Wei Biao Wu
Empirical processes for stationary, causal sequences are considered. We establish empirical central limit theorems for classes of indicators of left half lines, absolutely continuous functions and piecewise differentiable functions. Sample path properties of empirical distribution functions are also discussed. The results are applied to linear processes and Markov chains.
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In this paper, we obtain some uniform laws of large numbers and functional central limit theorems for sequential empirical measure processes indexed by classes of product functions satisfying appropriate Vapnik-Chervonenkis properties.
Wasserstein convergence rates for empirical measures of point processes
本文研究点过程经验测度在Wasserstein距离下的收敛速率，提出新度量并推导浓度结果，为点过程的假设检验和统计推断提供工具。
We present a new technique for proving empirical process invariance principle for stationary processes $(X_n)_{n\geq 0}$. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions $(f(X_n))_{n\geq 0}$, not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques.
We give two asymptotic results for the empirical distance covariance on separable metric spaces without any iid assumption on the samples. In particular, we show the almost sure convergence of the empirical distance covariance for any measure with finite first moments, provided that the samples form a strictly stationary and ergodic process. We further give a result concerning the asymptotic distribution of the empirical distance covariance under the assumption of absolute regularity of the samples and extend these results to certain types of pseudometric spaces. In the process, we derive a ge…
The class of Generalized $L$-statistics ($GL$-statistics) unifies a broad class of different estimators, for example scale estimators based on multivariate kernels. $GL$-statistics are functionals of $U$-quantiles and therefore the dimension of the kernel of the $U$-quantiles determines the kernel dimension of the estimator. Up to now only few results for multivariate kernels are known. Additionally, most theory was established under independence or for short range dependent processes. In this paper we establish a central limit theorem for $GL$-statistics of functionals of short range dependen…
This paper develops a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the sup-norm. We prove an abstract approximation theorem applicable to a wide variety of statistical problems, such as construction of uniform confidence bands for functions. Notably, the bound in the main approximation theorem is nonasymptotic and the theorem allows for functions that index the empirical process to be unbounded and have entropy divergent with the sample size. The…
Exact and asymptotic distribution theory for the empirical correlation of two AR(1) processes with Gaussian increments
本文研究了具有高斯增量的两个独立 AR(1) 过程经验相关性的精确分布，证明了其在 Wasserstein 距离和 Kolmogorov 距离下向标准高斯分布的收敛速率，并将结果扩展至具有相关高斯增量的 AR(1) 过程。
The aim of this paper is to extend the aggregation convergence results given in (Dacunha-Castelle and Fermin 2005, Dacunha-Castelle and Fermin 2008) to doubly stochastic linear and nonlinear processes with weakly dependent innovations. First, we introduce a weak dependence notion for doubly stochastic processes, based in the weak dependence definition given in (Doukhan and Louhichi 1999), and we exhibe several models satisfying this notion, such as: doubly stochastic Volterra processes and doubly stochastic Bernoulli scheme with weakly dependent innovations. Afterwards we derive a central limi…
We study the weak convergence of conditional empirical copula processes, when the conditioning event has a nonzero probability. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general - possibly conditional - multivariate dependence measures and their estimators. By applying our theoretical results, we prove the asymptotic normality of some estimators of such dependence measures.
We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are non-asymptotic, which allows us to work with classes of functions whose complexity increases with the sample size. The construction of couplings is not of the Hungarian type and is instead based on the Slepian-Stein methods and Gaussian comparison inequalities. The increasing complexity of classes of functions and non-centrality of the processes make the res…
The empirical copula process plays a central role for statistical inference on copulas. Recently, Segers (2011) investigated the asymptotic behavior of this process under non-restrictive smoothness assumptions for the case of i.i.d. random variables. In the present paper we extend his main result to the case of serial dependent random variables by means of the powerful and elegant functional delta method. Moreover, we utilize the functional delta method in order to obtain conditional consistency of certain bootstrap procedures. Finally, we extend the results to the more general sequential empi…
We establish a functional central limit theorem for Ripley's K-function for two classes of point processes. One is the class of point processes having exponential decay of correlations and further satisfying a conditional m-dependence condition. The other is a family of Gibbs point processes. We illustrate the use of our theorem for goodness-of-fit tests in simulations.
