arXiv subjects

Ester Mariucci

Publications and source records attributed to Ester Mariucci.

15 recordsLinked to original sources

arXiv2024

Nonparametric density estimation for the small jumps of Lévy processes

We consider the problem of estimating the density of the process associated with the small jumps of a pure jump Lévy process, possibly of infinite variation, from discrete observations of one trajectory. The interest of such a question lies on the observation that even when the Lévy measure is known, the density of the increments of the small jumps of the process cannot be computed in closed-form. We discuss results both from low and high frequency observations. In a low frequency setting, assuming the Lévy density associated with the jumps larger than $\varepsilon\in (0,1]$ in absolute value is known, a spectral estimator relying on the convolution structure of the problem achieves a parametric rate of convergence with respect to the integrated $L_2$ loss, up to a logarithmic factor. In a high frequency setting, we remove the assumption on the knowledge of the Lévy measure of the large jumps and show that the rate of convergence depends both on the sampling scheme and on the behaviour of the Lévy measure in a neighborhood of zero. We show that the rate we find is minimax up to a logarithmic factor. An adaptive penalized procedure is studied to select the cutoff parameter. These results are extended to encompass the case where a Brownian component is present in the Lévy process. Furthermore, we illustrate numerically the performances of our procedures.

math.ST↗

arXiv2024

Adaptive minimax estimation for discretely observed Lévy processes

In this paper, we study the nonparametric estimation of the density $f_Δ$ of an increment of a Lévy process $X$ based on $n$ observations with a sampling rate $Δ$. The class of Lévy processes considered is broad, including both processes with a Gaussian component and pure jump processes. A key focus is on processes where $f_Δ$ is smooth for all $Δ$. We introduce a spectral estimator of $f_Δ$ and derive both upper and lower bounds, showing that the estimator is minimax optimal in both low- and high-frequency regimes. Our results differ from existing work by offering weaker, easily verifiable assumptions and providing non-asymptotic results that explicitly depend on $Δ$. In low-frequency settings, we recover parametric convergence rates, while in high-frequency settings, we identify two regimes based on whether the Gaussian or jump components dominate. The rates of convergence are closely tied to the jump activity, with continuity between the Gaussian case and more general jump processes. Additionally, we propose a fully data-driven estimator with proven simplicity and rapid implementation, supported by numerical experiments.

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arXiv2020

Decompounding discrete distributions: A non-parametric Bayesian approach

Suppose that a compound Poisson process is observed discretely in time and assume that its jump distribution is supported on the set of natural numbers. In this paper we propose a non-parametric Bayesian approach to estimate the intensity of the underlying Poisson process and the distribution of the jumps. We provide a MCMC scheme for obtaining samples from the posterior. We apply our method on both simulated and real data examples, and compare its performance with the frequentist plug-in estimator proposed by Buchmann and Grübel. On a theoretical side, we study the posterior from the frequentist point of view and prove that as the sample size $n\rightarrow\infty$, it contracts around the `true', data-generating parameters at rate $1/\sqrt{n}$, up to a $\log n$ factor.

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arXiv2020

Spectral-free estimation of Lévy densities in high-frequency regime

We construct an estimator of the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy--Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of $L_p$ loss functions, $p\geq 1$, over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a large nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.

math.PR↗

arXiv2020

Non-asymptotic control of the cumulative distribution function of Lévy processes

We propose non-asymptotic controls of the cumulative distribution function $P(|X_{t}|\ge \varepsilon)$, for any $t>0$, $\varepsilon>0$ and any Lévy process $X$ such that its Lévy density is bounded from above by the density of an $α$-stable type Lévy process in a neighborhood of the origin. The results presented are non-asymptotic and optimal, they apply to a large class of Lévy processes.

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arXiv2019

Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps

It is common practice to treat small jumps of Lévy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of Lévy processes with infinite Lévy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a Lévy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of Lévy processes are provided. The theory is illustrated by concrete examples.

