The telegraph process $\{X(t), t>0\}$, is supposed to be observed at $n+1$ equidistant time points $t_i=iΔ_n,i=0,1,..., n$. The unknown value of $λ$, the underlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following:
$Δ_n \to 0$, $nΔ_n = T \to \infty$ as $n \to \infty$. We show that previously proposed moment type estimators are consistent and asymptotically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the additional assumption $nΔ_n^3 \to 0$ is required in order to obtain asymptotic normality. Finally, we propose a new estimator which is consistent, asymptotically normal and asymptotically efficient under no additional hypotheses.
The problem of parameter estimation by the observations of the two-state telegraph process in the presence of white Gaussian noise is considered. The properties of estimator of the method of moments are described in the asymptotics of large samples. Then this estimator is used as preliminary one to construct the one-step MLE-process, which provides the asymptotically normal and asymptotically efficient estimation of the unknown parameters.
The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity $+ v$ or $-v$. The changes of direction are governed by an homogeneous Poisson process with rate $λ>0.$ In this paper, we consider a change point estimation problem for the rate of the underlying Poisson process by means of least squares method. The consistency and the rate of convergence for the change point estimator are obtained and its asymptotic dist…
We consider a parameter estimation problem for one dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We prove that, the real valued parameter next to the Laplacian (the drift), and the constant parameter in front of the noise (the volatility) can be consistently estimated under somewhat surprisingly minimal information. Namely, it is enough to observe the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero. The proposed …
Parameter estimation in diffusion processes from discrete observations up to a first-hitting time is clearly of practical relevance, but does not seem to have been studied so far. In neuroscience, many models for the membrane potential evolution involve the presence of an upper threshold. Data are modeled as discretely observed diffusions which are killed when the threshold is reached. Statistical inference is often based on the misspecified likelihood ignoring the presence of the threshold causing severe bias, e.g. the bias incurred in the drift parameters of the Ornstein-Uhlenbeck model for …
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.
This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s.\@ continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is computationally efficient. It uses a recently developed technique for the exact simulation of diffusions, and involves no discretization error. We show that, under regularity conditions, the Monte Carlo MLE converges a.s. to the true MLE. For datasize $n\to\infty$, we show that the number of Monte Ca…
We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Lévy noises. We do not impose any moment condition on the driving Lévy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient $\varepsilon \to 0$ and $n \to \infty$ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to t…
The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consi…
This paper provides a semiparametric model of estimating states of the volatility defined as the squared diffusion coefficient of a stochastic differential equation. Without assuming any functional form of the volatility function, we consider the volatility state as an unobservable state in a state space model and estimate it by filtering. By using the estimates, we can draw the information on what kind of functions the volatility has behind the observable state of the process. We also prove the consistency of the model in the sense that estimated states converges to the true ones as the obser…
The problem of estimating the mean of random functions based on discretely sampled data arises naturally in functional data analysis. In this paper, we study optimal estimation of the mean function under both common and independent designs. Minimax rates of convergence are established and easily implementable rate-optimal estimators are introduced. The analysis reveals interesting and different phase transition phenomena in the two cases. Under the common design, the sampling frequency solely determines the optimal rate of convergence when it is relatively small and the sampling frequency has …
Birth-and-death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, because the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. Simple estimators may be based on an embedded Galton-Watson process, but this presupposes that the observation times are equi-spaced. We estimate the birth, death, and growth rates of a linear birth-and-death process whose population size is periodically observed via…
We study the problem of parameter estimation for reflected stochastic processes driven by a standard Brownian motion. The estimator is obtained using nonlinear least squares method based on discretely observed processes. Under some certain conditions, we obtain the consistency and give the asymptotic distribution of the estimator. Moreover, we briefly remark that our method can be extended to the one-sided reflected stochastic processes spontaneously. Numerical studies show that the proposed estimator is adequate for practical use.
Estimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are su…
Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of $Δ$-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for $Δ$-power variations of the iterated integrals of a fractional Brownian motion (fBm). These abstract results, besides being of independent interest, in the second part of the paper are applied to e…
We study the problem of estimating a mean pattern from a set of similar curves in the setting where the variability in the data is due to random geometric deformations and additive noise. We propose an estimator based on the notion of Frechet mean that is a generalization of the standard notion of averaging to non-Euclidean spaces. We derive a minimax rate for this estimation problem, and we show that our estimator achieves this optimal rate under the asymptotics where both the number of curves and the number of sampling points go to infinity.
