arXiv subjects

Cécile Durot

Publications and source records attributed to Cécile Durot.

14 recordsLinked to original sources

arXiv2026

Deconvolution in unlinked linear models

Unlinked regression, in which covariates and responses are observed separately without known correspondence, has recently gained increasing attention. Deconvolution, on the other hand, is a fundamental and challenging problem in nonparametric statistics with the aim of estimating the distribution of a latent random variable $Z$ based on observations contaminated by some additive noise. The complexity of this task is heavily influenced by the smoothness of the noise distribution and often leads to slow estimation rates. In this paper, we combine the recent unlinked linear regression problem with the classical deconvolution framework. Specifically, we study nonparametric deconvolution under the assumption that $Z$ is a linear function of an observable multidimensional covariate. This structural constraint allows us to introduce a nonparametric estimator of the distribution of $Z$ which achieves the parametric rate of convergence in the Wasserstein distance of order 1, where the smoothness of the noise does not affect the rate. Furthermore, we introduce nonparametric estimators for the unconditional density of $Z$ and the conditional density of $Z$ given an observed response. This allows us to study the problem of estimating the value of the latent linear predictor, whose link to the observed response is not accessible. Through several simulations, we illustrate the fast convergence rate of our deconvolution estimator and the performance of the proposed conditional estimators of the latent predictor in different simulation scenarios.

math.ST↗

arXiv2024

Harris recurrent Markov chains and nonlinear monotone cointegrated models

In this paper, we study a nonlinear cointegration-type model of the form $Z_t = f_0(X_t) + W_t$ where $f_0$ is a monotone function and $X_t$ is a Harris recurrent Markov chain. We use a nonparametric Least Square Estimator to locally estimate $f_0$, and under mild conditions, we show its strong consistency and obtain its rate of convergence. New results (of the Glivenko-Cantelli type) for localized null recurrent Markov chains are also proved.

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arXiv2021

Unlinked monotone regression

We consider so-called univariate unlinked (sometimes ``decoupled,'' or ``shuffled'') regression when the unknown regression curve is monotone. In standard monotone regression, one observes a pair $(X,Y)$ where a response $Y$ is linked to a covariate $X$ through the model $Y= m_0(X) + ε$, with $m_0$ the (unknown) monotone regression function and $ε$ the unobserved error (assumed to be independent of $X$). In the unlinked regression setting one gets only to observe a vector of realizations from both the response $Y$ and from the covariate $X$ where now $Y \stackrel{d}{=} m_0(X) + ε$. There is no (observed) pairing of $X$ and $Y$. Despite this, it is actually still possible to derive a consistent non-parametric estimator of $m_0$ under the assumption of monotonicity of $m_0$ and knowledge of the distribution of the noise $ε$. In this paper, we establish an upper bound on the rate of convergence of such an estimator under minimal assumption on the distribution of the covariate $X$. We discuss extensions to the case in which the distribution of the noise is unknown. We develop a second order algorithm for its computation, and we demonstrate its use on synthetic data. Finally, we apply our method (in a fully data driven way, without knowledge of the error distribution) on longitudinal data from the US Consumer Expenditure Survey.

stat.ME↗

arXiv2020

Results on standard estimators in the Cox model

We consider the Cox regression model and prove some properties of the maximum partial likelihood estimator $\hatβ_n$ and of the the Breslow estimator $Λ_n$. The asymptotic properties of these estimators have been widely studied in the literature but we are not aware of a reference where it is shown that they have uniformly bounded moments. These results are needed, for example, when studying global errors of shape restricted estimators of the baseline hazard function.

math.ST↗

arXiv2019

On the $L_p$-error of the Grenander-type estimator in the Cox model

We consider the Cox regression model and study the asymptotic global behavior of the Grenander-type estimator for a monotone baseline hazard function. This model is not included in the general setting of Durot (2007). However, we show that a similar central limit theorem holds for $L_p$-error of the Grenander-type estimator. We also propose a test procedure for a Weibull baseline distribution, based on the $L_p$-distance between the Grenander estimator and a parametric estimator of the baseline hazard. Simulation studies are performed to investigate the performance of this test.

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arXiv2017

Testing convexity of a discrete distribution

Based on the convex least-squares estimator, we propose two different procedures for testing convexity of a probability mass function supported on N with an unknown finite support. The procedures are shown to be asymptotically calibrated.

