Asymptotic unbiased density estimator

International audienceThis paper introduces a computationally tractable density estimator that has the same asymptotic variance as the classical Nadaraya-Watson density estimator but whose asymptotic bias is zero. We achieve this result using a two stage estimator that applies a multiplicative bias correction to an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator. Mathematics Subject Classification. 62G07, 62G20

Estimation de quantiles extrêmes et de probabilités d'événements rares

International audienceEtant donné une probabilité $\mu$ sur $\R^d$ ($d$ grand), on note $X$ un vecteur aléatoire générique de loi $\mu$ et $\Phi:\R^d\rightarrow\R$ une application ``boîte noire''. Un réel $q$ étant fixé, le but est de générer un échantillon i.i.d. $(X_1,\dots,X_N)$ tel que pour tout $i$ : $X_i\sim{\cal L}(X|\Phi(X)>q)$. Lorsque $q$ est grand comparé aux valeurs typiques de la variable $\Phi(X)$, la méthode Monte-Carlo classique devient trop coûteuse. Dans ce travail nous présentons et analysons une nouvelle approche pour ce problème. Celle-ci procède en plusieurs étapes, s'inspirant de l'algorithme de Metropolis-Hastings et des méthodes dites multi-niveaux en estimation d'événements rares. Deux problèmes peuvent être traités très facilement via cette nouvelle méthode : estimation de quantiles extrêmes et estimation d'événements rares. Les idées présentées seront illustrées sur un problème de tatouage numérique

Iterative Bias Reduction Multivariate Smoothing in R: The ibr Package

In multivariate nonparametric analysis curse of dimensionality forces one to use large smoothing parameters. This leads to a biased smoother. Instead of focusing on optimally selecting the smoothing parameter, we fix it to some reasonably large value to ensure an over-smoothing of the data. The resulting base smoother has a small variance but a substantial bias. In this paper, we propose an R package named ibr to iteratively correct the initial bias of the (base) estimator by an estimate of the bias obtained by smoothing the residuals. After a brief description of iterated bias reduction smoothers, we examine the base smoothers implemented in the package: Nadaraya-Watson kernel smoothers, Duchon splines smoothers and their low rank counterparts. Then, we explain the stopping rules available in the package and their implementation. Finally we illustrate the package on two examples: a toy example in R2 and the original Los Angeles ozone dataset

Inequity Measures for Evaluations of Environmental Justice: A Case Study of Close Proximity to Highways in NYC

Assessments of environmental and territorial justice are similar in that both assess whether empirical relations between the spatial arrangement of undesirable hazards (or desirable public goods and services) and socio-demographic groups are consistent with notions of social justice, evaluating the spatial distribution of benefits and burdens (outcome equity) and the process that produces observed differences (process equity. Using proximity to major highways in NYC as a case study, we review methodological issues pertinent to both fields and discuss choice and computation of exposure measures, but focus primarily on measures of inequity. We present inequity measures computed from the empirically estimated joint distribution of exposure and demographics and compare them to traditional measures such as linear regression, logistic regression and Theil’s entropy index. We find that measures computed from the full joint distribution provide more unified, transparent and intuitive operational definitions of inequity and show how the approach can be used to structure siting and decommissioning decisions