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This is a two faces paper. Firstly, it investigates diagnostic tests in situations when the observed variables are functional, that is, diagnostic tests that use functional variables as biomarkers. A procedure based on functional version of ROC analysis is proposed, the main question being linked with a suitable way for ranking the sample of functional data. The second facet of this paper is to present a general new way for ordering functional data in a self-contained way allowing for a wide scope of applications overpassing the former diagnostic test problem. Finite sample analysis highlight how this ranking procedure behaves for diagnostic test.

The paper deals with a test procedure able to state the compatibility of observed data with a reference model, by using an estimate of the volumetric part in the small-ball probability factorization which plays the role of a real complexity index. As a preliminary by-product we state some asymptotics for a new estimator of the complexity index. A suitable test statistic is derived and, referring to the U-statistics theory, its asymptotic null distribution is obtained. A study of level and power of the test for finite sample sizes and a comparison with a competitor are carried out by Monte Carlo simulations. The test procedure is performed over a financial time series.

This paper takes part on the current literature on semi-parametric regression modelling for statistical samples composed of multi-functional data. A new kind of partially linear model (so-called MFPLR model) is proposed. It allows for more than one functional covariate, for incorporating as well continuous and discrete effects of functional variables and for modelling these effects as well in a nonparametric as in a linear way. Based on the continuous specificity of functional data, a new method is proposed for variable selection (so-called PVS method). In addition, from this procedure, new estimates of the various parameters involved in the partial linear model are constructed. A simulation study illustrates the finite sample size behavior of the PVS procedure for selecting the influential variables. Through some real data analysis, it is shown how the method is reaching the three main objectives of any semi-parametric procedure. Firstly, the flexibility of the nonparametric component of the model allows to get nice predictive behavior; secondly, the linear component of the model allows to get interpretable outputs; thirdly, the low computational cost insures an easy applicability. Even if the intent is to be used in multi-functional problems, it will briefly discuss how it can also be used in uni-functional problems as a boosting tool for improving prediction power. Finally, note that the main feature of this paper is of applied nature but some basic asymptotics are also stated in a final "Appendix".

Given a functional regression model with scalar response, the aim is to present a methodology in order to approximate in a semi-parametric way the unknown regression operator through a single index approach, but taking possible structural changes into account. Our paper presents this methodology and illustrates its behaviour both on simulated and real curves datasets. It appears, from an example of interest in spectrometry, that the method provides a nice exploratory tool both for analyzing structural changes in the spectrum and for visualizing the most informative directions, still keeping good predictive power. Even if the main objective of this work is to discuss applied issues of the method, asymptotic behaviour is shortly described.

This paper investigates a semi-parametric model for functional data, based on partial linear ideas. A methodology is developped for testing the linear component of such a functional partial linear model. The behavior of the test is studied through some finite simulated samples before being applied to some chemometrical curves dataset. One interesting feature of the methodology is that it works with not necessarily independent data.

The aim of this paper is to present a nonparametric regression model with scalar response when the explanatory variables are curves. In this context, the crucial problem of dimension reduction is overriden by the use of an implicit fractal dimension hypothesis. For such a functional nonparametric regression model we introduce and study both practical and theoretical aspects of some kernel type estimator. After a simulation study, it is shown how this procedure is well adapted to some spectrometric data set. Asymptotic results are described and in conclusion it turns out that this method combines advantages of easy implementation and good mathematical properties.