B. L. S. Prakasa Rao
We investigate the asymptotic properties of maximum likelihood estimators of the drift parameter for fractional vasicek model driven by a sub-fractional Brownian motion.
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The mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model, has been widely used in modelling volatility, interest rate and exchange rate. Obviously, if some phenomenon are modeled by the mixed fractional Vasicek model, statistical inference for this process is of great interest. Based on continuous time observations, this paper considers the problem of estimating the drift parameters in the mixed fractional Vasicek model. We will propose the maximum likelihood estimators of the drift parameters in the mixed fractional Vasicek model with the Radon-Nikodym d…
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。
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.
This paper developed an inference problem for Vasicek model driven by a general Gaussian process. We construct a least squares estimator and a moment estimator for the drift parameters of the Vasicek model, and we prove the consistency and the asymptotic normality. Our approach extended the result of Xiao and Yu (2018) for the case when noise is a fractional Brownian motion with Hurst parameter H \in [1/2,1).
Let $Z$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac{1}{2}, 1)$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. When $q=1$, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as $q\geq 2$. In this paper, we deal with a Vasicek-type model driven by $Z$, of the form $dX_t = a(b - X_t)dt +dZ_t$. Here, $a > 0$ and $b \in \mathbb{R}$ are considered as unknown drift parameters. We provide estimators for $a$ and $b$ based on continuous-time observations. For all possible valu…
This paper deals with the problem of global parameter estimation of affine diffusions in $\mathbb{R}_+ \times \mathbb{R}^n$ denoted by $AD(1, n)$ where $n$ is a positive integer which is a subclass of affine diffusions introduced by Duffie et al in [14]. The $AD(1, n)$ model can be applied to the pricing of bond and stock options, which is illustrated for the Vasicek, Cox-Ingersoll-Ross and Heston models. Our first result is about the classification of $AD(1, n)$ processes according to the subcritical, critical and supercritical cases. Then, we give the stationarity and the ergodicity theorems…
We discuss maximum likelihood estimation of parameters for models governed by a stochastic differential equation driven by a mixed fractional Brownian motion with random effects.
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。
Parameters estimation of a Threshold Chan-Karolyi-Longstaff-Sanders process from continuous and discrete observations
针对阈值Chan-Karolyi-Longstaff-Sanders过程，研究连续与离散观测下漂移参数的极大似然及拟极大似然估计的渐近性质，证明高频观测下渐近正态性与连续观测一致，并讨论扩散系数估计，通过模拟与真实数据验证阈值模型合理性。
A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\geq 1/2$. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.
This paper deals with the problems of consistence and strong consistence of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. A central limit theorem for these estimators is also obtained by using the Malliavin calculus.
We study the problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $dX_t=(μ+θX_t)dt+dG_t,\ t\geq0$ with unknown parameters $θ>0$ and $μ\in\mathbb{R}$, where $G$ is a Gaussian process. We provide least square-type estimators $\widetildeθ_T$ and $\widetildeμ_T$ respectively for the drift parameters $θ$ and $μ$ based on continuous-time observations $\{X_t,\ t\in[0,T]\}$ as $T\rightarrow\infty$. Our aim is to derive some sufficient conditions on the driving Gaussian process $G$ in order to ensure that $\widetildeθ_T$ and $\widetildeμ_T$ are strongly consistent, t…
We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(α-βX_t)\,dt+γ\,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $α$ and $β$ in the non-ergodic case (when $β<0$) for arbitrary $x_0\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=α/β$, derive their asymptotic distributions and prove their asymptotic independence.
The Berry-Esséen upper bounds of moment estimators and least squares estimators of the mean and drift coefficients in Vasicek models driven by general Gaussian processes are studied. When studying the parameter estimation problem of Ornstein-Uhlenbeck (OU) process driven by fractional Brownian motion, the commonly used methods are mainly given by Kim and Park, they show the upper bound of Kolmogorov distance between the distribution of the ratio of two double Wiener-Itô stochastic integrals and the Normal distribution. The main innovation in this paper is extending the above ratio process, tha…
Consider an estimation of the Hurst parameter $H\in(0,1)$ and the volatility parameter $σ>0$ for a fractional Brownian motion with a drift term under high-frequency observations with a finite time interval. In the present paper, we propose a consistent estimator of the parameter $θ=(H,σ)$ combining the ideas of a quasi-likelihood function based on a local Gaussian approximation of a high-frequently observed time series and its frequency-domain approximation. Moreover, we prove an asymptotic normality property of the proposed estimator for all $H\in(0,1)$ when the drift process is constant.