Subordinate diffusions are constructed by time changing diffusion processes with an independent Lévy subordinator. This is a rich family of Markovian jump processes which exhibit a variety of jump behavior and have found many applications. This paper studies parametric inference of discretely observed ergodic subordinate diffusions. We solve the identifiability problem for these processes using spectral theory and propose a two-step estimation procedure based on estimating functions. In the first step, we use an estimating function that only involves diffusion parameters. In the second step, a…
Université Paris Saclay（巴黎萨克雷大学）；CEA（法国原子能和替代能源委员会）；List（List实验室（CEA下属））；Telecom SudParis（巴黎南电信学院）；Institut Polytechnique de Paris（巴黎理工学院）
In this paper we propose Discretely Indexed flows (DIF) as a new tool for solving variational estimation problems. Roughly speaking, DIF are built as an extension of Normalizing Flows (NF), in which the deterministic transport becomes stochastic, and more precisely discretely indexed. Due to the discrete nature of the underlying additional latent variable, DIF inherit the good computational behavior of NF: they benefit from both a tractable density as well as a straightforward sampling scheme, and can thus be used for the dual problems of Variational Inference (VI) and of Variational density e…
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 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.
In this paper, we investigate the parameter estimation problem for reflected OU processes. Both the estimates based on continuously observed processes and discretely observed processes are considered. The explicit formulas for the estimators are derived using the least squares method. Under some regular conditions, we obtain the consistency and establish the asymptotic normality for the estimators. Numerical results show that the proposed estimators perform well with moderate sample sizes.
We introduce a simple method for nearly simultaneous computation of all moments needed for quasi maximum likelihood estimation of parameters in discretely observed stochastic differential equations commonly seen in finance. The method proposed in this papers is not restricted to any particular dynamics of the differential equation and is virtually insensitive to the sampling interval. The key contribution of the paper is that computational complexity is sublinear in the number of observations as we compute all moments through a single operation. Furthermore, that operation can be done offline.…
We derive consistent and asymptotically normal estimators for the drift and volatility parameters of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space and the bounded domain. We establish the exact spatial regularity of the solution, which in turn, using power-variation arguments, allows building the desired estimators. We show that naive approximations of the derivatives appearing in the power-variation based estimators may create nontrivial biases, which we compute explicit…
The telegraph process $X(t)$, $t>0$, (Goldstein, 1951) and the geometric telegraph process $S(t) = s_0 \exp\{(μ-\frac12σ^2)t + σX(t)\}$ with $μ$ a known constant and $σ>0$ a parameter are supposed to be observed at $n+1$ equidistant time points $t_i=iΔ_n,i=0,1,..., n$. For both models $λ$, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also $σ>0$ has to be estimated. We propose different estimators of the parameters and we investigate their performance under the high frequency asymptotics, i.e. $Δ_n \to 0$, $nΔ= T<\infty$ as $n \to \infty$, w…
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,…
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…
This paper presents a central limit theorem for a pre-averaged version of the realized covariance estimator for the quadratic covariation of a discretely observed semimartingale with noise. The semimartingale possibly has jumps, while the observation times show irregularity, non-synchronicity, and some dependence on the observed process. It is shown that the observation times' effect on the asymptotic distribution of the estimator is only through two characteristics: the observation frequency and the covariance structure of the noise. This is completely different from the case of the realized …
This paper proposes a widely applicable method of approximate maximum-likelihood estimation for multivariate diffusion process from discretely sampled data. A closed-form asymptotic expansion for transition density is proposed and accompanied by an algorithm containing only basic and explicit calculations for delivering any arbitrary order of the expansion. The likelihood function is thus approximated explicitly and employed in statistical estimation. The performance of our method is demonstrated by Monte Carlo simulations from implementing several examples, which represent a wide range of com…
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: …
A maximum likelihood type estimation of the drift and volatility coefficient parameters in the CIR type model driven by $α$-stable noises is studied when the dispersion parameter $\varepsilon\to0$ and the discrete observations frequency $n\to\infty$ simultaneously.
