Given a discrete time sample $X_1,... X_n$ from a Lévy process $X=(X_t)_{t\geq 0}$ of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet $(γ,σ^2,ρ)$ corresponding to the process $X.$ Based on Fourier inversion and kernel smoothing, we propose estimators of $γ,σ^2$ and $ρ$ and study their asymptotic behaviour. The obtained results include derivation of upper bounds on the mean square error of the estimators of $γ$ and $σ^2$ and an upper bound on the mean integrated square error of an estimator of $ρ.$
Given a sample from a discretely observed Lévy process $X=(X_t)_{t\geq 0}$ of the finite jump activity, the problem of nonparametric estimation of the Lévy density $ρ$ corresponding to the process $X$ is studied. An estimator of $ρ$ is proposed that is based on a suitable inversion of the Lévy-Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of $ρ$ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.
Consider a scalar reflected diffusion $(X_t:t\geq 0)$, where the unknown drift function $b$ is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consists of $(X_0,X_Δ,...,X_{nΔ})$ for some fixed sampling distance $Δ>0$, the model satisfies the local asymptotic normality (LAN) property, assuming that $b$ satisfies some mild regularity assumptions. This is established by using the connections of diffusion processes to elliptic and parabolic PDEs. The key tools will be regularity estimates from the theory of parabolic PDEs as well as a detailed analysis …
A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as conf…
We study the nonparametric Nadaraya-Watson estimator of the drift function for ergodic stochastic processes driven by fractional Brownian motion of Hurst parameter H > 1/2. The estimator is based on the discretely observed stochastic processes. By using the ergodic properties and stochastic integral, we obtain the consistency of the proposed estimator.
Given a sample from a discretely observed compound Poisson process, we consider estimation of the density of the jump sizes. We propose a kernel type nonparametric density estimator and study its asymptotic properties. An order bound for the bias and an asymptotic expansion of the variance of the estimator are given. Pointwise weak consistency and asymptotic normality are established. The results show that, asymptotically, the estimator behaves very much like an ordinary kernel estimator.
This work develops change-point methods for statistics of high-frequency data. The main interest is in the volatility of an Itô semi-martingale, the latter being discretely observed over a fixed time horizon. We construct a minimax-optimal test to discriminate continuous paths from paths comprising volatility jumps. This is embedded into a more general theory to infer the smoothness of volatilities. In a high-frequency framework we prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we develop methods to infer changes in the Hurst param…
In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localisation of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analyzed by deriving new results on the weak convergence of a sequential empirical tail integral process and …
We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0,T]. We consider the regime when the sampling rate goes to 0. The main difficulty is that a renewal reward process is not a Levy process: the increments are non stationary and dependent. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss over Besov spaces. We achieve minimax rates of convergence for sampling rates that vanish with T at polynomial rate. In the same spirit as Buchmann and Grübel (2003) and Duval (2…
In the context of nonparametric regression, we study conditions under which the consistency (and rates of convergence) of estimators built from discretely sampled curves can be derived from the consistency of estimators based on the unobserved whole trajectories. As a consequence, we derive asymptotic results for most of the regularization techniques used in functional data analysis, including smoothing and basis representation.
We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.
Nonparametric density estimation is considered for a discretely observed stationary continuous-time process. For each of three given time sampling procedures either random or deterministic, we establish that histograms and frequency polygons can reach the same optimal $L_{2}$-rates as in the independent and identically distributed case. Moreover, thanks to a suitable "high frequency" sampling design, these rates are derived together with a minimized time of observation depending on the regularity of sample paths.
