LXLihu Xu

Lihu Xu

University of Macau
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Lihu Xu has a total of 79 co-authors. Here are the top co-authors by number of shared publications:

Latest Publications

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    Quantitative Diffusion Approximation for the Neutral r-Alleles Wright-Fisher Model with Mutations

    by Peng Chen, Jie Xiong, Lihu Xu, Jiayu Zheng

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    Open Access

    Joint robust estimation

    by Xiang Li, Jun S. Liu, Qiang Sun, Lihu Xu

    We introduce a joint robust estimation method for three parametric statistical models with heavy-tailed data: mean estimation, linear regression, and L2-penalized linear regression, where both the trend parameters and the error variance are unknown. Our approach is based on solving two coupled Catoni-type equations, one for estimating the trend parameters and the other for estimating the error variance. Notably, this joint estimation strategy cannot be obtained by minimizing a single loss function involving both the trend and variance parameters. The method offers four key advantages: (i) the length of the resulting (1 - epsilon) confidence interval scales as (log(1/epsilon))^{1/2}, matching the order achieved by classical estimators for sub-Gaussian data; (ii) it is tuning-free, eliminating the need for separate variance estimation; (iii) it allows flexible selection of Catoni-type functions tailored to the data; and (iv) it delivers strong performance for high-variance data, thanks to the explicit inclusion of the variance term in the denominators of both equations. We establish the consistency and asymptotic efficiency of the proposed joint robust estimators using new analytical techniques. The coupled equations are inherently complex, which makes the theoretical analysis of their solutions challenging. To address this, we employ the Poincare-Miranda theorem to show that the solutions lie within geometric regions, such as cylinders or cones, centered around the true parameter values. This methodology is of independent interest and extends to other statistical problems.

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    Functional central limit theorem for Euler--Maruyama scheme with decreasing step sizes

    by Qiyang Pei, Lihu Xu

    We consider the Euler--Maruyama (EM) scheme of a family of dissipative SDEs, whose step sizes $η_{1}\geη_{2}\ge \cdots$ are decreasing, and prove that the EM scheme weakly converges to a subordinated Brownian motion $\{B_{a(t)}\}_{0\le t\le 1}$ rather than $\{B_{t}\}_{0\le t\le 1}$, where $a(t)$ is an increasing function depending on $\{η_{k}\}_{k \ge 1}$, for instance, $a(t)=t^{1+α}$ if $η_k =k^{-α}$. Compared to the EM scheme with constant step size, there are substantial differences as the following: (i) the EM time series is inhomogeneous and weakly converges to the ergodic measure in a polynomial speed; (ii) we have a special number $T_n =\frac{1}{η_1 }+\cdots+\frac{1}{η_n }$ which roughly measures the dependence of the EM time series; (iii) the normalized number in the CLT is $T_n ^{-1/2}n$ rather than $\sqrt{n}$, in particular, $T_n ^{-1/2}n \propto n^{(1-β)/2}$ when $η_{k}=1/k^β$ with $β\in(0,1)$; (iv) in the critical choice $η_{k}=1/k$, we have $T_{n}^{-1/2}n=O(1)$ and thus conjecture that the CLT and FCLT do not hold. This conjecture has been verified by simulations. A key distinction arises between the constant and decreasing step size implementations of the EM scheme. Under a constant step size, the time series is homogeneous. This allows one to use a stationary initialization, which automatically eliminates several complex terms in the subsequent proof of the CLT. Conversely, the time series generated by the EM scheme with decreasing step sizes forms an inhomogeneous Markov chain. To manage the analogous difficult terms in this case, that is, when the test function $h$ is Lipschitz, we must instead establish a bound for the Wasserstein-2 distance $W_{2}(θ_k ,X_{t_k })$. This technique for handling the inhomogeneous case could be of independent interest beyond the current proof.