Exact maximum likelihood inference for drifted multi-sub-fractional Brownian motion at discrete observation
本文针对离散观测的漂移多子分数布朗运动，证明了完整有限样本似然理论的存在，给出显式极大似然估计量、精确水平的区间与检验，并验证了强相合性、渐近正态性及模拟效果。
This paper is devoted to parameter estimation of the mixed fractional Ornstein-Uhlenbeck process with a drift. Large sample asymptotical properties of the Maximum Likelihood Estimator is deduced using the Laplace transform computations or the Cameron-Martin formula with extra part from \cite{CK19}
We consider the problem of efficient estimation for the drift of fractional Brownian motion $B^H:=(B^H_t)_{t\in[0,T]}$ with hurst parameter $H$ less than 1/2. We also construct superefficient James-Stein type estimators which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.
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.
Stochastic differential equations and stochastic dynamics are good models to describe stochastic phenomena in real world. In this paper, we study N independent stochastic processes Xi(t) with real entries and the processes are determined by the stochastic differential equations with drift term relying on some random effects. We obtain the Girsanov-type formula of the stochastic differential equation driven by Fractional Brownian Motion through kernel transformation. Under some assumptions of the random effect, we estimate the parameter estimators by the maximum likelihood estimation and give s…
We consider the estimation of parametric fractional time series models in which not only is the memory parameter unknown, but one may not know whether it lies in the stationary/invertible region or the nonstationary or noninvertible regions. In these circumstances, a proof of consistency (which is a prerequisite for proving asymptotic normality) can be difficult owing to nonuniform convergence of the objective function over a large admissible parameter space. In particular, this is the case for the conditional sum of squares estimate, which can be expected to be asymptotically efficient under …
We construct the maximum likelihood estimator (MLE) of the unknown drift parameter $θ\in \mathbb{R}$ in the linear model $X_t=θt+σB^{H_1}(t)+B^{H_2}(t),\;t\in[0,T],$ where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $\frac12<H_1<H_2<1.$ The formula for MLE is based on the solution of the integral equation with weak polar kernel.
We discuss some extensions of results from the recent paper by Chernoyarov et al. (Ann. Inst. Stat. Math., October 2016) concerning limit distributions of Bayesian and maximum likelihood estimators in the model "signal plus white noise" with irregular cusp-type signals. Using a new representation of fractional Brownian motion (fBm) in terms of cusp functions we show that as the noise intensity tends to zero, the limit distributions are expressed in terms of fBm for the full range of asymmetric cusp-type signals correspondingly with the Hurst parameter H, 0<H<1. Simulation results for the densi…
We apply the techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of Hölder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.
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 study the problem of parametric estimation for continuously observed stochastic processes driven by additive small fractional Brownian motion with Hurst index 0<H<1/2 and 1/2<H<1. Under some assumptions on the drift coefficient, we obtain the asymptotic normality and moment convergence of maximum likelihood estimator of the drift parameter .
In this paper, we consider the problem of estimating the drift parameter of solution to the stochastic differential equation driven by a fractional Brownian motion with Hurst parameter less than $1/2$ under complete observation. We derive a formula for the likelihood ratio and prove local asymptotic normality when $H \in (1/4,1/2)$. Our result shows that the convergence rate is $T^{-1/2}$ for the parameters satisfying a certain equation and $T^{-(1-H)}$ for the others.
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 consider the problem of asymptotically efficient estimation of drift parameters of the ergodic fractional Ornstein-Uhlenbeck process under continuous observations when the Hurst parameter $H<1/2$ and the mean of its stationary distribution is not equal to zero. In this paper, we derive asymptotically efficient rates and variances of estimators of drift parameters and prove an asymptotic efficiency of a maximum likelihood estimator of drift parameters.
This paper deals with the problem of inference associated with linear fractional diffusion process with random effects in the drift. In particular we are concerned with the maximum likelihood estimators (MLE) of the random effect parameters. First of all, we estimate the Hurst parameter H from one single subject. Second, assuming the Hurst index H is known, we derive the MLE and examine their asymptotic behavior as the number of subjects under study becomes large, with random effects normally distributed.
Increasing Domain Infill Asymptotics for Stochastic Differential Equations Driven by Fractional Brownian Motion
针对分数布朗运动驱动的SDE推断难题，本文通过离散化建立增域填充渐近理论，证明乘性参数的经典与贝叶斯估计的一致性和渐近正态性，通过模拟和真实收盘价数据验证了贝叶斯分数SDE的优越性。
Consider an $(L,1)$ random walk in an i.i.d. random environment, whose environment involves certain parameter. We get the maximum likelihood estimator(MLE) of the environment parameter which can be written as functionals of a multitype branching process with immigration in a random environment(BPIRE). Because the offspring distributions of the involved multitype BPIRE are of the linear fractional type, the limit invariant distribution of the multitype BPIRE can be computed explicitly. As a result, we get the consistency of the MLE. Our result is a generalization of Comets et al. [Stochastic Pr…
We study the simple hypothesis testing problem for the drift coefficient for stochastic fractional heat equation driven by additive noise. We introduce the notion of asymptotically the most powerful test, and find explicit forms of such tests in two asymptotic regimes: large time asymptotics, and increasing number of Fourier modes. The proposed statistics are derived based on Maximum Likelihood Ratio. Additionally, we obtain a series of important technical results of independent interest: we find the cumulant generating function of the log-likelihood ratio; obtain sharp large deviation type re…
Parameter estimation for fractional stochastic heat equations : Berry-Esséen bounds in CLTs
本文研究加性时空噪声驱动的分数阶随机热方程漂移系数的 MLE，分别在连续观测与均匀离散观测下建立中心极限定理的 Wasserstein 距离 Berry-Esséen 界及一致性、渐近正态性。
We provide a comprehensive set of new results on the impact of mis-specifying the short run dynamics in fractionally integrated processes. We show that four alternative parametric estimators - frequency domain maximum likelihood, Whittle, time domain maximum likelihood and conditional sum of squares - converge to the same pseudo-true value under common mis-specification, and that they possess a common asymptotic distribution. The results are derived assuming a completely general parametric specification for the short run dynamics of the estimated (mis-specified) fractional model, and with long…
This paper investigates the asymptotic theory of the quasi-maximum exponential likelihood estimators (QMELE) for ARMA--GARCH models. Under only a fractional moment condition, the strong consistency and the asymptotic normality of the global self-weighted QMELE are obtained. Based on this self-weighted QMELE, the local QMELE is showed to be asymptotically normal for the ARMA model with GARCH (finite variance) and IGARCH errors. A formal comparison of two estimators is given for some cases. A simulation study is carried out to assess the performance of these estimators, and a real example on the…
Statistical Inference for the Rough Homogenization Limit of Multiscale Fractional Ornstein-Uhlenbeck Processes
本文研究分数多尺度系统均匀化极限的参数估计，将 Brown 情形下 MLE 的子采样一致性结果推广至缩放分数动力学 Brown 运动，并通过控制离散分数 Gaussian 噪声协方差逆矩阵谱范数证明扩散系数 MLE 的收敛性。
In this work we introduce the class of beta autoregressive fractionally integrated moving average models for continuous random variables taking values in the continuous unit interval $(0,1)$. The proposed model accommodates a set of regressors and a long-range dependent time series structure. We derive the partial likelihood estimator for the parameters of the proposed model, obtain the associated score vector and Fisher information matrix. We also prove the consistency and asymptotic normality of the estimator under mild conditions. Hypotheses testing, diagnostic tools and forecasting are als…
Quasi-maximum Likelihood Inference for Linear Double Autoregressive Models
本文针对线性双自回归模型，在仅分数矩条件下提出两种拟极大似然估计量及配套模型选择准则与诊断检验，通过模拟和实例验证了其有效性。
In this paper we are interested in the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order $p$ with regular stationary Gaussian noise. We exhibit the large sample asymptotical properties of the MLE under very mild conditions. Simulations are done for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).
Maximum likelihood, neural networks and posterior mean for inferring fractional Brownian motion parameters
该研究针对分数布朗运动参数推断，结合极大似然估计、神经网络后验均值方法，分析了细胞内囊泡扩散的动态切换特性。
Random effects estimation in a fractional diffusion model based on continuous observations
本研究针对分数阶扩散模型的随机效应，结合最大似然参数估计与Bernstein多项式非参数逼近构建混合估计方法，推导其渐近性质，并通过数值模拟验证其密度估计性能优于标准核估计。
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…
Statistical Guarantees for Data-driven Posterior Tempering
本文研究了数据驱动后验退火中α参数的选择及分数后验的统计保证，提出了关键阈值α≈1/√n，揭示了分数后验的渐近性质。
The aim of the present paper is to estimate and control the Type I and Type II errors of a simple hypothesis testing problem of the drift/viscosity coefficient for stochastic fractional heat equation driven by additive noise. Assuming that one path of the first $N$ Fourier modes of the solution is observed continuously over a finite time interval $[0,T]$, we propose a new class of rejection regions and provide computable thresholds for $T$, and $N$, that guarantee that the statistical errors are smaller than a given upper bound. The considered tests are of likelihood ratio type. The main ideas…