We study nonparametric density estimation problems where error is measured in the Wasserstein distance, a metric on probability distributions popular in many areas of statistics and machine learning. We give the first minimax-optimal rates for this problem for general Wasserstein distances, and show that, unlike classical nonparametric density estimation, these rates depend on whether the densities in question are bounded below. Motivated by variational problems involving the Wasserstein distance, we also show how to construct discretely supported measures, suitable for computational purposes,…
Stochastic volatility modelling of financial processes has become increasingly popular. The proposed models usually contain a stationary volatility process. We will motivate and review several nonparametric methods for estimation of the density of the volatility process. Both models based on discretely sampled continuous time processes and discrete time models will be discussed. The key insight for the analysis is a transformation of the volatility density estimation problem to a deconvolution model for which standard methods exist. Three type of nonparametric density estimators are reviewed: …
We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + σ(X_t) dW_t,$ with discretely sampled observations $X_0, X_Δ, \dots, X_{nΔ}$. We analyse the nonlinear inverse problem corresponding to the `low frequency sampling' regime where $Δ>0$ is fixed and $n \to \infty$. A general theorem is proved that gives conditions for prior distributions $Π$ on the diffusion coefficient $σ$ and the drift function $b$ that ensure minimax optimal contraction rates of the posterior distribution over Hölder-Sobolev smoothness classes. These conditions are verified for nat…
We introduce verifiable criteria for weak posterior consistency of identifiable Bayesian nonparametric inference for jump diffusions with unit diffusion coefficient and uniformly Lipschitz drift and jump coefficients in arbitrary dimension. The criteria are expressed in terms of coefficients of the SDEs describing the process, and do not depend on intractable quantities such as transition densities. We also show that products of discrete net and Dirichlet mixture model priors satisfy our conditions, again under an identifiability assumption. This generalises known results by incorporating jump…
We consider the problem of the Bayesian inference of drift and diffusion coefficient functions in a stochastic differential equation given discrete observations of a realisation of its solution. We give conditions for the well-posedness and stable approximations of the posterior measure. These conditions in particular allow for priors with unbounded support. Our proof relies on the explicit construction of transition probability densities using the parametrix method for general parabolic equations. We then study an application of these results in inferring the rates of Birth-and-Death processe…
Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_0$ and intensity $λ_0$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_0,λ_0)$ at these rates. To the best of our knowledge, construction of nonparametric density est…
This paper is concerned with nonparametric estimation of the weighted stochastic block model. We first show that the model implies a set of multilinear restrictions on the joint distribution of edge weights of certain subgraphs involving (in its simplest form) triplets and quadruples of nodes. From this system of equations the unknown components of the model can be recovered nonparametrically, up to the usual labeling ambiguity. We introduce a simple and computationally-attractive manner to do this. Estimators then follow from the analogy principle. Limit theory is provided. We find that compo…
The spectral density function describes the second-order properties of a stationary stochastic process on $\mathbb{R}^d$. This paper considers the nonparametric estimation of the spectral density of a continuous-time stochastic process taking values in a separable Hilbert space. Our estimator is based on kernel smoothing and can be applied to a wide variety of spatial sampling schemes including those in which data are observed at irregular spatial locations. Thus, it finds immediate applications in Spatial Statistics, where irregularly sampled data naturally arise. The rates for the bias and v…
We consider nonparametric estimation of the transition operator $P$ of a Markov chain and its transition density $p$ where the singular values of $P$ are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency. We investigate the performance of a spectral hard thresholded Galerkin-type estimator for $P$ and ${p}$, discarding most of the estimated singular triplets. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establish…
We propose nonparametric estimators of the occupation measure and the occupation density of the diffusion coefficient (stochastic volatility) of a discretely observed Itô semimartingale on a fixed interval when the mesh of the observation grid shrinks to zero asymptotically. In a first step we estimate the volatility locally over blocks of shrinking length, and then in a second step we use these estimates to construct a sample analogue of the volatility occupation time and a kernel-based estimator of its density. We prove the consistency of our estimators and further derive bounds for their ra…
We propose localized spectral estimators for the quadratic covariation and the spot covolatility of diffusion processes which are observed discretely with additive observation noise. The eligibility of this approach to lead to an appropriate estimation for time-varying volatilities stems from an asymptotic equivalence of the underlying statistical model to a white noise model with correlation and volatility processes being constant over small intervals. The asymptotic equivalence of the continuous-time and the discrete-time experiments are proved by a construction with linear interpolation in …
We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the data and construct a nonparametric estimator of the invariant density of the process. In the second step, which is referred to as a matching step, we exploit a characterisation of the invariant density as a solution of a certain ordinary differential equation, replace the invariant density in this …
Who Drives the System? Classifying Mean-Field Particle Systems from Trajectory Data
SAMM, Université Paris 1（SAMM，巴黎第一大学）；LPSM, Sorbonne Université, Université Paris Cité, CNRS（LPSM，索邦大学，巴黎西岱大学，法国国家科学研究中心）；CMAP, Ecole Polytechnique, IP Paris（CMAP，巴黎综合理工学院，巴黎理工学院）；CEREMADE, Université Paris Dauphine 1（CEREMADE，巴黎第九大学）
