arXiv subjects

Fabienne Comte

Publications and source records attributed to Fabienne Comte.

28 records · Page 2Linked to original sources

arXiv2009

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of 1-periodic, second order stationary random functions X1,...,Xn. We consider an orthogonal series estimator of the slope function, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. Wepropose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill posedness to be known. Then we generalize the procedure to a random set of admissible m's and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in term of a general weighted L2-risk. This means that we provide adaptive estimators of both the slope function and its derivatives.

math.ST↗

arXiv2009

Cumulative distribution function estimation under interval censoring case 1

We consider projection methods for the estimation of the cumulative distribution function under interval censoring, case 1. Such censored data also known as current status data, arise when the only information available on the variable of interest is whether it is greater or less than an observed random time. Two types of adaptive estimators are investigated. The first one is a two-step estimator built as a quotient estimator. The second estimator results from a mean square regression contrast. Both estimators are proved to achieve automatically the standard optimal rate associated with the unknown regularity of the function, but with some restriction for the quotient estimator. Simulation experiments are presented to illustrate and compare the methods.

math.ST↗

arXiv2008

Nonparametric adaptive estimation for pure jump Lévy processes

This paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at $n$ discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the ${\mathbb L}^2$-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the Lévy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.

math.ST↗

arXiv2007

Nonparametric estimation for a stochastic volatility model

Consider discrete time observations (X_{\ellδ})_{1\leq \ell \leq n+1}$ of the process $X$ satisfying $dX_t= \sqrt{V_t} dB_t$, with $V_t$ a one-dimensional positive diffusion process independent of the Brownian motion $B$. For both the drift and the diffusion coefficient of the unobserved diffusion $V$, we propose nonparametric least square estimators, and provide bounds for theirrisk. Estimators are chosen among a collection of functions belonging to a finite dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.

stat.ME↗

arXiv2007

Penalized nonparametric mean square estimation of the coefficients of diffusion processes

We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $Δ$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms.

math.ST↗

arXiv2006

Adaptive density estimation for general ARCH models

We consider a model $Y\_t=σ\_tη\_t$ in which $(σ\_t)$ is not independent of the noise process $(η\_t)$, but $σ\_t$ is independent of $η\_t$ for each $t$. We assume that $(σ\_t)$ is stationary and we propose an adaptive estimator of the density of $\ln(σ^2\_t)$ based on the observations $Y\_t$. Under various dependence structures, the rates of this nonparametric estimator coincide with the minimax rates obtained in the i.i.d. case when $(σ\_t)$ and $(η\_t)$ are independent, in all cases where these minimax rates are known. The results apply to various linear and non linear ARCH processes.

math.ST↗

arXiv2006

Adaptive density deconvolution with dependent inputs

In the convolution model $Z\_i=X\_i+ ε\_i$, we give a model selection procedure to estimate the density of the unobserved variables $(X\_i)\_{1 \leq i \leq n}$, when the sequence $(X\_i)\_{i \geq 1}$ is strictly stationary but not necessarily independent. This procedure depends on wether the density of $ε\_i$ is super smooth or ordinary smooth. The rates of convergence of the penalized contrast estimators are the same as in the independent framework, and are minimax over most classes of regularity on ${\mathbb R}$. Our results apply to mixing sequences, but also to many other dependent sequences. When the errors are super smooth, the condition on the dependence coefficients is the minimal condition of that type ensuring that the sequence $(X\_i)\_{i \geq 1}$ is not a long-memory process.

math.ST↗

arXiv2006

Penalized contrast estimator for adaptive density deconvolution

The authors consider the problem of estimating the density $g$ of independent and identically distributed variables $X\_i$, from a sample $Z\_1, ..., Z\_n$ where $Z\_i=X\_i+σε\_i$, $i=1, ..., n$, $ε$ is a noise independent of $X$, with $σε$ having known distribution. They present a model selection procedure allowing to construct an adaptive estimator of $g$ and to find non-asymptotic bounds for its $\mathbb{L}\_2(\mathbb{R})$-risk. The estimator achieves the minimax rate of convergence, in most cases where lowers bounds are available. A simulation study gives an illustration of the good practical performances of the method.

math.ST↗

arXiv2006

Finite sample penalization in adaptive density deconvolution

We consider the problem of estimating the density $g$ of identically distributed variables $X\_i$, from a sample $Z\_1, ..., Z\_n$ where $Z\_i=X\_i+σε\_i$, $i=1, ..., n$ and $σε\_i$ is a noise independent of $X\_i$ with known density $ σ^{-1}f\_ε(./σ)$. We generalize adaptive estimators, constructed by a model selection procedure, described in Comte et al. (2005). We study numerically their properties in various contexts and we test their robustness. Comparisons are made with respect to deconvolution kernel estimators, misspecification of errors, dependency,... It appears that our estimation algorithm, based on a fast procedure, performs very well in all contexts.

math.ST↗

arXiv2005

Nonparametric Estimation of the Regression Function in an Errors-in-Variables Model

We consider the regression model with errors-in-variables where we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f(X)+ξ, Z=X+σε$, involving independent and unobserved random variables $X,ξ,ε$. The density $g$ of $X$ is unknown, whereas the density of $σε$ is completely known. Using the observations $(Y\_i, Z\_i)$, $i=1,...,n$, we propose an estimator of the regression function $f$, built as the ratio of two penalized minimum contrast estimators of $\ell=fg$ and $g$, without any prior knowledge on their smoothness. We prove that its $\mathbb{L}\_2$-risk on a compact set is bounded by the sum of the two $\mathbb{L}\_2(\mathbb{R})$-risks of the estimators of $\ell$ and $g$, and give the rate of convergence of such estimators for various smoothness classes for $\ell$ and $g$, when the errors $ε$ are either ordinary smooth or super smooth. The resulting rate is optimal in a minimax sense in all cases where lower bounds are available.

math.ST↗