Density deconvolution from repeated measurements without symmetry assumption on the errors

We consider deconvolution from repeated observations with unknown error distribution. So far, this model has mostly been studied under the additional assumption that the errors are symmetric. We construct an estimator for the non-symmetric error case and study its theoretical properties and practical performance. It is interesting to note that we can improve substantially upon the rates of convergence which have so far been presented in the literature and, at the same time, dispose of most of the extremely restrictive assumptions which have been imposed so far

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.Comment: Published in at http://dx.doi.org/10.1214/08-EJS209 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation for L\'{e}vy processes from high frequency data within a long time interval

In this paper, we study nonparametric estimation of the L\'{e}vy density for L\'{e}vy processes, with and without Brownian component. For this, we consider $n$ discrete time observations with step $\Delta$. The asymptotic framework is: $n$ tends to infinity, $\Delta=\Delta_n$ tends to zero while $n\Delta_n$ tends to infinity. We use a Fourier approach to construct an adaptive nonparametric estimator of the L\'{e}vy density and to provide a bound for the global ${\mathbb{L}}^2$-risk. Estimators of the drift and of the variance of the Gaussian component are also studied. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.Comment: Published in at http://dx.doi.org/10.1214/10-AOS856 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 $Z\_i=X\_i+\sigma\epsilon\_i$, $i=1, ..., n$, $\epsilon$ is a noise independent of $X$, with $\sigma\epsilon$ 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

Adaptive density estimation for general ARCH models

We consider a model $Y\_t=\sigma\_t\eta\_t$ in which $(\sigma\_t)$ is not independent of the noise process $(\eta\_t)$, but $\sigma\_t$ is independent of $\eta\_t$ for each $t$. We assume that $(\sigma\_t)$ is stationary and we propose an adaptive estimator of the density of $\ln(\sigma^2\_t)$ 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 $(\sigma\_t)$ and $(\eta\_t)$ are independent, in all cases where these minimax rates are known. The results apply to various linear and non linear ARCH processes

Adaptive density deconvolution with dependent inputs

In the convolution model $Z\_i=X\_i+ \epsilon\_i$, 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 $\epsilon\_i$ 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