We provide a framework for empirical process theory of locally stationary processes using the functional dependence measure. Our results extend known results for stationary Markov chains and mixing sequences by another common possibility to measure dependence and allow for additional time dependence. Our main result is a functional central limit theorem for locally stationary processes. Moreover, maximal inequalities for expectations of sums are developed. We show the applicability of our theory in some examples, for instance we provide uniform convergence rates for nonparametric regression wi…
The purpose of this note is to provide an approximation for the generalized bootstrapped empirical process achieving the rate in Kolmos et al. (1975). The proof is based on much the same arguments as in Horvath et al. (2000). As a consequence, we establish an approximation of the bootstrapped kernel-type density estimator
The theory for multiplier empirical processes has been one of the central topics in the development of the classical theory of empirical processes, due to its wide applicability to various statistical problems. In this paper, we develop theory and tools for studying multiplier $U$-processes, a natural higher-order generalization of the multiplier empirical processes. To this end, we develop a multiplier inequality that quantifies the moduli of continuity of the multiplier $U$-process in terms of that of the (decoupled) symmetrized $U$-process. The new inequality finds a variety of applications…
Regularly varying stochastic processes are able to model extremal dependence between process values at locations in random fields. We investigate the empirical extremogram as an estimator of dependence in the extremes. We provide conditions to ensure asymptotic normality of the empirical extremogram centred by a pre-asymptotic version. For max-stable processes with Fr{é}chet margins we provide conditions such that the empirical extremogram centred by its true version is asymptotically normal. The results of this paper apply to a variety of spatial and space-time processes, and to time series m…
Generalized linear statistics are an unifying class that contains U-statistics, U-quantiles, L-statistics as well as trimmed and winsorized U-statistics. For example, many commonly used estimators of scale fall into this class. GL-statistics only have been studied under independence; in this paper, we develop an asymptotic theory for GL-statistics of sequences which are strongly mixing or L^1 near epoch dependent on an absolutely regular process. For this purpose, we prove an almost sure approximation of the empirical U-process by a Gaussian process. With the help of a generalized Bahadur repr…
This paper develops asymptotic theory of integrals of empirical quantile functions with respect to random weight functions, which is an extension of classical $L$-statistics. They appear when sample trimming or Winsorization is applied to asymptotically linear estimators. The key idea is to consider empirical processes in the spaces appropriate for integration. First, we characterize weak convergence of empirical distribution functions and random weight functions in the space of bounded integrable functions. Second, we establish the delta method for empirical quantile functions as integrable f…
Conditional Delta-Method for Resampling Empirical Processes in Multiple Sample Problems
针对经验过程重抽样程序缺乏通用条件泛函delta方法的问题，本文提出适用于多样本问题的通用条件delta方法，并通过多类示例验证其应用灵活性，可改善基于极限结果的统计推断的有限样本性质。
We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with $n$ and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.
