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Jean-François Coeurjolly

Publications and source records attributed to Jean-François Coeurjolly.

At least 37 records · Page 2Linked to original sources
arXiv2019

The median of a jittered Poisson distribution

Let $N_λ$ and $U$ be two independent random variables respectively distributed as a Poisson distribution with parameter $λ>0$ and a uniform distribution on $(0,1)$. This paper establishes that the median, say $M$, of $N_λ+U$ is close to $λ+1/3$ and more precisely that $M-λ-1/3=o(λ^{-1})$ as $λ\to \infty$. This result is used to construt a very simple robust estimator of $λ$ which is consistent and asymptotically normal. Compared to known robust estimates, this one can still be used with large datasets ($n\simeq 10^9$).

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arXiv2019

Second-order variational equations for spatial point processes with a view to pair correlation function estimation

Second-order variational type equations for spatial point processes are established. In case of log linear parametric models for pair correlation functions, it is demonstrated that the variational equations can be applied to construct estimating equations with closed form solutions for the parameter estimates. This result is used to fit orthogonal series expansions of log pair correlation functions of general form.

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arXiv2017

Spatial point processes intensity estimation with a diverging number of covariates

Feature selection procedures for spatial point processes parametric intensity estimation have been recently developed since more and more applications involve a large number of covariates. In this paper, we investigate the setting where the number of covariates diverges as the domain of observation increases. In particular, we consider estimating equations based on Campbell theorems derived from Poisson and logistic regression likelihoods regularized by a general penalty function. We prove that, under some conditions, the consistency, the sparsity, and the asymptotic normality are valid for such a setting. We support the theoretical results by numerical ones obtained from simulation experiments and an application to forestry datasets.

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arXiv2017

Approximation intensity for pairwise interaction Gibbs point processes using determinantal point processes

The intensity of a Gibbs point process is usually an intractable function of the model parameters. For repulsive pairwise interaction point processes, this intensity can be expressed as the Laplace transform of some particular function. Baddeley and Nair (2002) developped the Poisson-saddlepoint approximation which consists, for basic models, in calculating this Laplace transform with respect to a homogeneous Poisson point process. In this paper, we develop an approximation which consists in calculating the same Laplace transform with respect to a specific determinantal point process. This new approximation is efficiently implemented and turns out to be more accurate than the Poisson-saddlepoint approximation, as demonstrated by some numerical examples.

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arXiv2017

Convex and non-convex regularization methods for spatial point processes intensity estimation

This paper deals with feature selection procedures for spatial point processes intensity estimation. We consider regularized versions of estimating equations based on Campbell theorem derived from two classical functions: Poisson likelihood and logistic regression likelihood. We provide general conditions on the spatial point processes and on penalty functions which ensure consistency, sparsity and asymptotic normality. We discuss the numerical implementation and assess finite sample properties in a simulation study. Finally, an application to tropical forestry datasets illustrates the use of the proposed methods.

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arXiv2016

Towards optimal Takacs--Fiksel estimation

The Takacs--Fiksel method is a general approach to estimate the parameters of a spatial Gibbs point process. This method embraces standard procedures such as the pseudolikelihood and is defined via weight functions. In this paper we propose a general procedure to find weight functions which reduce the Godambe information and thus outperform pseudolikelihood in certain situations. The new procedure is applied to a standard dataset and to a recent neuroscience replicated point pattern dataset. Finally, the performance of the new procedure is investigated in a simulation study.

math.ST↗
arXiv2016

A tutorial on Palm distributions for spatial point processes

This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes , we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally we discuss some examples of Palm distributions for specific models and some applications.

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arXiv2016

Palm distributions for log Gaussian Cox processes

This paper establishes a remarkable result regarding Palmdistributions for a log Gaussian Cox process: the reduced Palmdistribution for a log Gaussian Cox process is itself a log Gaussian Coxprocess which only differs from the original log Gaussian Cox processin the intensity function. This new result is used to study functionalsummaries for log Gaussian Cox processes.

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arXiv2016

Standard and robust intensity parameter estimation for stationary determinantal point processes

This work is concerned with the estimation of the intensity parameter of a stationary determinantal point process. We consider the standard estimator, corresponding to the number of observed points per unit volume and a recently introduced median-based estimator more robust to outliers. The consistency and asymptotic normality of estimators are obtained under mild assumptions on the determinantal point process. We illustrate the efficiency of the procedures in a simulation study.

