Limit Behaviour of Sequential Empirical Measure Processes

A Sequential Density-Based Empirical Likelihood Ratio Test for Treatment Effects

In health-related experiments, treatment effects can be identified using paired data that consist of pre- and post-treatment measurements. In this framework, sequential testing strategies are widely accepted statistical tools in practice. Since performances of parametric sequential testing procedures vitally depend on the validity of the parametric assumptions regarding underlying data distributions, we focus on distribution-free mechanisms for sequentially evaluating treatment effects. In fixed sample size designs, the density-based empirical likelihood (DBEL) methods provide powerful nonpara…

Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes

In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localisation of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analyzed by deriving new results on the weak convergence of a sequential empirical tail integral process and …

On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae under Verifiable Conditions

In the following article we consider the time-stability associated to the sequential Monte Carlo (SMC) estimate of the backward interpretation of Feynman-Kac Formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models (HMMs). We prove a central limit theorem (CLT) under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression, for additive functionals grows at most linearly in time, under hypotheses that are weaker than those currently existing in the literature. The assumptions are verifie…

Asymptotics of an empirical bridge of regression on induced order statistics

We propose a class of tests for linear regression on concomitants (induced order statistics). These tests are based on sequential sums of regression residuals. We self-center and self-normalize these sums. The resulting process is called an empirical bridge. We prove weak convergence of the empirical bridge in uniform metrics to a centered Gaussian process. The proposed tests are of chi-square type.

多级动态因子模型中顺序最小二乘因子的经验分布

The empirical distribution of sequential LS factors in Multi-level Dynamic Factor Models

本文探讨了多级动态因子模型中顺序最小二乘因子的渐近分布近似问题，通过蒙特卡洛实验验证了主成分因子分布的适用性，并提出了最优的均方误差估计方法。

Consistent nonparametric change point detection combining CUSUM and marked empirical processes

A weakly dependent time series regression model with multivariate covariates and univariate observations is considered, for which we develop a procedure to detect whether the nonparametric conditional mean function is stable in time against change point alternatives. Our proposal is based on a modified CUSUM type test procedure, which uses a sequential marked empirical process of residuals. We show weak convergence of the considered process to a centered Gaussian process under the null hypothesis of no change in the mean function and a stationarity assumption. This requires some sophisticated …

Martingale Methods for Sequential Estimation of Convex Functionals and Divergences

We present a unified technique for sequential estimation of convex divergences between distributions, including integral probability metrics like the kernel maximum mean discrepancy, $φ$-divergences like the Kullback-Leibler divergence, and optimal transport costs, such as powers of Wasserstein distances. This is achieved by observing that empirical convex divergences are (partially ordered) reverse submartingales with respect to the exchangeable filtration, coupled with maximal inequalities for such processes. These techniques appear to be complementary and powerful additions to the existing …

分组序贯设计中的学生化置换检验

A studentized permutation test in group sequential designs

本文提出一种学生化置换检验，用于分组序贯设计，以改善小样本下的第一类错误率控制，同时保持渐近性质，模拟研究验证了其有效性。

Empirical likelihood test for high-dimensional two-sample model

A non parametric method based on the empirical likelihood is proposed for detecting the change in the coefficients of high-dimensional linear model where the number of model variables may increase as the sample size increases. This amounts to testing the null hypothesis of no change against the alternative of one change in the regression coefficients. Based on the theoretical asymptotic behaviour of the empirical likelihood ratio statistic, we propose, for a fixed design, a simpler test statistic, easier to use in practice. The asymptotic normality of the proposed test statistic under the null…

A note on weak convergence of the sequential multivariate empirical process under strong mixing

This article investigates weak convergence of the sequential $d$-dimensional empirical process under strong mixing. Weak convergence is established for mixing rates $α_n = O(n^{-a})$, where $a>1$, which slightly improves upon existing results in the literature that are based on mixing rates depending on the dimension $d$.