Empirical likelihood approach is one of non-parametric statistical methods, which is applied to the hypothesis testing or construction of confidence regions for pivotal unknown quantities. This method has been applied to the case of independent identically distributed random variables and second order stationary processes. In recent years, we observe heavy-tailed data in many fields. To model such data suitably, we consider symmetric scalar and multivariate $α$-stable linear processes generated by infinite variance innovation sequence. We use a Whittle likelihood type estimating function in th…
We study sequences of scaled edge-corrected empirical (generalized) K-functions (modifying Ripley's K-function) each of them constructed from a single observation of a $d$-dimensional fourth-order stationary point process in a sampling window W_n which grows together with some scaling rate unboundedly as n --> infty. Under some natural assumptions it is shown that the normalized difference between scaled empirical and scaled theoretical K-function converges weakly to a mean zero Gaussian process with simple covariance function. This result suggests discrepancy measures between empirical and th…
In the extreme value analysis of time series, not only the tail behavior is of interest, but also the serial dependence plays a crucial role. Drees and Rootzén (2010) established limit theorems for a general class of empirical processes of so-called cluster functionals which can be used to analyse various aspects of the extreme value behavior of mixing time series. However, usually the limit distribution is too complex to enable a direct construction of confidence regions. Therefore, we suggest a multiplier block bootstrap analog to the empirical processes of cluster functionals. It is shown t…
This paper considers the maximum generalized empirical likelihood (GEL) estimation and inference on parameters identified by high dimensional moment restrictions with weakly dependent data when the dimensions of the moment restrictions and the parameters diverge along with the sample size. The consistency with rates and the asymptotic normality of the GEL estimator are obtained by properly restricting the growth rates of the dimensions of the parameters and the moment restrictions, as well as the degree of data dependence. It is shown that even in the high dimensional time series setting, the …
Markov processes are used in a wide range of disciplines, including finance. The transition densities of these processes are often unknown. However, the conditional characteristic functions are more likely to be available, especially for Lévy-driven processes. We propose an empirical likelihood approach, for both parameter estimation and model specification testing, based on the conditional characteristic function for processes with either continuous or discontinuous sample paths. Theoretical properties of the empirical likelihood estimator for parameters and a smoothed empirical likelihood ra…
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 …
In this paper, weak convergences of marked empirical processes in $L^2(\mathbb{R},ν)$ and their applications to statistical goodness-of-fit tests are provided, where $L^2(\mathbb{R},ν)$ is the set of equivalence classes of the square integrable functions on $\mathbb{R}$ with respect to a finite Borel measure $ν$. The results obtained in our framework of weak convergences are, in the topological sense, weaker than those in the Skorokhod topology on a space of cádlág functions or the uniform topology on a space of bounded functions, which have been well studied in previous works. However, our re…
In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bound…
It is well known that the empirical likelihood ratio confidence region suffers finite sample under-coverage issue, and this severely hampers its application in statistical inferences.} The root cause of this under-coverage is an upper limit imposed by the convex hull of the estimating functions that is used in the construction of the profile empirical likelihood. For i.i.d data, various methods have been proposed to solve this issue by modifying the convex hull, but it is not clear how well these methods perform when the data are no longer independent. In this paper, we propose an adjusted blo…
False discovery rate (FDR) has been widely used as an error measure in large scale multiple testing problems, but most research in the area has been focused on procedures for controlling the FDR based on independent test statistics or the properties of such procedures for test statistics with certain types of stochastic dependence. Based on an approach proposed in Tang and Zhang (2005), we further develop in this paper empirical Bayes methods for controlling the FDR with dependent data. We implement our methodology in a time series model and report the results of a simulation study to demonstr…
We consider marked empirical processes indexed by a randomly projected functional covariate to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-n convergence rates and circumvent the curse of dimensionality. The weak convergence of the empirical process is obtained conditionally on a random direction, whilst the almost surely equivalence between the testing for significance expressed on the original and on the …
We correct an error in a technical lemma of Drees and Rootzén (2010) [arXiv:0910.0343] and discuss consequences for applications.
We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm.
While effective concentration inequalities for suprema of empirical processes exist under boundedness or strict tail assumptions, no comparable results have been available under considerably weaker assumptions. In this paper, we derive concentration inequalities assuming only low moments for an envelope of the empirical process. These concentration inequalities are beneficial even when the envelope is much larger than the single functions under consideration.
A moderate deviation principle for functionals, with at most quadratic growth, of moving average processes is established. The main assumptions on the moving average process are a Logarithmic Sobolev inequality for the driving random variables and the continuity, or weaker, of the spectral density of the moving average process. We also obtain the moderate deviations for the empirical spectral density, exhibiting an interesting new form of the rate function, i.e. with a correction term compared to the Gaussian rate functionnal.
We consider long-range dependent data. It is shown that the bootstrapped empirical process of these data converges to a semi-degenerate limit. The random part of this limit is always Gaussian. Thus the bootstrap might fail when the original empirical process accomplishes a noncentral limit theorem.