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    Unadjusted Langevin algorithms for SDEs with Hölder drift

    by Xiang Li, Fengyu Wang, Lihu Xu

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    Distribution estimation for time series via DNN-based GANs with an application to change-point estimation

    by Jianya Lu, Yingjun Mo, Zhijie Xiao, Lihu Xu, Qiuran Yao

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    An Approximation to the Invariant Measure of the Limiting Diffusion of G/Ph/n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics

    by Xinghu Jin, Guodong Pang, Lihu Xu, Xin Xu

    In this paper, we develop a stochastic algorithm based on the Euler–Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of [ ] queues in the Halfin–Whitt regime. Specifically, we prove a nonasymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein’s method for multidimensional diffusions, in which the regularity of Stein’s equation obtained by the partial differential equation (PDE) theory plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of [ ] queues and its Euler–Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein’s equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role. Funding: X. Jin is supported in part by the Fundamental Research Funds for the Central Universities [Grants JZ2022HGQA0148 and JZ2023HGTA0170]. G. Pang is supported in part by the U.S. National Science Foundation [Grants DMS-1715875 and DMS-2216765]. L. Xu is supported in part by the National Nature Science Foundation of China [Grant 12071499], Macao Special Administrative Region [Grant FDCT 0090/2019/A2], and the University of Macau [Grant MYRG2018-00133-FST]. This work was supported by U.S. National Science Foundation [Grant DMS-2108683].

    Cited by 7
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    Approximation of the invariant measure for stable stochastic differential equations by the Euler–Maruyama scheme with decreasing step sizes

    by Peng Chen, Xinghu Jin, Yimin Xiao, Lihu Xu

    Let $(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation \[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\] where $b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Lipschitz-continuous function, $A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, $(Z_t)_{t\geqslant 0}$ is a d -dimensional rotationally symmetric $\alpha$ -stable Lévy process with $\alpha \in (1,2)$ and $x\in\mathbb{R}^{d}$ . We use two Euler–Maruyama schemes with decreasing step sizes $\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of $(X_t)_{t \geqslant 0}$ : one uses independent and identically distributed $\alpha$ -stable random variables as innovations, and the other employs independent and identically distributed Pareto random variables. We study the convergence rates of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, under the assumption that the function b is Lipschitz and satisfies a certain dissipation condition, we demonstrate a convergence rate of $\gamma^{\frac{1}{\alpha}}_n$ . This convergence rate can be improved to $\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$ for any $\kappa \in [1,\alpha)$ , provided b has the additional regularity of bounded second-order directional derivatives. For the second scheme, where the function b is assumed to be twice continuously differentiable, we establish a convergence rate of $\gamma^{\frac{2-\alpha}{\alpha}}_n$ ; moreover, we show that this rate is optimal for the one-dimensional stable Ornstein–Uhlenbeck process. Our theorems indicate that the recent significant result of [34] concerning the unadjusted Langevin algorithm with additive innovations can be extended to stochastic differential equations driven by an $\alpha$ -stable Lévy process and that the corresponding convergence rate exhibits similar behaviour. Compared with the result in [6], our assumptions have relaxed the second-order differentiability condition, requiring only a Lipschitz condition for the first scheme, which broadens the applicability of our approach.

    Cited by 3
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    Stable Central Limit Theorem in Total Variation Distance

    by Xiang Li, Lihu Xu, Haoran Yang

    Cited by 4
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    Open Access

    $W_{\bf d}$-convergence rate of EM schemes for invariant measures of supercritical stable SDEs

    by Peng Chen, Lihu Xu, Xiaolong Zhang, Xicheng Zhang

    By establishing the regularity estimates for nonlocal Stein/Poisson equations under $γ$-order Hölder and dissipative conditions on the coefficients, we derive the $W_{\bf d}$-convergence rate for the Euler-Maruyama schemes applied to the invariant measure of SDEs driven by multiplicative $α$-stable noises with $α\in (\frac{1}{2}, 2)$, where $W_{\bf d}$ denotes the Wasserstein metric with ${\bf d}(x,y)=|x-y|^γ\wedge 1$ and $γ\in ((1-α)_+, 1]$.

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    Robust estimation for high-dimensional time series with heavy tails

    by Yu Wang, Guodong Li, Zhijie Xiao, Lihu Xu, Wenyang Zhang

    We study in this paper the problem of least absolute deviation (LAD) regression for high-dimensional heavy-tailed time series which have finite $α$-th moment with $α\in (1,2]$. To handle the heavy-tailed dependent data, we propose a Catoni type truncated minimization problem framework and obtain an $\mathcal{O}\big( \big( (d_1+d_2) (d_1\land d_2) \log^2 n / n \big)^{(α- 1)/α} \big)$ order excess risk, where $d_1$ and $d_2$ are the dimensionality and $n$ is the number of samples. We apply our result to study the LAD regression on high-dimensional heavy-tailed vector autoregressive (VAR) process. Simulations for the VAR($p$) model show that our new estimator with truncation are essential because the risk of the classical LAD has a tendency to blow up. We further apply our estimation to the real data and find that ours fits the data better than the classical LAD.

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