Magalie Fromont

Model selection by bootstrap penalization for classification

We consider the binary classification problem. Given an i.i.d. sample drawn from the distribution of an χ×{0,1}−valued random pair, we propose to estimate the so-called Bayes classifier by minimizing the sum of the empirical classification error and a penalty term based on Efron’s or i.i.d. weighted bootstrap samples of the data. We obtain exponential inequalities for such bootstrap type penalties, which allow us to derive non-asymptotic properties for the corresponding estimators. In particular, we prove that these estimators achieve the global minimax risk over sets of functions built from Vapnik-Chervonenkis classes. The obtained results generalize Koltchinskii (2001) and Bartlett et al.’s (2002) ones for Rademacher penalties that can thus be seen as special examples of bootstrap type penalties. To illustrate this, we carry out an experimental study in which we compare the different methods for an intervals model selection problem.

Adaptive goodness-of-fit tests in a density model

Given an i.i.d. sample drawn from a density f, we propose to test that f equals some prescribed density f 0 or that f belongs to some translation/scale family. We introduce a multiple testing procedure based on an estimation of the L 2 -distance between f and f 0 or between f and the parametric family that we consider. For each sample size n, our test has level of significance a. In the case of simple hypotheses, we prove that our test is adaptive: it achieves the optimal rates of testing established by Ingster [J. Math. Sci. 99 (2000) 1110-1119] over various classes of smooth functions simultaneously. As for composite hypotheses, we obtain similar results up to a logarithmic factor. We carry out a simulation study to compare our procedures with the Kolmogorov-Smimov tests, or with goodness-of-fit tests proposed by Bickel and Ritov [in Nonparametric Statistics and Related Topics (1992) 51-57] and by Kallenberg and Ledwina [Ann. Statist. 23 (1995) 1594-1608].

Functional classification with margin conditions

Let (X,Y) be a

Adaptive tests of homogeneity for a Poisson process

\mathcal{X}

Model Selection by Bootstrap Penalization for Classification

× 0,1 valued random pair and consider a sample (X1,Y1),...,(Xn,Yn) drawn from the distribution of (X,Y). We aim at constructing from this sample a classifier that is a function which would predict the value of Y from the observation of X. The special case where

Bootstrap and permutation tests of independence for point processes

\mathcal{X}

The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach

is a functional space is of particular interest due to the so called curse of dimensionality. In a recent paper, Biau et al. [1] propose to filter the Xi’s in the Fourier basis and to apply the classical k–Nearest Neighbor rule to the first d coefficients of the expansion. The selection of both k and d is made automatically via a penalized criterion. We extend this study, and note here the penalty used by Biau et al. is too heavy when we consider the minimax point of view under some margin type assumptions. We prove that using a penalty of smaller order or equal to zero is preferable both in theory and practice. Our experimental study furthermore shows that the introduction of a small-order penalty stabilizes the selection process, while preserving rather good performances.

Family-Wise Separation Rates for multiple testing

We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in

Kernels Based Tests with Non-asymptotic Bootstrap Approaches for Two-sample Problems

\mathbb{L}^2

Adaptive tests for periodic signal detection with applications to laser vibrometry

norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non asymptotic and nonparametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.