In this paper we study the asymptotic behaviour of empirical processes when parameters are estimated, assuming that the underlying sequence of random variables is long-range dependent. We show completely different phenomena compared to i.i.d. situation, as well as compared to ordinary empirical processes of long range dependent sequences. Applications include Kolmogorov-Smirnov and Cramer-Smirnov-von Mises goodness-of-fit statistics.
We offer an umbrella type result which extends weak convergence of the classical empirical process on the line to that of more general processes indexed by functions of bounded variation. This extension is not contingent on the type of dependence of the underlying sequence of random variables. As a consequence we establish weak convergence for stationary empirical processes indexed by general classes of functions under alpha mixing conditions.
For stochastic processes $\{X_t:t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_t\le y}-\operatorname{Pr}(X_t\le y):t\in E,y\in\mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E,y\in\mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E=[0,1]$.
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.
In a paper of Jason Swanson, a CLT for the sample median of independent Brownian motions with value 0 at 0 was proved. Here we extend this result in two ways. We prove such a result for a collection of self-similar processes which include the fractional Brownian motions, all stationary, independent increment symmetric stable processes tied down at 0 as well as iterated and integrated Brownian motions. Second, our results hold uniformly over all quantiles in a compact sub-interval of (0,1). We also examine sample function properties connected with these CLTs.
A reinforcement algorithm introduced by H.A. Simon \cite{Simon} produces a sequence of uniform random variables with memory as follows. At each step, with a fixed probability $p\in(0,1)$, $\hat U_{n+1}$ is sampled uniformly from $\hat U_1, \ldots, \hat U_n$, and with complementary probability $1-p$, $\hat U_{n+1}$ is a new independent uniform variable. The Glivenko-Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when…
We consider multivariate copula-based stationary time-series under Gaussian subordination. Observed time series are subordinated to long-range dependent Gaussian processes and characterized by arbitrary marginal copula distributions. First of all, we establish limit theorems for the marginal and quantile marginal empirical processes of multivariate stationary long-range dependent sequences under Gaussian subordination. Furthermore, we establish the asymptotic behavior of sequential empirical copula processes under non-restrictive smoothness assumptions. The limiting processes in the case of lo…
We establish empirical quantile process CLTs based on $n$ independent copies of a stochastic process $\{X_t: t \in E\}$ that are uniform in $t \in E$ and quantile levels $α\in I$, where $I$ is a closed sub-interval of $(0,1)$. Typically $E=[0,T]$, or a finite product of such intervals. Also included are CLT's for the empirical process based on $\{I_{X_t \le y} - \rm {Pr}(X_t \le y): t \in E, y \in R \}$ that are uniform in $t \in E, y \in R$. The process $\{X_t: t \in E\}$ may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment …
Following the works of Berthet (1997), we first obtain exact clustering rates in the functional law of the iterated logarithm for the uniform empirical and quantile processes and for their increments. In a second time, we obtain functional Chung-type limit laws for the local empirical process for a class of target functions on the border of the Strassen set.
We investigate a particular form of weak convergence of the local empirical process.
Given a random sample from a continuous multivariate distribution, Stute's representation is obtained for empirical copula processes constructed from a broad class of smooth, possibly data-adaptive nonparametric copula estimators. The latter class contains for instance empirical Bernstein copulas introduced by Sancetta and Satchell and thus the empirical beta copula proposed by Segers, Sibuya and Tsukahara. The almost sure rate in Stute's representation is expressed in terms of a parameter controlling the speed at which the spread of the smoothing region decreases as the sample size increases.
There are all kinds of weak dependence. For example, strong mixing. Short-range dependence (SRD) is also a form of weak dependence. It occurs in the context of processes that are subordinated to the Gaussian. Is a SRD process strong mixing if the underlying Gaussian process is long-range dependent? We show that this is not necessarily the case.
We consider a general class of empirical-type likelihoods and develop higher order asymptotics with a view to characterizing members thereof that allow the existence of possibly data-dependent probability matching priors ensuring approximate frequentist validity of posterior quantiles. In particular, for the usual empirical likelihood, positive results are obtained. This is in contrast with what happens if only data-free priors are entertained.
