Wenceslao González Manteiga

Editorial: Statistics for Functional Data

Functional data analysis is an active field of research in Statistics. This Special Issue on Statistics for Functional Data contains a selected set of contributions which covers a scope, as wide as possible, of this many-facetted discipline. The diversity of this field of statistics is highlighted by the wide scope of methodological problems discussed in this special issue. Also, the large set of applied scientific disciplines concerned with functional data appears through the numerous curves data set analyzed in these contributions. This introductory paper presents these contributions by emphasizing on how they are taking place in the actual development of statistical methods for analyzing functional data. A special, but not exclusive, place is given to the three more current kinds of problems: factorial analysis of functional data, regression with functional variables and curves classification. The links between functional data analysis and nonparametric statistics deserve a special attention.

PREDICTION OF SO2 LEVELS USING NEURAL NETWORKS

Abstract In this paper, we present an adaptation of the air pollution control help system in the neighborhood of a power plant in As Pontes (A Coruña, Spain), property of Endesa Generación S.A., to the European Council Directive 1999/30/CE. This system contains a statistic prediction made half an hour before the measurement, and it helps the staff in the power plant prevent air quality level episodes. The prediction is made using neural network models. This prediction is compared with one made by a semiparametric model.

Asymptotic normality of generalized functional estimators dependent on covariables

Abstract Given a random vector (X,Y) with distribution function H(x,y) such that the marginal random variable X has support S, different statistical parameters θ(x) associated with Y conditioned by X=x can be defined by means of functionals T:R×Θ× F →R (Θ a parameter space, F a space of distribution functions) as those for which ʃ T(y,θ(x),F(·|x))dF(y|x)=0 , where F(·|x) represents the conditional distribution of the random variable Y given X=x. This paper shows the asymptotic normality of the general class of estimators θn(x) (x∈S) defined as the solutions of ʃT(y, θ n (x), F n (·|x))d F n (y|x)=0 where Fn(y|x) is a non-parametric estimator of F(y|x). This result is applied to several particular cases.

Kernel smoothers and bootstrapping for semiparametric mixed effects models

While today linear mixed effects models are frequently used tools in different fields of statistics, in particular for studying data with clusters, longitudinal or multi-level structure, the nonparametric formulation of mixed effects models is still quite recent. In this paper we discuss and compare different nonparametric estimation methods. In this context we introduce a computationally inexpensive bootstrap method, which is used to estimate local mean squared errors, to construct confidence intervals and to find locally optimal smoothing parameters. The theoretical considerations are accompanied by the provision of algorithms and simulation studies of the finite sample behavior of the methods. We show that our confidence intervals have good coverage probabilities, and that our bandwidth selection method succeeds to minimize the mean squared error for the nonparametric function locally.

Nearest neighbor smoothing in linear regression

A new class of estimators is introduced for estimating the parameter ([theta]10, [theta]20) in the linear regression model y = E[YX = x] = [theta]10 + [theta]20x. Given independent copies {(X1, Y1),..., (Xn, Yn)} of the two-dimensional random vector (X, Y), these estimators are derived from minimizing the functional [psi]n([theta]) = [integral operator] (mn(x) - [theta]1 - [theta]2x)2[nu]n(dx), where mn(x) is a nearest neighbor type estimator of m(x) = E[YX = x] and [nu]n is an empirical measure. Strong consistency and asymptotic normality are proved under weak assumptions on (X, Y). Also a small sample comparison with LSE is incluced.

Empirical likelihood based testing for regression

Consider a random vector (X, Y ) and let m(x) = E(Y|X = x). We are interested in testing H0 : m ∈ MΘ,G = {γ (·, θ, g) : θ ∈ Θ, g ∈ G } for some known function γ , some compact set Θ ⊂ IRp and some function set G of real valued functions. Specific examples of this general hypothesis include testing for a parametric regression model, a generalized linear model, a partial linear model, a single index model, but also the selection of explanatory variables can be considered as a special case of this hypothesis. To test this null hypothesis, we make use of the so-called marked empirical process introduced by Diebolt (1995) and studied by Stute (1997) for the particular case of parametric regression, in combination with the modern technique of empirical likelihood theory in order to obtain a powerful testing procedure. The asymptotic validity of the proposed test is established, and its finite sample performance is compared with other existing tests by means of a simulation study.

Parametric conditional mean and variance testing with censored data

Suppose the random vector (X, Y ) satisfies the heteroscedastic regression model Y = m ( X ) + c.(X)&, where m(.) = E(YI.) , uz( ( ) = Var(Y1.) and E (with mean zero and variance one) is independent of X. The response Y is subject to random right censoring and the covariate X is completely observed. New goodness-of-fit testing procedures for m and uz(.) are proposed. They are based on a modified integrated regression function technique which uses the method of [Heuchenne and Van Keilegom, 2006b] to construct new versions of functions of the data points. Asymptotic representations of the processes are obtained and weak convergence to gaussian processes is deduced.

SUAVIZACION NO PARAMETRICA EN FIABILIDAD

ResumenEn este trabajo consideramos estimaciones no paramétricas de las funciones de razón de fallo y supervivencia en fiabilidad haciendo uso de suavizaciones no paramétricas de la función de distribución empírica (datos no censurados) y de la distribución de Kaplan-Meier (datos censurados). Se obtienen sesgos, varianzas y distribuciones asintóticas de los estimadores aquí propuestos probándose mediante técnicas de expansiones de segundo orden la eficiencia de éstos respecto de otras estimaciones introducidas en la literatura hasta la actualidad.SummaryIn this paper the nonparametric estimation of the hazard rate and survival functions is considered using one nonparametric smoothing of the empirical distribution (without censoring) and of the Kaplan-Meier distribution (with censoring). Bias, variances and asymptotic distributions are obtained from the estimators here introduced and the efficiency with respect to other estimators introduced up to now is proved using second order expressions for the bias and variance.

Aplicación de la suavización no paramétrica del tipo "K-puntos próximos" a modelos de regresión lineal

ResumenEn el modelo de regresión linealy=E(Y/X=x)=θx, donde (X, Y) es un vector aleatorio bidimensional, del que se dispone de una muestra {(X1,Y1),...,(Xn,Yn)}, se han introducido recientemente una clase general de estimadores para θ definida como aquellos valores que minimizan el funcional: