S. Nakajima, S. Nakamura, Y. Shimizu
We study the problem of parameter estimation for discretely observed stochastic differential equations driven by small fractional noise. Under some conditions, we obtain strong consistency and rate of convergence of the least square estimator(LSE) when small dispersion coefficient converges to 0 and sample size converges to infty.
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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.
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.
This paper provides rate-efficient estimators of the volatility parameter in the presence of Lévy jumps
This paper proposes consistent and asymptotically Gaussian estimators for the drift, the diffusion coefficient and the Hurst exponent of the discretely observed fractional Ornstein-Uhlenbeck process. For the estimation of the drift, the results are obtained only in the case when 1/2 < H < 3/4. This paper also provides ready-to-use software for the R statistical environment based on the YUIMA package.
We consider a multidimensional diffusion X with drift coefficient b(α,X(t)) and diffusion coefficient εσ(β,X(t)). The diffusion is discretely observed at times t_k=kΔ for k=1..n on a fixed interval [0,T]. We study minimum contrast estimators derived from the Gaussian process approximating X for small ε. We obtain consistent and asymptotically normal estimators of α for fixed Δ and ε\rightarrow0 and of (α,β) for Δ\rightarrow0 and ε\rightarrow0. We compare the estimators obtained with various methods and for various magnitudes of Δ and ε based on simulation studies. Finally, we investigate the i…
In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are d…
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…
We extend the theoretical results for any FOU(p) processes for the case in which the Hurst parameter is less than 1/2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to obtain consistent estimators of the parameters when the process is observed in a discretized and equispaced interval [0, T ]. Also we will show that the FOU(p) processes can be used to model a wide range of time series varying from short range dependence to large range dependence with similar results as the ARMA or ARFIMA models, and in several cases outperforms …
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 $ρ.$
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…
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.
Estimation of several parameters in discretely-observed Stochastic Differential Equations with additive fractional noise
本文针对由加性分数布朗运动驱动的随机微分方程，基于离散观测构造Hurst参数、扩散参数和漂移项的联合估计量，证明一致性并给出收敛速率，且验证了分数Ornstein-Uhlenbeck过程满足可识别性假设。
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…
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…
A Theory of Nonparametric Covariance Function Estimation for Discretely Observed Data
本文研究了在离散位置观测功能数据时的非参数协方差函数估计问题，提出了一种统一的自适应估计器理论，揭示了深度学习估计器在高维设置中的适应性与方差权衡。
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 assu…
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…
Estimating parameters of drift and diffusion coefficients for multidimensional stochastic delay equations with small noise are considered. The delay structure is written as an integral form with respect to a delay measure. Our contrast function is based on a local-Gauss approximation to the transition probability density of the process. We show consistency and asymptotic normality of the minimum-contrast estimator when the dispersion coefficient goes to zero and the sample size goes to infinity, simultaneously.
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.
Estimation of parameters and local times in a discretely observed threshold diffusion model
针对带固定阈值的分段常数均值回复扩散过程，研究离散观测下的参数与局部时估计，推导不同观测频率下估计量的渐近性质，发现三类局部时估计量的收敛表现因观测间隔假设不同而有差异。
We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random. We introduce a new operator, the generalized infinitesimal generator, to obtain Taylor expansions of the asymptotic moments of the estimators. As a special case, our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval. We include as specific examples estimators based on maximum-likelihood a…
Robust Functional Regression with Discretely Sampled Predictors
本文提出了一种针对部分观测功能数据的鲁棒函数回归估计器，通过薄板样条和高效二次惩罚，提升对异常数据的鲁棒性并改善收敛速率。
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…
We aim at estimating in a non-parametric way the density $π$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$, for $d \ge 2$, from the discrete observations of a finite sample $X_{t_0}$, ... , $X_{t_n}$ with $0= t_0 < t_1 < ... < t_n =: T_n$. We propose a kernel density estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as …
In various practical situations, we encounter data from stochastic processes which can be efficiently modelled by an appropriate parametric model for subsequent statistical analyses. Unfortunately, the most common estimation and inference methods based on the maximum likelihood (ML) principle are susceptible to minor deviations from assumed model or data contamination due to their well known lack of robustness. Since the alternative non-parametric procedures often lose significant efficiency, in this paper, we develop a robust parameter estimation procedure for discretely observed data from a …
We treat the change point problem in ergodic diffusion processes from discrete observations. Tonaki et al. (2020) proposed adaptive tests for detecting changes in the diffusion and drift parameters in ergodic diffusion models. When any changes are detected by this method, the next question to be considered is where the change point is. Therefore, we propose the method to estimate the change point of the parameter for two cases: the case where there is a change in the diffusion parameter, and the case where there is no change in the diffusion parameter but a change in the drift parameter. Furth…
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…
This paper studies the identification of the Lévy jump measure of a discretely-sampled semimartingale. We define successive Blumenthal-Getoor indices of jump activity, and show that the leading index can always be identified, but that higher order indices are only identifiable if they are sufficiently close to the previous one, even if the path is fully observed. This result establishes a clear boundary on which aspects of the jump measure can be identified on the basis of discrete observations, and which cannot. We then propose an estimation procedure for the identifiable indices and compare …
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 …
We model two time and space scales discrete observations by using a unique continuous diffusion process with time dependent coefficient. We define new parameters for the large scale model as functions of the small scale distribution cumulants. We use the non - uniform distribution of the observation time intervals to obtain consistent and unbiased estimators for these parameters. Closed form expressions for migration proportions between spatial domains are derived as functions of these parameters. The models are applied to estimate migration patterns from satellite tag data.
Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime
本文研究离散观测一维扩散过程在中间情形下的不变密度估计，利用核估计与 Malliavin 得分表示证明其 $n^{-2β/(2β+1)}$ 速率最优。
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.
Adaptive minimax estimation for discretely observed Lévy processes
本文针对离散观测的Lévy过程增量密度非参数估计问题，提出谱估计量，证明其在高低频场景下均达极小极大最优，还提出易实现的数据驱动估计量并通过数值实验验证。
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…
Parameter estimation of discretely observed interacting particle systems
本文针对随机McKean-Vlasov方程及其相互作用粒子系统，提出基于伪似然的对比函数，在离散观测下实现漂移与扩散系数的联合估计，并证明估计量的一致性与渐近正态性。
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.
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 …
Properties of mixed fractional Brownian motion has been discussed by Cheridito (2001) and Zili (2006). We have proposed an estimator of volatility parameter for a model driven by MFBM. In our article we have shown that the estimator has some desirable asymptotic properties.
Drift Estimation for Diffusion Processes Using Neural Networks Based on Discretely Observed Independent Paths
本文提出基于神经网络的非参数估计方法，用于估计扩散过程中在紧致域上的漂移函数，通过高频离散观测获得收敛率，并在高维情况下展示出优于B-样条方法的性能。
We study the Nadaraya-Watson (N-W) estimator for the drift function of two-sided reflected stochastic processes. We propose a discrete-type N-W estimator and a continuous-type N-W estimator based on the discretely observed processes and continuously observed processes respectively. Under some regular conditions, we obtain the consistency and give the asymptotic distributions for the two estimators. Furthermore, we briefly remark that our method can be applied to the one-sided reflected stochastic processes spontaneously. Numerical studies show that the proposed estimators is adequate for pract…
In some non-regular statistical estimation problems, the limiting likelihood processes are functionals of fractional Brownian motion (fBm) with Hurst's parameter H; 0 < H <=? 1. In this paper we present several analytical and numerical results on the moments of Pitman estimators represented in the form of integral functionals of fBm. We also provide Monte Carlo simulation results for variances of Pitman and asymptotic maximum likelihood estimators.
We construct an adaptive wavelet estimator that attains minimax near-optimal rates in a wide range of Besov balls. The convergence rates are affected only by the weakest dependence amongst the channels, and take into account both noise sources.
We study the parameter estimation problem of Vasicek Model driven by sub-fractional Brownian processes from discrete observations, and let {S_t^H,t>=0} denote a sub-fractional Brownian motion whose Hurst parameter 1/2<H<1 . The studies are as follows: firstly, two unknown parameters in the model are estimated by the least squares method. Secondly, the strong consistency and the asymptotic distribution of the estimators are studied respectively. Finally, our estimators are validated by numerical simulation.
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.
Inference for SDEs driven by Hermite processes
本文针对由Hermite过程驱动的非线性随机微分方程，基于离散观测轨迹，利用加权二次变差提出了Hurst参数、Hermite阶数及噪声强度等参数的估计量，并在填充渐近下证明了其弱相合性。
Projection Estimators of the Stationary Density of a Differential Equation Driven by the Fractional Brownian Motion
本文研究在漂移函数耗散性条件下，由分数布朗运动驱动的微分方程平稳解密度的投影估计量，通过模型选择方法并借助集中不等式，为自适应估计量建立了 oracle 不等式。
We consider adaptive maximum-likelihood-type estimators and adaptive Bayes-type ones for discretely observed ergodic diffusion processes with observation noise whose variance is constant. The quasi-likelihood functions for the diffusion and drift parameters are introduced and the polynomial-type large deviation inequalities for those quasi-likelihoods are shown to see the convergence of moments for those estimators.
Functional data are typically modeled as sample paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The smoothness assumption is imposed to allow for the use of smoothing techniques that annihilate the noise. At the same time, imposing the smoothness assumption excludes a considerable range of stochastic processes, most notably diffusion processes. Under perfect observation of the sample paths, such processes would not need to be excluded from the realm of functional data analysis. In thi…
We define a generalized index of jump activity, propose estimators of that index for a discretely sampled process and derive the estimators' properties. These estimators are applicable despite the presence of Brownian volatility in the process, which makes it more challenging to infer the characteristics of the small, infinite activity jumps. When the method is applied to high frequency stock returns, we find evidence of infinitely active jumps in the data and estimate their index of activity.
Evolving privacy: drift parameter estimation for discretely observed i.i.d. diffusion processes under LDP
本文研究在局部差分隐私约束下，基于离散观测的独立同分布随机微分方程的漂移参数估计问题，提出基于伪似然的对比函数并加入Laplace噪声，证明了估计量的一致性与渐近正态性。
Maximum likelihood estimation of mean functions for Gaussian processes under small noise asymptotics
针对高斯过程时变均值函数估计问题，本文在连续观测下提出显式似然的极大似然估计量，通过小噪声渐近得到局部渐近正态性与效率，还引入离散M估计量及拟信息准则，为均值函数估计与模型选择提供方法与理论支持。
Asymptotic properties of parameter estimators in Vasicek model driven by tempered fractional Brownian motion
本文推导了由缓和分数布朗运动驱动的Vasicek模型中未知漂移参数最小二乘估计量的渐近分布，延续了此前关于该估计量强相合性的研究。
Linear fractional stable motion, denoted by $\{X_{H,\al}(t)\}_{t\in \R}$, is one of the most classical stable processes; it depends on two parameters $H\in (0,1)$ and $\al\in (0,2)$. The parameter $H$ characterizes the self-similarity property of $\{X_{H,\al}(t)\}_{t\in \R}$ while the parameter $\al$ governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that $H>1/\al$ and that $H$ is known. We show that, on the interval $[0,1]$, the asymptotic behaviour of the maximum, at a given scale $j$, of absolute …
We consider a reflected Ornstein-Uhlenbeck process $X$ driven by a fractional Brownian motion with Hurst parameter $H\in (0, \frac12) \cup (\frac12, 1)$. Our goal is to estimate an unknown drift parameter $α\in (-\infty,\infty)$ on the basis of continuous observation of the state process. We establish Girsanov theorem for the process $X$, derive the standard maximum likelihood estimator of the drift parameter $α$, and prove its strong consistency and asymptotic normality. As an improved estimator, we obtain the explicit formulas for the sequential maximum likelihood estimator and its mean squa…
We construct a least squares estimator for the drift parameters of a fractional Ornstein Uhlenbeck process with periodic mean function and long range dependence. For this estimator we prove consistency and asymptotic normality. In contrast to the classical fractional Ornstein Uhlenbeck process without periodic mean function the rate of convergence is slower depending on the Hurst parameter $H$, namely $n^{1-H}$.
The maximum likelihood type estimator of SDEs with fractional Brownian motion under small noise asymptotics in the rough case
针对分数布朗运动驱动的随机微分方程，在漂移和扩散系数满足一定假设时，构造最大似然估计量，证明小噪声下漂移参数的渐近正态性与矩收敛。
Mean and Covariance Estimation for Discretely Observed High-Dimensional Functional Data: Rates of Convergence and Division of Observational Regimes
本文研究了高维函数型数据下均值与协方差局部线性估计量的收敛速度，提出通用加权方案并证明观测数与维度及样本量的关系将数据划分为三个区制，后两区制可达高维参数收敛速度。
Parameter estimation for fractional autoregressive process with periodic structure
本文提出由分数高斯噪声驱动的周期分数自回归过程（PFAR）以建模降水蒸散发，结合GLSE和GPH方法构建初始估计并经One-Step程序优化，证明了估计量的一致性与渐近正态性。
In this article, we study the problem of parameter estimation for a discrete Ornstein - Uhlenbeck model driven by Poisson fractional noise. Based on random walk approximation for the noise, we study least squares and maximum likelihood estimators. Thus, asymptotic behaviours of the estimator is carried out, and a simulation study is shown to illustrate our results.
We propose a nonparametric estimation for a class of fractional stochastic differential equations (FSDE) with random effects. We precisely consider general linear fractional stochastic differential equations with drift depending on random effects and non-random diffusion. We build ordinary kernel estimators and histogram estimators and study their Lp-risk (p =1 or 2), when H>1/2. Asymptotic results are evaluated as both T = T(N) and N tend to infinity.
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