基于独立与序贯e值的真、假发现研究

True and false discoveries with independent and sequential e-values

该研究针对多重假设检验场景，提出基于独立与序贯e值的新算法，经模拟、实证及理论验证，其性能优于现有独立p值算法及无独立性假设的e值算法。

序列核回归的更紧置信界

Tighter Confidence Bounds for Sequential Kernel Regression

本文利用鞅尾不等式为序列核回归提出更紧的置信界，通过锥规划对偶高效计算，应用于核老虎机问题可提升KernelUCB算法性能。

非平稳条件下函数索引序列经验过程及其平滑类似物的弱收敛性研究

On the Weak Convergence of the Function-Indexed Sequential Empirical Process and its Smoothed Analogue under Nonstationarity

本文研究了非平稳时间序列下函数索引序列经验过程及其平滑类似物的弱收敛性，通过新型极大不等式和一致链式论证，在温和假设下建立了渐近等连续性条件，并详细应用于非平稳α-混合序列。

Weak Convergence of the Sequential Empirical Process of some Long-Range Dependent Sequences with Respect to a Weighted Norm

Let $(X_k)_{k\geq1}$ be a Gaussian long-range dependent process with $EX_1=0$, $EX_1^2=1$ and covariance function $r(k)=k^{-D}L(k)$. For any measurable function $G$ let $(Y_k)_{k\geq1}=(G(X_k))_{k\geq1}$. We study the asymptotic behaviour of the associated sequential empirical process $\left(R_N(x,t)\right)$ with respect to a weighted norm $\|\cdot\|_w$. We show that, after an appropriate normalization, $\left(R_N(x,t)\right)$ converges weakly in the space of càdlàg functions with finite weighted norm to a Hermite process.

The Function-Indexed Sequential Empirical Process under Long-Range Dependence

Let $(X_j)_{j\geq1}$ be a multivariate long-range dependent Gaussian process. We study the asymptotic behavior of the corresponding sequential empirical process indexed by a class of functions. If some entropy condition is satisfied we have weak convergence to a linear combination of Hermite processes.

The Sequential Empirical Process of Nonlinear Long-Range Dependent Random Vectors

Let $(G(X_j))_{j\geq1}$ be a multivariate subordinated Gaussian process, which exhibits long-range dependence. We study the asymptotic behaviour of the corresponding sequential empirical process under two different types of subordination. The limiting process is either a product of a deterministic function and a Hermite process as in the one-dimensional case or a sum of various processes of this kind.

A weak convergence result for sequential empirical processes under weak dependence

The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $α$-mixing sequence of random variables. In particular the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained, as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as po…

The Sequential Empirical Process of a Random Walk in Random Scenery

A random walk in random scenery $(Y_n)_{n\in\mathbb{N}}$ is given by $Y_n=ξ_{S_n}$ for a random walk $(S_n)_{n\in\mathbb{N}}$ and iid random variables $(ξ_n)_{n\in\mathbb{Z}}$. In this paper, we will show the weak convergence of the sequential empirical process, i.e. the centered and rescaled empirical distribution function. The limit process shows a new type of behavior, combining properties of the limit in the independent case (roughness of the paths) and in the long range dependent case (self-similarity).

A Sequential Empirical Central Limit Theorem for Multiple Mixing Processes with Application to B-Geometrically Ergodic Markov Chains

We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average.

On the behaviour of large empirical autocovariance matrices between the past and the future

The asymptotic behaviour of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established the distribution behaves as a deterministic probability measure whose support S is characterized. It is also established that the singular values to the square are almost surely located in a neighbourhood of S.

面向具有遍历行为的随机混合系统的近似贝叶斯计算

Approximate Bayesian computation for stochastic hybrid systems with empirically observed ergodic behaviour

针对分段扩散马尔可夫过程（PDifMPs）参数估计难题，提出基于近似贝叶斯计算的推断框架，配套仿真算法与适配混合特性的汇总统计量，在多类遍历PDifMP算例中实现了可靠的参数恢复。

Large Time Behaviour and the Second Eigenvalue Problem for Finite State Mean-Field Interacting Particle Systems

This article examines large time behaviour of finite state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) on the time required for convergence of the empirical measure process of the $N$-particle system to its invariant measure; we show that when time is of the order of $\exp\{NΛ\}$ for a suitable constant $Λ\geq 0$, the process has mixed well and it is close to its invariant measure. We then obtain large-$N$ asymptotics of the second largest eigenvalue of the generator associated with the empirical measure process when it is rever…