mauritanian Exclusive Economic Zone during the period 1987-2010 and realistics simulations of theses data. In the rst case, we rst built homogeneous lists of counts (replicates), by gathering observations of each species with respect to typical environments obtained by clustering the sampled stations. The best estimation of (K, P) was generally obtained by Penalized Minimum Hellinger Distance Estimation. Interestingly, the parameters of most of the correctly sampled species seem compatible with a classical birth-and-dead model of population growth with immigration of Kendall (1948).

Given a spatial random process (Xi; Yi) 2 E R; i 2 ZN , we investigate a nonparametric estimate of the conditional expectation of the real random variable Yi given the functional random field Xi valued in a semi-metric space E. The weak and strong consistencies of the estimate are shown and almost sure rates of convergence are given. Special attention is paid to apply the regression estimate introduced to spatial prediction problems.

Nous nous intéressons à l'estimation de la fonction de régression $r(x)=Eleft(Y_{mathbfu}|X_{mathbfu}=xright)$ à partir d'observations d'un processus $left{ Z_{mathbfi}=left(X_{mathbfi}, Y_{mathbfi}right),,mathbfiinmathbbZ^Nright}. On suppose que les variables $Z_{mathbfi}$ sont de même distribution que $Z=(X,Y)$, où $Y$ est une variable réelle, intégrable et $X$ un vecteur aléatoire à valeurs dans un espace séparable $mathcalE$ muni (éventuellement de dimension infinie). Dans ce travail, la convergence nos estimateurs est étudiée sous conditions de mélange à partir d'observations dans une région rectangulaire de $mathbbZ^N$. Nous illustrerons nos résultats par des simulations. L'application de nos méthodes à la prédiction spatiale sera également abordée.

In this paper, we propose a dimension reduction model for spatially dependent variables. Namely, we investigate an extension of the emph{inverse regression} method under strong mixing condition. This method is based on estimation of the matrix of covariance of the expectation of the explanatory given the dependent variable, called the emph{inverse regression}. Then, we study, under strong mixing condition, the weak and strong consistency of this estimate, using a kernel estimate of the emph{inverse regression}. We provide the asymptotic behaviour of this estimate. A spatial predictor based on this dimension reduction approach is also proposed. This latter appears as an alternative to the spatial non-parametric predictor.