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arXiv2015

Parametric estimation of pairwise Gibbs point processes with infinite range interaction

This paper is concerned with statistical inference for infinite range interaction Gibbs point processes and in particular for the large class of Ruelle superstable and lower regular pairwise interaction models. We extend classical statistical methodologies such as the pseudolikelihood and the logistic regression methods, originally defined and studied for finite range models. Then we prove that the associated estimators are strongly consistent and satisfy a central limit theorem, provided the pairwise interaction function tends sufficiently fast to zero. To this end, we introduce a new central limit theorem for almost conditionally centered triangular arrays of random fields.

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arXiv2015

Stein estimation of the intensity of a spatial homogeneous Poisson point process

In this paper, we revisit the original ideas of Stein and propose an estimator of the intensity parameter of a homogeneous Poisson point process defined in $\R^d$ and observed in a bounded window. The procedure is based on a new general integration by parts formula for Poisson point processes. We show that our Stein estimator outperforms the maximum likelihood estimator in terms of mean squared error. In particular, we show that in many practical situations we have a gain larger than 30\%.

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arXiv2015

Median-based estimation of the intensity of a spatial point process

This paper is concerned with a robust estimator of the intensity of a stationary spatial point process. The estimator corresponds to the median of a jittered sample of the number of points, computed from a tessellation of the observation domain. We show that this median-based estimator satisfies a Bahadur representation from which we deduce its consistency and asymptotic normality under mild assumptions on the spatial point process. Through a simulation study, we compare the new estimator with the standard one counting the mean number of points per unit volume. The empirical study verifies the asymptotic properties established and shows that the median-based estimator is more robust to outliers than the standard estimator.

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arXiv2014

Variational approach for spatial point process intensity estimation

We introduce a new variational estimator for the intensity function of an inhomogeneous spatial point process with points in the $d$-dimensional Euclidean space and observed within a bounded region. The variational estimator applies in a simple and general setting when the intensity function is assumed to be of log-linear form $β+{θ}^{\top}z(u)$ where $z$ is a spatial covariate function and the focus is on estimating ${θ}$. The variational estimator is very simple to implement and quicker than alternative estimation procedures. We establish its strong consistency and asymptotic normality. We also discuss its finite-sample properties in comparison with the maximum first order composite likelihood estimator when considering various inhomogeneous spatial point process models and dimensions as well as settings were $z$ is completely or only partially known.

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arXiv2012

Basic properties of the Multivariate Fractional Brownian Motion

This paper reviews and extends some recent results on the multivariate fractional Brownian motion (mfBm) and its increment process. A characterization of the mfBm through its covariance function is obtained. Similarly, the correlation and spectral analyses of the increments are investigated. On the other hand we show that (almost) all mfBm's may be reached as the limit of partial sums of (super)linear processes. Finally, an algorithm to perfectly simulate the mfBm is presented and illustrated by some simulations.

math.PR↗
arXiv2011

Expectiles for subordinated Gaussian processes with applications

In this paper, we introduce a new class of estimators of the Hurst exponent of the fractional Brownian motion (fBm) process. These estimators are based on sample expectiles of discrete variations of a sample path of the fBm process. In order to derive the statistical properties of the proposed estimators, we establish asymptotic results for sample expectiles of subordinated stationary Gaussian processes with unit variance and correlation function satisfying $ρ(i)\sim κ|i|^{-α}$ ($κ\in \RR$) with $α>0$. Via a simulation study, we demonstrate the relevance of the expectile-based estimation method and show that the suggested estimators are more robust to data rounding than their sample quantile-based counterparts.

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arXiv2011

Wavelet analysis of the multivariate fractional Brownian motion

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behavior of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral density is also considered in a second part. Its existence is proved and its evaluation is performed using a von Bahr-Essen like representation of the function $\sign(t) |t|^α$. The behavior of the cross-spectral density of the wavelet field at the zero frequency is also developed and confirms the results provided by the asymptotic analysis of the correlation.

math.PR↗
arXiv2011

Geodesic Normal distribution on the circle

This paper is concerned with the study of a circular random distribution called geodesic Normal distribution recently proposed for general manifolds. This distribution, parameterized by two real numbers associated to some specific location and dispersion concepts, looks like a standard Gaussian on the real line except that the support of this variable is $[0,2π)$ and that the Euclidean distance is replaced by the geodesic distance on the circle. Some properties are studied and comparisons with the von Mises distribution in terms of intrinsic and extrinsic means and variances are provided. Finally, the problem of estimating the parameters through the maximum likelihood method is investigated and illustrated with some simulations.

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