Frédéric Lavancier

Covariance function of vector self-similar processes

The paper obtains the general form of the cross-covariance function of vector fractional Brownian motions with correlated components having different self-similarity indices.

Long memory random fields

A random field X = (Xn)n∈Zd is usually said to exhibit long memory, or strong dependence, or long-range dependence, when its covariance function r(n), n ∈ Z, is not absolutely summable : ∑ n∈Zd |r(n)| = ∞. An alternative definition involves spectral properties : a random field is said to be strongly dependent if its spectral density is unbounded. These two points of view are closely related but not equivalent. Generalizing a hypothesis widely used in dimension 1, most studies on long-range dependent random fields assume that the covariance function behaves at infinity as

Takacs-Fiksel method for stationary marked Gibbs point processes

This paper studies a method to estimate the parameters governing the distribution of a stationary marked Gibbs point process. This method, known as the Takacs-Fiksel method, is based on the estimation of the left and right hand sides of the Georgii-Nguyen-Zessin formula and leads to a family of estimators due to the possible choices of test functions. We propose several examples illustrating the interest and flexibility of this procedure. We also provide sufficient conditions based on the model and the test functions to derive asymptotic properties (consistency and asymptotic normality) of the resulting estimator. The different assumptions are discussed for exponential family models and for a large class of test functions.

A general procedure to combine estimators

A general method to combine several estimators of the same quantity is investigated. In the spirit of model and forecast averaging, the final estimator is computed as a weighted average of the initial ones, where the weights are constrained to sum to one. In this framework, the optimal weights, minimizing the quadratic loss, are entirely determined by the mean squared error matrix of the vector of initial estimators. The averaging estimator is built using an estimation of this matrix, which can be computed from the same dataset. A non-asymptotic error bound on the averaging estimator is derived, leading to asymptotic optimality under mild conditions on the estimated mean squared error matrix. This method is illustrated on standard statistical problems in parametric and semi-parametric models where the averaging estimator outperforms the initial estimators in most cases.

Quantifying repulsiveness of determinantal point processes

Determinantal point processes (DPPs) have recently proved to be a useful class of models in several areas of statistics, including spatial statistics, statistical learning and telecommunications networks. They are models for repulsive (or regular, or inhibitive) point processes, in the sense that nearby points of the process tend to repel each other. We consider two ways to quantify the repulsiveness of a point process, both based on its second order properties, and we address the question of how repulsive a stationary DPP can be. We determine the most repulsive stationary DPP, when the intensity is fixed, and we investigate repulsiveness in the subclass of R-dependent stationary DPPs (for a given R > 0), i.e. stationary DPPs with R-compactly supported kernels. Finally, in both the general case and the R-dependent case, we present some new parametric families of stationary DPPs that can cover a large range of DPPs, from the homogeneous Poisson process (which induces no interaction) to the most repulsive DPP.

A two-sample test for comparison of long memory parameters

We construct a two-sample test for comparison of long memory parameters based on ratios of two rescaled variance (V/S) statistics studied in Giraitis et al. [L. Giraitis, R. Leipus, A. Philippe, A test for stationarity versus trends and unit roots for a wide class of dependent errors, Econometric Theory 21 (2006) 989-1029]. The two samples have the same length and can be mutually independent or dependent. In the latter case, the test statistic is modified to make it asymptotically free of the long-run correlation coefficient between the samples. To diminish the sensitivity of the test on the choice of the bandwidth parameter, an adaptive formula for the bandwidth parameter is derived using the asymptotic expansion in Abadir et al. [K. Abadir, W. Distaso, L. Giraitis, Two estimators of the long-run variance: beyond short memory, Journal of Econometrics 150 (2009) 56-70]. A simulation study shows that the above choice of bandwidth leads to a good size of our comparison test for most values of fractional and ARMA parameters of the simulated series.

Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction

General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed as extensions of Ords process, are considered. The interaction may occur on each cell of the tessellation and between neighbour cells. The tessellation may also be subjected to a geometric hardcore interaction, forcing the cells not to be too large, too small, or too flat. This setting, natural for applications, introduces some theoretical difficulties since the interaction is not necessarily hereditary. Mathematical results available for studying these models are reviewed and further outcomes are provided. They concern the existence, the simulation and the estimation of such tessellations. Based on these results, tools to handle these objects in practice are presented: how to simulate them, estimate their parameters and validate the fitted model. Some examples of simulated tessellations are studied in detail.

Residuals and goodness‐of‐fit tests for stationary marked Gibbs point processes

The inspection of residuals is a fundamental step to investigate the quality of adjustment of a parametric model to data. For spatial point processes, the concept of residuals has been recently proposed by Baddeley et al. (2005) as an empirical counterpart of the {\it Campbell equilibrium} equation for marked Gibbs point processes. The present paper focuses on stationary marked Gibbs point processes and deals with asymptotic properties of residuals for such processes. In particular, the consistency and the asymptotic normality are obtained for a wide class of residuals including the classical ones (raw residuals, inverse residuals, Pearson residuals). Based on these asymptotic results, we define goodness-of-fit tests with Type-I error theoretically controlled. One of these tests constitutes an extension of the quadrat counting test widely used to test the null hypothesis of a homogeneous Poisson point process.

Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes

In this paper, we study Gibbs point processes involving a hardcore interaction which is not necessarily hereditary. We first extend the famous Campbell equilibrium equation, initially proposed by Nguyen and Zessin [Math. Nachr. 88 (1979) 105-115], to the non-hereditary setting and consequently introduce the new concept of removable points. A modified version of the pseudo-likelihood estimator is then proposed, which involves these removable points. We consider the following two-step estimation procedure: first estimate the hardcore parameter, then estimate the smooth interaction parameter by pseudo-likelihood, where the hard core parameter estimator is plugged in. We prove the consistency of this procedure in both the hereditary and non-hereditary settings.

Brillinger mixing of determinantal point processes and statistical applications

Stationary determinantal point processes are proved to be Brillinger mixing. This property is an important step towards asymptotic statistics for these processes. As an important example, a central limit theorem for a wide class of functionals of determinantal point processes is established. This result yields in particular the asymptotic normality of the estimator of the intensity of a stationary determinantal point process and of the kernel estimator of its pair correlation.