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arXiv2014

A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation

We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\hat{F}_n$ of a naive estimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\hat{F}_n$ and $F_n$ is of the order $O_p(n^{-1}\log n)^{2/(4-τ)}$, for some $τ\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_n$. In typical examples, the approximating process is Gaussian and $τ=1$, in which case the convergence rate is $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz (1976) for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of $F_n$, in which case $τ=2$, leading to a faster rate $n^{-1}\log n$, also found by Wang and Woodfroofe (2007). As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order.

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arXiv2014

Nonparametric species richness estimation under convexity constraint

We consider the estimation of the total number $N$ of species based on the abundances of species that have been observed. We adopt a non parametric approach where the true abundance distribution $p$ is only supposed to be convex. From this assumption, we propose a definition for convex abundance distributions. We use a least-squares estimate of the truncated version of $p$ under the convexity constraint. We deduce two estimators of the total number of species, the asymptotic distribution of which are derived. We propose three different procedures, including a bootstrap one, to obtain a confidence interval for $N$. The performances of the estimators are assessed in a simulation study and compared with competitors. The proposed method is illustrated on several examples.

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arXiv2014

On asymptotics of the discrete convex LSE of a pmf

In this article, we derive the weak limiting distribution of the least squares estimator (LSE) of a convex probability mass function (pmf) with a finite support. We show that it can be defined via a certain convex projection of a Gaussian vector. Furthermore, samples of any given size from this limit distribution can be generated using an efficient Dykstra-like algorithm.

math.ST↗

arXiv2014

Testing monotonicity via local least concave majorants

We propose a new testing procedure for detecting localized departures from monotonicity of a signal embedded in white noise. In fact, we perform simultaneously several tests that aim at detecting departures from concavity for the integrated signal over various intervals of different sizes and localizations. Each of these local tests relies on estimating the distance between the restriction of the integrated signal to some interval and its least concave majorant. Our test can be easily implemented and is proved to achieve the optimal uniform separation rate simultaneously for a wide range of Hölderian alternatives. Moreover, we show how this test can be extended to a Gaussian regression framework with unknown variance. A simulation study confirms the good performance of our procedure in practice.

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arXiv2013

Testing equality of functions under monotonicity constraints

We consider the problem of testing equality of functions $f_j:[0,1]\to \mathbb{R}$ for $j=1,2,...,J$ the basis of $J$ independent samples from possibly different distributions under the assumption that the functions are monotone. We provide a uniform approach that covers testing equality of monotone regression curves, equality of monotone densities and equality of monotone hazards in the random censorship model. Two test statistics are proposed based on $L_1$-distances. We show that both statistics are asymptotically normal and we provide bootstrap implementations, which are shown to have critical regions with asymptotic level $α$.

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arXiv2012

The limit distribution of the $L_{\infty}$-error of Grenander-type estimators

Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\log n)^{-1/3}$ and the limiting distribution to be Gumbel.

math.ST↗

arXiv2012

Estimation of a convex discrete distribution

Non-parametric estimation of a convex discrete distribution may be of interest in several applications, such as the estimation of species abundance distribution in ecology. In this paper we study the least squares estimator of a discrete distribution under the constraint of convexity. We show that this estimator exists and is unique, and that it always outperforms the classical empirical estimator in terms of the $\ell_{2}$-distance. We provide an algorithm for its computation, based on the support reduction algorithm. We compare its performance to those of the empirical estimator, on the basis of a simulation study.

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arXiv2007

On the $\mathbb{L}_p$-error of monotonicity constrained estimators

We aim at estimating a function $λ:[0,1]\to \mathbb {R}$, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the $\mathbb {L}_p$-loss of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of $λ$, based on $n$ observations. Our main task is to prove that the $\mathbb {L}_p$-loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local $\mathbb {L}_p$-risk at a fixed point and the global $\mathbb {L}_p$-risk are of order $n^{-p/3}$. Applying the results to the density and regression models, we recover and generalize known results about Grenander and Brunk estimators. Also, we obtain new results for the Huang--Wellner estimator of a monotone failure rate in the random censorship model, and for an estimator of the monotone intensity function of an inhomogeneous Poisson process.

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