A Fractional Model for Earthquakes
本文扩展了现有的分数Hawkes过程以更好地建模地震主震-余震序列。通过最大似然估计验证了模型的一致性，并在南加州三个地震序列中与ETAS模型进行比较，结果显示分数Hawkes过程表现更优。
GARTFIMA Models: A Class of Observation-Driven Models with Tempered Fractional Dynamics
研究提出一类含缓和分数差分项的观测驱动模型（GARTFIMA模型），用部分最大似然法进行参数估计等，经蒙特卡罗模拟和实证应用，展现其在有限样本性能及实际应用中的优势。
Exact finite-sample inference for multi-mixed fractional Brownian motion with drift
本文针对带漂移的多重混合分数布朗运动，推导出漂移和尺度的闭式最大似然估计量及其精确有限样本联合分布，并证明其具有均匀最小方差无偏性、强相合性和渐近正态性，且精度由观测窗口长度决定。
Statistical inference with nonresponse is quite challenging, especially when the response mechanism is nonignorable. The existing methods often require correct model specifications for both outcome and response models. However, due to nonresponse, both models cannot be verified from data directly and model misspecification can lead to a seriously biased inference. To overcome this limitation, we develop a robust and efficient semiparametric method based on the profile likelihood. The proposed method uses the robust semiparametric response model, in which fully unspecified function of study var…
Parameter Estimation of Incomplete Gamma Subordinators
本文基于不完全伽马、不完全伽马-ε和T不完全伽马次序器，改进矩法估计其参数，并讨论最大似然估计的渐近正态性。
Finite-Rank Logistic Gaussian Processes with Exact Likelihood for Conditional Density Estimation
本文提出ExFR-LGP方法，通过有限秩高斯过程先验实现条件密度估计的精确似然，获得闭式解，并证明其最优收缩速率，在合成数据和ADNI数据上验证了有效性。
Fractional binomial regression model for count data with excess zeros
针对含过多零值的计数数据，提出基于分数二项分布的新型回归模型，验证了估计量一致性，其通用性与拟合效果优于部分现有零膨胀模型，但存在计算障碍待解决。
We use the jackknife to bias correct the log-periodogram regression(LPR) estimator of the fractional parameter in a stationary fractionally integrated model. The weights for the jackknife estimator are chosen in such a way that bias reduction is achieved without the usual increase in asymptotic variance, with the estimator viewed as `optimal' in this sense. The theoretical results are valid under both the non-overlapping and moving-block sub-sampling schemes that can be used in the jackknife technique, and do not require the assumption of Gaussianity for the data generating process. A Monte Ca…
Volume limitations and low yield thresholds of biological fluids have led to widespread use of passive microparticle rheology. The mean-squared-displacement (MSD) statistics of bead position time series (bead paths) are either applied directly to determine the creep compliance [Xu et al (1998)] or transformed to determine dynamic storage and loss moduli [Mason & Weitz (1995)]. A prevalent hurdle arises when there is a non-diffusive experimental drift in the data. Commensurate with the magnitude of drift relative to diffusive mobility, quantified by a Péclet number, the MSD statistics are disto…
We study parameter estimation problem for diagonalizable stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter $H\in(0,1)$. Two classes of estimators are investigated: traditional maximum likelihood type estimators, and a new class called closed-form exact estimators. Finally the general results are applied to stochastic heat equation driven by a fractional Brownian motion.
Based on Malliavin calculus tools and approximation results, we show how to compute a maximum likelihood type estimator for a rather general differential equation driven by a fractional Brownian motion with Hurst parameter H>1/2. Rates of convergence for the approximation task are provided, and numerical experiments show that our procedure leads to good results in terms of estimation.
Regularization in Paired Comparison Models via Pseudo-Games and Phantom Players
针对成对比较模型中最大似然估计不稳定的问题，本文提出两种数据增广正则化方法：添加伪博弈和引入幻影玩家，产生有限收缩估计并解决位置不可识别性，在Bradley-Terry模型中与岭正则化效果相当。
This work proposes a new minimum distance estimator (MDE) for the parameters of short and long memory models. This bias corrected minimum distance estimator (BCMDE) considers a correction in the usual MDE to account for the bias of the sample autocorrelation function when the mean is unknown. We prove the weak consistency of the BCMDE for the general fractional autoregressive moving average (ARFIMA(p, d, q)) model and derive its asymptotic distribution for some particular cases. Simulation studies show that the BCMDE presents a good performance compared to other procedures frequently used in t…
We construct an estimator of the unknown drift parameter $θ\in {\mathbb{R}}$ in the linear model \[X_t=θt+σ_1B^{H_1}(t)+σ_2B^{H_2}(t),\;t\in[0,T],\] where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $H_1$ and $H_2$ satisfying the condition $\frac{1}{2}\leq H_1<H_2<1.$ Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular…
We obtain a Berry-Esseen type bound for the distribution of the maximum likelihood estimator of the drift parameter for fractional Ornstein-uhlenbeck type process driven by sub-fractional Brownian motion.
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