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Chunhao Cai

Publications and source records attributed to Chunhao Cai.

34 records · Page 2Linked to original sources
arXiv2025

Parameter estimation for fractional autoregressive process with periodic structure

This paper introduces a new periodic fractional autoregressive process (PFAR) driven by fractional Gaussian noise (fGn) to model time series of precipitation evapotranspiration. Compared with the similar model in [\emph{Water Resources Research}, \textbf{20} (1984) 1898--1908], the new model incorporates a periodic structure via specialized varying coefficients and captures long memory and rough voltality through fGn for $0<H<1$, rather than via fractional differencing. In this work, Generalized Least Squares Estimation (GLSE) and the GPH method are employed to construct an initial estimator for the joint estimation of model parameters. A One-Step procedure is then used to obtain a more asymptotically efficient estimator. The paper proves that both estimators are consistent and asymptotically normal, and their performance is demonstrated via Monte Carlo simulations with finite-size samples. Simulation studies suggest that, while both estimation methods can accurately estimate the model parameters, the One-Step estimator outperforms the initial estimator.

math.ST↗
arXiv2025

RNN-BSDE method for high-dimensional fractional backward stochastic differential equations with Wick-Itô integrals

Fractional Brownian motions(fBMs) are not semimartingales so the classical theory of Itô integral can't apply to fBMs. Wick integration as one of the applications of Malliavin calculus to stochastic analysis is a fine definition for fBMs. We consider the fractional forward backward stochastic differential equations(fFBSDEs) driven by a fBM that have the Hurst parameter in (1/2,1) where $\int_{0}^{t} f_s \, dB_s^H$ is in the sense of a Wick integral, and relate our fFBSDEs to the system of partial differential equations by using an analogue of the Itô formula for Wick integrals. And we develop a deep learning algorithm referred to as the RNN-BSDE method based on recurrent neural networks which is exactly designed for solving high-dimensional fractional BSDEs and their corresponding partial differential equations.

math.PR↗
arXiv2025

Parameter estimation for fractional autoregressive process with seasonal structure

This paper introduces a new kind of seasonal fractional autoregressive process (SFAR) driven by fractional Gaussian noise (fGn). The new model includes a standard seasonal AR model and fGn. {The estimation of the parameters of this new model has to solve two problems: nonstationarity from the seasonal structure and long memory from fGn. We innovatively solve these by getting a stationary subsequence, making a stationary additive sequence, and then obtaining their spectral density. Then, we use one-step procedure for Generalized Least Squares Estimator (GLSE) and the Geweke Porter-Hudak (GPH) method to get better results. We prove that both the initial and one-step estimators are consistent and asymptotically normal. Finally, we use Monte Carlo simulations with finite-sized samples to demonstrate the performance of these estimators. Moreover, through empirical analysis, it is shown that the SFAR model can simulate some real world phenomena better than general models.

stat.AP↗
arXiv2022

Uncertainty-based Network for Few-shot Image Classification

The transductive inference is an effective technique in the few-shot learning task, where query sets update prototypes to improve themselves. However, these methods optimize the model by considering only the classification scores of the query instances as confidence while ignoring the uncertainty of these classification scores. In this paper, we propose a novel method called Uncertainty-Based Network, which models the uncertainty of classification results with the help of mutual information. Specifically, we first data augment and classify the query instance and calculate the mutual information of these classification scores. Then, mutual information is used as uncertainty to assign weights to classification scores, and the iterative update strategy based on classification scores and uncertainties assigns the optimal weights to query instances in prototype optimization. Extensive results on four benchmarks show that Uncertainty-Based Network achieves comparable performance in classification accuracy compared to state-of-the-art method.

cs.CV↗
arXiv2021

Learning Class-level Prototypes for Few-shot Learning

Few-shot learning aims to recognize new categories using very few labeled samples. Although few-shot learning has witnessed promising development in recent years, most existing methods adopt an average operation to calculate prototypes, thus limited by the outlier samples. In this work, we propose a simple yet effective framework for few-shot classification, which can learn to generate preferable prototypes from few support data, with the help of an episodic prototype generator module. The generated prototype is meant to be close to a certain \textit{\targetproto{}} and is less influenced by outlier samples. Extensive experiments demonstrate the effectiveness of this module, and our approach gets a significant raise over baseline models, and get a competitive result compared to previous methods on \textit{mini}ImageNet, \textit{tiered}ImageNet, and cross-domain (\textit{mini}ImageNet $\rightarrow$ CUB-200-2011) datasets.

cs.CV↗
arXiv2021

Mixed sub-fractional Brownian motion and drift estimation of related Ornstein-Uhlenbeck process

In this paper, we will first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk. In order to verify the rationality of this simulation, we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein-Uhlenbeck process, and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter $H>1/2$.

math.PR↗
arXiv2020

Controlled Mean-Reverting Estimation for The AR(1) Model with Stationary Gaussian Noise

This paper deals with the maximum likelihood estimator for the mean-reverting parameter of a first order autoregressive models with exogenous variables, which are stationary Gaussian noises (Colored noise). Using the method of the Laplace transform, both the asymptotic properties and the asymptotic design problem of the maximum likelihood estimator are investigated. The numerical simulation results confirm the theoretical analysis and show that the proposed maximum likelihood estimator performs well in finite sample.

math.ST↗
arXiv2020

Maximum likelihood estimation for mixed fractional Vasicek processes

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 derivative for a mixed fractional Brownian motion. Using the fundamental martingale and the Laplace transform, both the strong consistency and the asymptotic normality of the maximum likelihood estimators have been established for all $H\in(0,1)$, $H\neq 1/2$.

math.PR↗
arXiv2020

Simulation of Integro-Differential Equation and Application in Estimation of Ruin Probability with Mixed Fractional Brownian Motion

In this paper, we are concerned with the numerical solution of one type integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we will try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical Lévy process but by the mixed fractional Brownian motion.

math.PR↗
arXiv2019

Experiment design for controlled partially observed fractional diffusion process

We consider a controlled second order differential equation which is partially observed with an additional fractional noise. we study the asymptotic (for large observation time) design problem of the input and give an efficient estimator of the unknown signal drift parameter. When the input depends on the unknow parameter, we will try the one-step estimation procedure using the Newton-Raphson method.

math.PR↗
arXiv2016

Mixed Gaussian processes: A filtering approach

This paper presents a new approach to the analysis of mixed processes \[X_t=B_t+G_t,\qquad t\in[0,T],\] where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation representation of $X$, using linear filtering theory. When the kernel \[K(s,t)=\frac{\partial^2}{\partial s\,\partial t}\mathbb{E}G_tG_s,\qquad s\ne t\] has a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with additional "fractional" structure, including the mixed fractional Brownian motion from mathematical finance. We show how previously-known measure equivalence relations and semimartingale properties follow from our canonical representation in a unified way, and complement them with new formulas for Radon-Nikodym densities.

math.PR↗
arXiv2016

Non-parametric threshold estimation for classical risk process perturbed by diffusion

In this paper,we consider a macro approximation of the flow of a risk reserve, The process is observed at discrete time points. Because we cannot directly observe each jump time and size then we will make use of a technique for identifying the times when jumps larger than a suitably defined threshold occurred. We estimate the jump size and survival probability of our risk process from discrete observations.

math.ST↗
arXiv2013

Asymptotic properties of the MLE for the autoregressive process coefficients under stationary Gaussian noise

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)).

math.ST↗