Mustapha Rachdi

Functional data: local linear estimation of the conditional density and its application

In this paper, we introduce a new nonparametric estimation procedure of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish both the pointwise and the uniform almost-complete consistencies with convergence rates of the conditional density estimator related to this estimation procedure. Moreover, we give some particular cases of our results which can also be considered as novel in the finite-dimensional setting. Notice also that the results of this paper are used to derive some asymptotic properties of the local linear estimator of the conditional mode.

Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data

The problem of the nonparametric local linear estimation of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space is considered. Some theoretical and practical asymptotic properties of this estimator are established. The usefulness of the estimator is highlighted through the exact expression involved in the leading terms of the quadratic error, and by conducting a computational investigation to show the superiority of this estimation method for the conditional density and then for the conditional mode. Moreover, in order to verify the pertinence of the technique, from a practical point of view, it is applied to a real dataset.

Demography and diffusion in epidemics: malaria and black death spread.

The classical models of epidemics dynamics by Ross and McKendrick have to be revisited in order to incorporate elements coming from the demography (fecundity, mortality and migration) both of host and vector populations and from the diffusion and mutation of infectious agents. The classical approach is indeed dealing with populations supposed to be constant during the epidemic wave, but the presently observed pandemics show duration of their spread during years imposing to take into account the host and vector population changes as well as the transient or permanent migration and diffusion of hosts (susceptible or infected), as well as vectors and infectious agents. Two examples are presented, one concerning the malaria in Mali and the other the plague at the middle-age.

Local Weighted Average Estimation of the Regression Operator for Functional Data

We are concerned with the problem of local weighted average estimation of the regression operator when the responses are real-valued random variables, the explanatory data are of functional fixed-design type, and the errors consist of an independent and identically distributed variables. In this article, our main contributions on the local linear functional estimation concern from one part, the situation when the data are of functional fixed-design kind, and from the other part, in deriving uniform asymptotic results on the behavior of this estimator with respect to the topological properties of the space data (normed or semi-metric).

Demographic and Spatial Factors as Causes of an Epidemic Spread, the Copule Approach: Application to the Retro-prediction of the Black Death Epidemy of 1346

The classical models by Ross and McKendrick have to be revisited in order to incorporate dynamical elements coming from the demography and from the spatial aspects of epidemics. The classical approach is dealing with populations supposed to be constant during the epidemic wave, but the present pandemics show duration during years imposing now to take into account the population growth as well as the transient or permanent migrations of hosts susceptible or infected, and of vectors and infectious agents. Two examples are studied, concerning malaria in Mali and plague at the middle-age.

Uniform in bandwidth consistency for various kernel estimators involving functional data

ABSTRACT The paper investigates various nonparametric models including regression, conditional distribution, conditional density and conditional hazard function, when the covariates are infinite dimensional. The main contribution is to prove uniform in bandwidth asymptotic results for kernel estimators of these functional operators. Then, the application issues, involving data-driven bandwidth selection, are discussed.

Data-driven kNN estimation in nonparametric functional data analysis

Kernel nearest-neighbor (kNN) estimators are introduced for the nonparametric analysis of statistical samples involving functional data. Asymptotic theory is provided for several different target operators including regression, conditional density, conditional distribution and hazard operators. The main point of the paper is to consider data-driven methods of selecting the number of neighbors in order to make the proposed methods fully automatic. As a by-product of our proofs we state consistency results for kNN functional estimators which are uniform in the number of neighbors (UINN). Some simulated experiences illustrate the feasibility and the finite-sample behavior of the method.

Relative-error prediction in nonparametric functional statistics

In this paper, an alternative kernel estimator of the regression operator of a scalar response variable Y given a random variable X taking values in a semi-metric space is considered. The constructed estimator is based on the minimization of the mean squared relative error. This technique is useful in analyzing data with positive responses, such as stock prices or life times. Least squares or least absolute deviation are among the most widely used criteria in statistical estimation for regression models. However, in many practical applications, especially in treating, for example, the stock price data, the size of the relative error rather than that of the error itself, is the central concern of the practitioners. This paper offers then an alternative to traditional estimation methods by considering the minimization of the least absolute relative error for operatorial regression models. We prove the strong and the uniform consistencies (with rates) of the constructed estimator. Moreover, the mean squared convergence rate is given and the asymptotic normality of the proposed estimator is proved. Finally, supportive evidence is shown by simulation studies and an application on some economic data was performed.

Kernel Conditional Density Estimation When the Regressor is Valued in a Semi-Metric Space

This article deals with the conditional density estimation when the explanatory variable is functional. In fact, nonparametric kernel type estimator of the conditional density has been recently introduced when the regressor is valued in a semi-metric space. This estimator depends on a smoothing parameter which controls its behavior. Thus, we aim to construct and study the asymptotic properties of a data-driven criterion for choosing automatically and optimally this smoothing parameter. This criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown conditional density, it is proved that this rule is asymptotically optimal. A simulation study and an application on real data are carried out to illustrate, for finite samples, the behavior of our method. Finally, we mention that our results can also be considered as novel in the finite dimensional setting and several other open questions are raised in this article.

Random modelling of contagious diseases.

Modelling contagious diseases needs to include a mechanistic knowledge about contacts between hosts and pathogens as specific as possible, e.g., by incorporating in the model information about social networks through which the disease spreads. The unknown part concerning the contact mechanism can be modelled using a stochastic approach. For that purpose, we revisit SIR models by introducing first a microscopic stochastic version of the contacts between individuals of different populations (namely Susceptible, Infective and Recovering), then by adding a random perturbation in the vicinity of the endemic fixed point of the SIR model and eventually by introducing the definition of various types of random social networks. We propose as example of application to contagious diseases the HIV, and we show that a micro-simulation of individual based modelling (IBM) type can reproduce the current stable incidence of the HIV epidemic in a population of HIV-positive men having sex with men (MSM).