The uniform approach to calculation of MISE for histogram and density box-spline estimators gives us a possibility to obtain estimators of derivatives of densities and the asymptotic constant.

We prove the central limit theorem for the integrated square error of multivariate box-spline density estimators.

The aim of this paper is to apply the appropriate numerical, statistical and computer techniques to the construction of approximate solutions to nonlinear 2nd order stochastic differential equations modeling some engineering systems subject to large random external disturbances. This provides us with quantitative results on their asymptotic behavior.

The aim of this paper is to demonstrate how the appropriate numerical, statistical and computer techniques can be successfully applied to the construction of approximate solutions of stochastic differential equations modeling some engineering systems subject to large disturbances. In particular, the evolution in time of densities of stochastic processes solving such problems is discussed.

Se define un estimador no paramétrico, recursivo, de la función de regresión r(x) = E(Y/X = x), que se calcula a partir de un conjunto de n observaciones {(X1,Yi): i = 1, ..., n} del vector aleatorio (X,Y). Bajo la hipótesis de que los datos son idénticamente distribuidos pero no necesariamente independientes, lo que permite utilizar el estimador definido para estimar la función de autorregresión de una serie de tiempo, se obtienen resultados sobre la consistencia puntual débil (en probabilidad)...

The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,..., for the observation model $y_i=f\left(x_i\right)+{\eta }_i$, i=1,...,n, where the ${\eta }_i$ are independent random variables with zero mean value and finite variance, and the observation points $x_i\in [a,b]$, i=1,...,n, form a random sample from a distribution with density $\varrho \in L^1[a,b]$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors $\parallel f-{\widehat{f}}_N\parallel ,|f\left(x\right){-}_N\left(x\right)|$, $x\in [a,b]$,...

In the companion paper [C. Maugis and B. Michel, A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM: P&S 15 (2011) 41–68] , a penalized likelihood criterion is proposed to select a Gaussian mixture model among a specific model collection. This criterion depends on unknown constants which have to be calibrated in practical situations. A “slope heuristics” method is described and experimented to deal with this practical problem. In a model-based clustering context,...

In the companion paper [C. Maugis and B. Michel, A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM: P&S15 (2011) 41–68] , a penalized likelihood criterion is proposed to select a Gaussian mixture model among a specific model collection. This criterion depends on unknown constants which have to be calibrated in practical situations. A “slope heuristics” method is described and experimented to deal with this practical problem. In a model-based clustering context, the...

We study the density deconvolution problem when the random variables of interest are an associated strictly stationary sequence and the random noises are i.i.d. with a nonstandard density. Based on a nonparametric strategy, we introduce an estimator depending on two parameters. This estimator is shown to be consistent with respect to the mean integrated squared error. Under additional regularity assumptions on the target function as well as on the density of noises, some error estimates are derived....

We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^2$-approximation of a probability density function $f$. If $f$ itself is a step-function the number of jumps may be unknown.

We propose a feature selection method for density estimation with quadratic loss. This method relies on the study of unidimensional approximation models and on the definition of confidence regions for the density thanks to these models. It is quite general and includes cases of interest like detection of relevant wavelets coefficients or selection of support vectors in SVM. In the general case, we prove that every selected feature actually improves the performance of the estimator. In the case...

In this paper we consider a smoothness parameter estimation problem for a density function. The smoothness parameter of a function is defined in terms of Besov spaces. This paper is an extension of recent results (K. Dziedziul, M. Kucharska, B. Wolnik, Estimation of the smoothness parameter). The construction of the estimator is based on wavelets coefficients. Although we believe that the effective estimation of the smoothness parameter is impossible in general case, we can show that it becomes...

In this paper, a very useful lemma (in two versions) is proved: it simplifies notably the essential step to establish a Lindeberg central limit theorem for dependent processes. Then, applying this lemma to weakly dependent processes introduced in Doukhan and Louhichi (1999), a new central limit theorem is obtained for sample mean or kernel density estimator. Moreover, by using the subsampling, extensions under weaker assumptions of these central limit theorems are provided. All the usual causal...

This work deals with a class of discrete-time zero-sum Markov games whose state process $\left\{x_t\right\}$ evolves according to the equation $x_{t+1}=F(x_t,a_t,b_t,{\xi }_t),$ where $a_t$ and $b_t$ represent the actions of player 1 and 2, respectively, and $\left\{{\xi }_t\right\}$ is a sequence of independent and identically distributed random variables with unknown distribution $\theta$. Assuming possibly unbounded payoff, and using the empirical distribution to estimate $\theta$, we introduce approximation schemes for the value of the game as well as for optimal strategies considering both,...

Se definen en este trabajo r-desarrollos de Neumann y se prueba que toda densidad de probabilidad f admite un desarrollo r-convergente a f.Los resultados obtenidos se aplican a la estimación de f sin la suposición de que sea de cuadrado integrable, estudiándose propiedades asintóticas de los estimadores e ilustrándose con un ejemplo de aplicación.