Sélection automatique du paramètre de lissage pour l'estimation non paramétrique de la régression pour des données fonctionnelles

We study regression estimation when the explanatory variable is functional. Nonparametric estimates of the regression operator have been recently introduced. They depend on a smoothing factor which controls its behaviour, and the aim of our Note is to construct some data-driven criterion for choosing this smoothing parameter. The criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown regression operator, it is seen that this rule is asymptotically optimal. As by-products of this result, we state asymptotic equivalences for several measures of accuracy for nonparametric estimate of the regression operator. To cite this article: M. Rachdi, P. Vieu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Cardinal splines in nonparametric regression

We discuss a nonparametric regression model on an equidistant grid of the real line. A class of kernel type estimates based on the so-called fundamental cardinal splines will be introduced. Asymptotic optimality of these estimates will be established for certain functional classes. This model explains the often mentioned heuristic fact that cubic splines are adequate for most practical applications.      

Likelihood-Based Local Polynomial Fitting for Single-Index Models

The parametric generalized linear model assumes that the conditional distribution of a response Y given a d-dimensional covariate X belongs to an exponential family and that a known transformation of the regression function is linear in X. In this paper we relax the latter assumption by considering a nonparametric function of the linear combination β^TX, say η₀(β^TX). To estimate the coefficient vector β and the nonparametric component η₀ we consider local polynomial fits based on kernel weighted conditional likelihoods. We then obtain an estimator of the regression function by simply replacing β and η₀ in η₀(β^TX) by these estimators. We derive the asymptotic distributions of these estimators and give the results of some numerical experiments.     

On the complexity of a class of combinatorial optimization problems with uncertainty

We consider a robust (minmax-regret) version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements. We present a polynomial algorithm with the order of complexity O((min {p,m-p})² m) for the case where uncertainty is represented by means of interval estimates for the weights. We show that the problem is NP-hard in the case of an arbitrary finite set of possible scenarios, even if there are only two possible scenarios. This is the first known example of a robust combinatorial optimization problem that is NP-hard in the case of scenario-represented uncertainty but is polynomially solvable in the case of the interval representation of uncertainty. Received: July 1998 / Accepted: May 2000?Published online March 22, 2001     

Tests de structure en régression sur variable fonctionnelle

In this Note we introduce a general approach to construct structural testing procedures in regression on functional variables. In the case of multivariate explanatory variables a well-known method consists in a comparison between a nonparametric estimator and a particular one. We adapt this approach to the case of functional explanatory variables. We give the asymptotic law of the proposed test statistic. The general approach used allows us to cover a large scope of possible applications as tests for no-effect, tests for linearity, …. To cite this article: L. Delsol, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Random weights, robust lattice rules and the geometry of the cbcrc algorithm

In this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube [0,1] ^s from a weighted reproducing kernel Hilbert space. We assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications: (i) It is usually not known in practice how to choose the weights. Thus by assuming that the weights are random variables, we obtain robust constructions (with respect to the weights) of lattice rules. This, to some extend, removes the necessity to carefully choose the weights. (ii) In practice it is convenient to use the same lattice rule for many different integrands. The best choice of weights for each integrand may vary to some degree, hence considering the weights random variables does justice to how lattice rules are used in applications. In this paper the worst-case error is therefore a random variable depending on random weights. We show how one can construct lattice rules which perform well for weights taken from a set with large measure. Such lattice rules are therefore robust with respect to certain changes in the weights. The construction algorithm uses the component-by-component (cbc) idea based on two criteria, one using the mean of the worst case error and the second criterion using a bound on the variance of the worst-case error. We call the new algorithm the cbc2c (component-by-component with 2 constraints) algorithm. We also study a generalized version which uses r constraints which we call the cbcrc (component-by-component with r constraints) algorithm. We show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights ?? ⁽¹⁾, . . . , ?? ^(r). Thus, in applications, instead of finding one set of weights, it is enough to find a convex polytope in which the optimal weights lie. The price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule. Thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules. Numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights.     

Convergence de l'estimateur à noyau des k plus proches voisins en régression fonctionnelle non-paramétrique

This note focuses on the k nearest neighbor method when one regresses a real random variable on a functional random variable (i.e. valued in an infinite-dimensional space). More precisely, we consider a kernel estimator of the regression based on a local bandwidth using exactly the k nearest neighbors. Although it is frequently used in functional data analysis, this method has not given any theoretical result so far. The aim of this Note is to show the pointwise almost-complete convergence of the k nearest neighbor kernel estimator in nonparametric functional regression. To cite this article: F. Burba et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Régression non-paramétrique pour des variables aléatoires fonctionnelles mélangeantesNonparametric regression for mixing functional random variables

This paper concerns a nonparametric regression model for dependent functional variables. The aim is to explain a r.r.v. Y by a functional regressor namely a variable X which is valued in some semi-normed vector space. A kernel type estimator is proposed and asymptotics with rates are proved under strong mixing assumption. To cite this article: F. Ferraty, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 217–220.     

Adaptive nonparametric estimation of a multivariate regression function

We consider the kernel estimation of a multivariate regression function at a point. Theoretical choices of the bandwidth are possible for attaining minimum mean squared error or for local scaling, in the sense of asymptotic distribution. However, these choices are not available in practice. We follow the approach of Krieger and Pickands (Ann. Statist.9 (1981) 1066–1078) and Abramson (J. Multivariate Anal.12 (1982), 562–567) in constructing adaptive estimates after demonstrating the weak convergence of some error process. As consequences, efficient data-driven consistent estimation is feasible, and data-driven local scaling is also feasible. In the latter instance, nearest-neighbor-type estimates and variance-stabilizing estimates are obtained as special cases.     