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arXiv2018

A Bayesian nonparametric approach to log-concave density estimation

The estimation of a log-concave density on $\mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We also present two computationally more feasible approximations and a more practical empirical Bayes approach, which are illustrated numerically via simulations.

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arXiv2018

Wasserstein and total variation distance between marginals of Lévy processes

We present upper bounds for the Wasserstein distance of order $p$ between the marginals of Lévy processes, including Gaussian approximations for jumps of infinite activity. Using the convolution structure, we further derive upper bounds for the total variation distance between the marginals of Lévy processes. Connections to other metrics like Zolotarev and Toscani-Fourier distances are established. The theory is illustrated by concrete examples and an application to statistical lower bounds.

math.PR↗

arXiv2016

Le cam theory on the comparison of statistical models

We recall the main concepts of the Le Cam theory of statistical experiments , especially the notion of Le Cam distance and its properties. We also review classical tools for bounding such a distance before presenting some examples. A proof of the classical equivalence result between density estimation problems and Gaussian white noise models will be analyzed.

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arXiv2015

Asymptotic equivalence for pure jump Lévy processes with unknown Lévy density and Gaussian white noise

The aim of this paper is to establish a global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a L{é}vy process and a Gaussian white noise experiment observed up to a time T, with T tending to $\infty$. These approximations are given in the sense of the Le Cam distance, under some smoothness conditions on the unknown L{é}vy density. All the asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.

math.PR↗

arXiv2015

Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise

We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jump-diffusion process and a Gaussian white noise experiment. Here, the considered parameter is the drift function, and we suppose that the observation time $T$ tends to $\infty$. The approximation is given in the sense of the Le Cam $Δ$-distance, under smoothness conditions on the unknown drift function. These asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.

math.PR↗

arXiv2015

Asymptotic equivalence for density estimation and gaussian white noise: An extension

The aim of this paper is to present an extension of the well-known as-ymptotic equivalence between density estimation experiments and a Gaussian white noise model. Our extension consists in enlarging the nonparametric class of the admissible densities. More precisely, we propose a way to allow densities defined on any subinterval of R, and also some discontinuous or unbounded densities are considered (so long as the discontinuity and unboundedness patterns are somehow known a priori). The concept of equivalence that we shall adopt is in the sense of the Le Cam distance between statistical models. The results are constructive: all the asymptotic equivalences are established by constructing explicit Markov kernels.

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arXiv2015

Asymptotic equivalence of discretely observed diffusion processes and their Euler scheme: small variance case

This paper establishes the global asymptotic equivalence, in the sense of the Le Cam $Δ$-distance, between scalar diffusion models with unknown drift function and small variance on the one side, and nonparametric autoregressive models on the other side. The time horizon $T$ is kept fixed and both the cases of discrete and continuous observation of the path are treated. We allow non constant diffusion coefficient, bounded but possibly tending to zero. The asymptotic equivalences are established by constructing explicit equivalence mappings.

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arXiv2014

L_1-distance for additive processes with time-homogeneous Lévy measures

We give an explicit bound for the $L_1$-distance between two additive processes of local characteristics $(f_j(\cdot),σ^2(\cdot),ν_j)$, $j = 1,2$. The cases $σ=0$ and $σ> 0$ are both treated. We allow $ν_1$ and $ν_2$ to be equivalent time-homogeneous Lévy measures, possibly with infinite variation. Some examples of possible applications are discussed.

math.PR↗

arXiv2013

Asymptotic equivalence of jumps Lévy processes and their discrete counterpart

We establish the global asymptotic equivalence between a pure jumps Lévy process $\{X_t\}$ on the time interval $[0,T]$ with unknown Lévy measure $ν$ belonging to a non-parametric class and the observation of $2m^2$ Poisson independent random variables with parameters linked with the Lévy measure $ν$. The equivalence result is asymptotic as $m$ tends to infinity. The time $T$ is kept fixed and the sample path is continuously observed. This result justifies the idea that, from a statistical point of view, knowing how many jumps fall into a grid of intervals gives asymptotically the same amount of information as observing $\{X_t\}$.

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