The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. A\"ıt-Sahalia [J. Finance 54 (1999) 1361--1395; Econometrica 70 (2002) 223--262] proposed asymptotic expansions to the transition densities of diffusion processes, which lead to an approximate maximum likelihood estimation (AMLE) for parameters. Built on A\"ıt-Sahalia's [Econometrica 70 (2002) 223--262; Ann. Statist. 36 (2008) 906--937] proposal and analysis on the AMLE, we establish the consistenc…
We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate Lévy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models a…
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…
Brownian motion whose infinitesimal variance changes according to a three-state continuous time Markov Chain is studied. This Markov Chain can be viewed as a telegraph process with one on state and two off states. We first derive the distribution of occupation time of the on state. Then the result is used to develop a likelihood estimation procedure when the stochastic process at hand is observed at discrete, possibly irregularly spaced time points. The likelihood function is evaluated with the forward algorithm in the general framework of hidden Markov models. The analytic results are confirm…
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.
As a starting point we prove a functional central limit theorem for estimators of the invariant measure of a geometrically ergodic Harris-recurrent Markov chain in a multi-scale space. This allows to construct confidence bands for the invariant density with optimal (up to undersmoothing) $L^{\infty}$-diameter by using wavelet projection estimators. In addition our setting applies to the drift estimation of diffusions observed discretely with fixed observation distance. We prove a functional central limit theorem for estimators of the drift function and finally construct adaptive confidence ban…
Given the importance of continuous-time stochastic volatility models to describe the dynamics of interest rates, we propose a goodness-of-fit test for the parametric form of the drift and diffusion functions, based on a marked empirical process of the residuals. The test statistics are constructed using a continuous functional (Kolmogorov-Smirnov and Cramér-von Mises) over the empirical processes. In order to evaluate the proposed tests, we implement a simulation study, where a bootstrap method is considered for the calibration of the tests. As the estimation of diffusion models with stochasti…
Is it possible to detect if the sample paths of a stochastic process almost surely admit a finite expansion with respect to some/any basis? The determination is to be made on the basis of a finite collection of discretely/noisily observed sample paths. We show that it is indeed possible to construct a hypothesis testing scheme that is almost surely guaranteed to make only finitely many incorrect decisions as more data are collected. Said differently, our scheme almost certainly detects whether the process has a finite or infinite basis expansion for all sufficiently large sample sizes. Our app…
We consider the moderate deviations behaviors for two (co-) volatility estima-tors: generalised bipower variation, Hayashi-Yoshida estimator. The results are obtained by using a new result about the moderate deviations principle for m-dependent random variables based on the Chen-Ledoux type condition. In the last decade there has been a considerable development of the asymptotic theory for processes observed at a high frequency. This was mainly motivated by financial applications , where the data, such as stock prices or currencies, are observed very frequently. As under the no-arbitrage assum…
This paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function, and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate. Here we suppose that covariates are discretely observed and sampling points may differ across subjects, where the number of measurements per subject increases as the sample size. Also, we allow the quantile index to vary over a given subset of the open unit interval, so the slope function is a function of two variables: (typically) time an…
This paper investigates the Gaussian quasi-likelihood estimation of an exponentially ergodic multidimensional Markov process, which is expressed as a solution to a Lévy driven stochastic differential equation whose coefficients are known except for the finite-dimensional parameters to be estimated, where the diffusion coefficient may be degenerate or even null. We suppose that the process is discretely observed under the rapidly increasing experimental design with step size $h_n$. By means of the polynomial-type large deviation inequality, convergence of the corresponding statistical random fi…
In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process by Alòs et {\it al.} \cite{AE08} and Altmayer et {\it al.} \cite{AN14} together with the $L^p$-n…
The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian li…
We consider the problem of statistical inference for a class of partially-observed diffusion processes, with discretely-observed data and finite-dimensional parameters. We construct unbiased estimators of the score function, i.e. the gradient of the log-likelihood function with respect to parameters, with no time-discretization bias. These estimators can be straightforwardly employed within stochastic gradient methods to perform maximum likelihood estimation or Bayesian inference. As our proposed methodology only requires access to a time-discretization scheme such as the Euler-Maruyama method…