On the distribution of simple zeros of polynomials

Erd s and Turán discussed in (Ann. of Math. 41 (1940), 162–173; 51 (1950), 105–119) the distribution of the zeros of monic polynomials if their Chebyshev norm on [−1, 1] or on the unit disk is known. We sharpen this result to the case that all zeros of the polynomials are simple. As applications, estimates for the distribution of the zeros of orthogonal polynomials and the distribution of the alternation points in Chebyshev polynomial approximation are given. This last result sharpens a well-known error bound of Kadec (Amer. Math. Soc. Transl. 26 (1963), 231–234).     

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH: ∫^π_−π K{f(λ)} dλ=cagainstA: ∫^π_−π K{f(λ)} dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testT_nbased on ∫^π_−π K{f(λ)} dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofT_nunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf₄^Zofz(t). If it does not depend onf₄^Z, we say thatT_nis non-Gaussian robust. We will give sufficient conditions forT_nto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.     

A Note on the Robust 1-Center Problem on Trees

We consider the robust 1-center problem on trees with uncertainty in vertex weights and edge lengths. The weights of the vertices and the lengths of the edges can take any value in prespecified intervals with unknown distribution. We show that this problem can be solved in O(n ³ logn) time thus improving on Averbakh and Berman's algorithm with time complexity O(n ⁶). For the case when the vertices of the tree have weights equal to 1 we show that the robust 1-center problem can be solved in O(nlogn) time, again improving on Averbakh and Berman's time complexity of O(n ² logn).     

Estimation of the distribution of random shifts deformation

Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution μ modeling the variability in the response of each individual. Our aim is to construct a nonparametric estimator of the density of these random translation deformations using semiparametric preliminary estimates of the shifts. Based on the results of Dalalyan et al. [7], semiparametric estimators are obtained in our discrete framework and their performance studied. From these estimates we construct a nonparametric estimator of the target density. Both rates of convergence and an algorithm to construct the estimator are provided.      

Estimating the direction in which a data set is most interesting

Summary Estimation of orientation is a key operation at each step in projection pursuit. Since projection pursuit is a nonparametric algorithm, and since even low-dimensional approximations to the target function must converge to their limits at rates considerably slower than n _-1 ² (where n is sample size), then it might be thought that the same is true of orientation estimates. It is shown in the present paper that this is not the case, and that estimation of orientation is a parametric operation, in the sense that, under mild nonparametric assumptions, correctly-chosen kernel-type orientation estimates converge to their limits at rate n _-1 ² . This property is not enjoyed by standard projection pursuit orientation estimates, which converge at a slower rate than n _-1 ² . Most attention in the present paper is focussed on the case of projection pursuit density approximation, but it is pointed out that our arguments hold generally. An important practical conclusion is that data should be smoothed less when estimating orientation than when constructing the final projection pursuit approximation.     

Interpolation approximations based on Gauss–Lobatto–Legendre–Birkhoff quadrature

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss–Lobatto–Legendre–Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a user-oriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach allows an exact imposition of Neumann boundary conditions, and is as efficient as the pseudospectral methods based on Gauss–Lobatto quadrature for PDEs with Dirichlet boundary conditions.     

On estimation of the L r norm of a regression function

f be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, L _r norms ||f|| _r of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L ₂ ) functional or estimating a singular functional like the value of f at certain point or the maximum of f. In the first case, the convergence rate typically is n ^−1/2, n being the number of observations. In the second case, the rate of convergence coincides with the one of estimating the function f itself in the corresponding norm. We show that the case of estimating ||f|| _r is in some sense intermediate between the above extremes. The optimal rate of convergence is worse than n ^−1/2 but is better than the rate of convergence of nonparametric estimates of f. The results depend on the value of r. For r even integer, the rate occurs to be n ^{−β/(2β+1−1/r)} where β is the degree of smoothness. If r is not an even integer, then the nonparametric rate n ^{−β/(2β+1)} can be improved, but only by a logarithmic in n factor. Received: 6 February 1996hinspaceairsp/Revised version: 10 June 1998     

Application of semi-definite programming to robust stability of delay systems

We study here robust stability of linear systems with several uncertain incommensurate delays, more precisely the property usually called delay-dependent stability. The main result of this paper consists in establishing that the latter is equivalent to the feasibility of some Linear Matrix Inequality (LMI), a convex optimization problem whose numerical solution is well documented.The method is based on two main techniques:

• use of Padé approximation to transform the system into some singularly perturbed finite-dimensional system, for which robust dichotomy has to be checked;

• recursive applications of Generalized Kalman–Yakubovich–Popov (KYP) lemma to characterise by an LMI the previous property.

Vitesses de convergence uniforme presque sûre d'estimateurs non-paramétriques de la régression

We obtain rates of strong uniform consistency for some nonparametric regression estimators, including the local linear regression and some wavelet estimators. Our method of proof relies on recent empirical process theory developed by Deheuvels and Mason (Statist. Inference Stoch. Process. 7 (3) (2004) 225–277). To cite this article: D. Blondin et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A strong consistency of a nonparametric estimate of entropy under random censorship

The purpose of this Note is to provide the rate of strong consistency for a nonparametric estimator of entropy under random censorship. We also establish an uniform-in-bandwidth consistency for this estimator. To cite this article: S. Bouzebda, I. Elhattab, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Some Functional Large Deviations Principles in?Nonparametric Function Estimation

In this paper, we investigate functional large deviation behaviors of some nonparametric function estimates. As a first step, we define a a vector process W _n and study its large deviation behavior in the space L ₁×L ₁×L ₁ with respect to the weak convergence topology. As by-products, we derive large deviation principles in the L ₁-space equipped with the weak convergence topology simultaneously for several density and regression estimators built up using the delta-sequence estimation method.