A general law of moment convergence rates for uniform empirical process

Let {X _n ; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process $F_n (t) = n^{ - \tfrac{1} {2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right| $F_n (t) = n^{ - \tfrac{1} {2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right| . In this paper, the exact convergence rates of a general law of weighted infinite series of E {‖F _n ‖ − ɛg ^s (n)}₊ are obtained.     

Rényi-type empirical processes

We develop a theory of asymptotics for Rényi-type weighted empirical and quantile processes and statistics via characterising their possible limiting behaviour in the middle and on the tails. In case of moderate weight functions tail limiting behaviour is found to be Gaussian, while heavily weighted tail empirical and uniform quantile processes are characterised by their respective Poisson process and exponential sums like asymptotic behaviour.     

Quantile Processes in the Presence of Auxiliary Information

We employ the empirical likelihood method to propose a modified quantile process under a nonparametric model in which we have some auxiliary information about the population distribution. Furthermore, we propose a modified bootstrap method for estimating the sampling distribution of the modified quantile process. To explore the asymptotic behavior of the modified quantile process and to justify the bootstrapping of this process, we establish the weak convergence of the modified quantile process to a Gaussian process and the almost-sure weak convergence of the modified bootstrapped quantile process to the same Gaussian process. These results are demonstrated to be applicable, in the presence of auxiliary information, to the construction of asymptotic bootstrap confidence bands for the quantile function. Moreover, we consider estimating the population semi-interquartile range on the basis of the modified quantile process. Results from a simulation study assessing the finite-sample performance of the proposed semi-interquartile range estimator are included.     

On Rescaled Poisson Processes and the Brownian Bridge

The process obtained by rescaling a homogeneous Poisson process by the maximum likelihood estimate of its intensity is shown to have surprisingly strong self-correcting behavior. Formulas for the conditional intensity and moments of the rescaled Poisson process are derived, and its behavior is demonstrated using simulations. Relationships to the Brownian bridge are explored, and implications for point process residual analysis are discussed.     

Komlos-Major-Tusnady approximation for the general empirical process and Haar expansions of classes of functions

Rates of convergence of the Komlos-Major-Tusnady type in the invariance principle for empirical processes indexed by functions are obtained. The conditions are given in terms of the empirical entropy and the accuracy of the Haar type approximation for the corresponding classes of functions. The recent results of Massart⁽²⁵⁾ as well as some new results for empirical characteristic functions are obtained as a corollary.     

Exit identities for diusion processes observed at Poisson arrival times

For diffusion processes, we extend various two-sided exit identities to the situation when the process is only observed at arrival times of an independent Poisson process. The results are expressed in terms of solutions to the differential equations associated with the diffusions generators.     

最近核邻型的光滑条件分位过程强逼近及其应用

文中证明了核邻型的光滑条件分位过程的强逼近，获得了其一致逼近速度．并由此结果推导出了光滑条件分位估计的渐近正态性、弱收敛和对数律等深刻结果．     

The central limit theorem for weighted empirical processes indexed by sets

Sufficient conditions are found for the weak convergence of a weighted empirical process {(ν_n(C)/q(P(C))) 1 _{[P(C) λn]}: C }, indexed by a class of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q, determine the fastest rate at which one may allow λ_n → 0.     

Large deviations for empirical measures of switching diffusion processes with small parameters

We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.     

一类带干扰风险过程的破产概率的估计

In this paper,a class of risk processes perturbed by diffusion are considered. The Lundberg inequalities for the ruin probability are obtained. The size of the Lundberg exponents for different kinds of risk model is compared. The numerical illustration for the impact of the parameters on the ruin probability is given.     

Local empirical processes near boundaries of convex bodies

We investigate the behaviour of Poisson point processes in the neighbourhood of the boundary ∂K of a convex body K in ,d ≥ 2. Making use of the geometry of K, we show various limit results as the intensity of the Poisson process increases and the neighbourhood shrinks to ∂K. As we shall see, the limit processes live on a cylinder generated by the normal bundle of K and have intensity measures expressed in terms of the support measures of K. We apply our limit results to a spatial version of the classical change-point problem, in which random point patterns are considered which have different distributions inside and outside a fixed, but unknown convex body K.     