The well-known Komlós-Major-Tusnády inequalities [Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] provide sharp inequalities to partial sums of iid standard exponential random variables by a sequence of standard Brownian motions. In this paper, we employ these results to establish Gaussian approximations to weighted increments of uniform empirical and quantile processes. This approach provides rates to the approximations which, among others, have direct applications to statistics of extreme values for randomly censored data.
Weak convergence of the empirical copula process indexed by a class of functions is established. Two scenarios are considered in which either some smoothness of these functions or smoothness of the underlying copula function is required. A novel integration by parts formula for multivariate, right continuous functions of bounded variation, which is perhaps of independent interest, is proved. It is a key ingredient in proving weak convergence of a general empirical process indexed by functions of bounded variation.
The validity of various bootstrapping methods has been proved for the sample mean of strongly mixing data. But in many applications, there appear nonlinear statistics of processes that are not strongly mixing. We investigate the nonoverlapping block bootstrap sequences which are near epoch dependent on strong mixing or absolutely regular processes. This includes ARMA and GARCH-processes as well as data from chaotic dynamical systems. We establish the strong consistency of the bootstrap distribution estimator not only for the sample mean, but also for U-statistics, which include examples as Gin…
Consider the empirical autocovariance matrix at a given non-zero time lag based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measures in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices…
Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 1819--1837] obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 0--1 point processes and then applied Stein's method to the latter. In this paper, we adopt a different approach to the problem by using Palm theory and Stein's method, thereby expressing the error bounds in terms of the mean …
For fixed $t\in [0,1)$ and $h>0$, consider the local uniform empirical process $$\DD_{n,h,t}(s):=n^{-1/2}\coo\sliin 1_{[t,t+hs]}(U_i)-hs\cff,\;s\in [0,1],$$ where the $U_i$ are independent and uniformly distributed on $[0,1]$. We investigate the functional limit behaviour of $\DD_{n,h,t}$ uniformly in $\wth_n\le h\le h_n$ when $n\wth_n/\log\log n\rar \infty$ and $h_n\rar 0$.
We study Dirichlet process-based models for sets of predictor-dependent probability distributions, where the domain and predictor space are general Polish spaces. We generalize the definition of dependent Dirichlet processes, originally constructed on Euclidean spaces, to more general Polish spaces. We provide sufficient conditions under which dependent Dirichlet processes have appealing properties regarding continuity (weak and strong), association structure, and support (under different topologies). We also provide sufficient conditions under which mixture models induced by dependent Dirichl…
Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes
针对图依赖观测数据，提出一种结合图适应依赖系数与块分区着色的耦合构造，导出经验过程的极大不等式，并应用于网络自回归、非线性局部传播等模型。
This paper analyzes the limit properties of the empirical process of $α$-stable random variables with long range dependence. The $α$-stable random variables are constructed by non-linear transformations of bivariate sequences of strongly dependent gaussian processes. The approach followed allows an analysis of the empirical process by means of expansions in terms of bivariate Hermite polynomials for the full range $0<α<2$. A weak uniform reduction principle is provided and it is shown that the limiting process is gaussian. The results of the paper different substantailly from those available f…
Maximal Inequalities for Empirical Processes under General Mixing Conditions
本文在一般混合过程中研究样本均值的上界，提出基于浓度速率和新复杂度度量的界，揭示混合速度影响浓度速率并导致相变，应用于获得新的Glivenko-Cantelli型结果。
We consider the convergence of empirical processes indexed by functions that depend on an estimated parameter $η$ and give several alternative conditions under which the ``estimated parameter'' $η_n$ can be replaced by its natural limit $η_0$ uniformly in some other indexing set $Θ$. In particular we reconsider some examples treated by Ghoudi and Remillard [Asymptotic Methods in Probability and Statistics (1998) 171--197, Fields Inst. Commun. 44 (2004) 381--406]. We recast their examples in terms of empirical process theory, and provide an alternative general view which should be of wide appli…
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