On the strong approximation of bootstrapped empirical copula processes with applications

The purpose of the present paper is to provide a strong invariance principle for the generalized bootstrapped empirical copula processwith the rate of the approximation for multivariate empirical processes. As a by-product, we obtain a uniform-in-bandwidth consistency result for kernel-type estimators of copula derivatives, which is of its own interest. We introduce also the delta-sequence estimators of the copula derivatives. The applications discussed here are change-point detection in multivariate copula models, nonparametric tests of stochastic vectorial independence and the law of iterated logarithm for the generalized bootstrapped empirical copula process. Finally, a general notion of bootstrapped empirical copula process constructed by exchangeably weighting the sample is presented.     

Estimation for nonhomogeneous Poisson processes from aggregated data

A well-known heuristic for estimating the rate function or cumulative rate function of a nonhomogeneous Poisson process assumes that the rate function is piecewise constant on a set of data-independent intervals. We investigate the asymptotic (as the amount of data grows) behavior of this estimator in the case of equal interval widths, and show that it can be transformed into a consistent estimator if the interval lengths shrink at an appropriate rate as the amount of data grows.     

Constructions for planar point processes using concentric circles

Some point processes are obtained by generalising the well-known construction for a two-dimensional Poisson process which locates an event on each of a sequence of concentric circles in a particular way. The constructions considered here have, in general, a random number of events on each circle. Under certain sufficient conditions, the constructed processes are asymptotically Poisson, far from the origin. The obvious regularity in the structure of these processes can be removed at least superficially, by displacing the events independently off the concentric circles.     

Weak convergence of the sequential empirical processes of residuals in TAR models

This paper studies the weak convergence of the sequential empirical process K n of the residuals in the threshold autoregressive(TAR)model of order p.Under some mild conditions,it is shown that K n converges weakly to a Kiefer process plus a random variable which converges to a multivariate normal.This differs from that given by Bai(1994)for a stationary autoregressive and moving average(ARMA)model.     

The order of magnitude of the moments of the modulus of continuity of multiparameter poisson and empirical processes

In this note we derive the exact order of magnitude of the moments of the modulus of continuity for multiparameter Poisson and, almost as a corollary, for multivariate empirical processes.     

An Lp-view of the Bahadur-Kiefer Theorem

Summary Let ]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>]]>\alpha_n$ and $\beta_n$ be respectively the uniform empirical and quantile processes, and define $R_n = \alpha_n + \beta_n$, which usually is referred to as the Bahadur--Kiefer process. The well-known Bahadur-Kiefer theorem confirms the following remarkable equivalence: $\|R_n\| /\sqrt{\| \alpha_n \| }\, \sim \, n^{-1/4} (\log n)^{1/2}$ almost surely, as $n$ goes to infinity, where $\| f\| =\sup_{0\le t\le 1} |f(t)|$ is the $L^\infty$-norm. We prove that $\|R_n\|_2 /\sqrt{\| \alpha_n \|_1}\, \sim \, n^{-1/4}$ almost surely, where $\| \, \cdot \, \|_p$ is the $L^p$-norm. It is interesting to note that there is no longer any logarithmic term in the normalizing function. More generally, we show that $n^{1/4} \|R_n\|_p /\sqrt{\| \alpha_n \|_{(p/2)}}$ converges almost surely to a finite positive constant whose value is explicitly known.     

An empirical extension of the M/G/1 heavy traffic approximation

Numerical evaluation of waiting time distributions for M/G/1 systems is somewhat difficult. This paper examines a simple variation of the heavy traffic formula which may be useful at modest levels of traffic intensity. One can justify the heavy traffic approximation by expressing the Laplace transform of the service time distribution as a Maclaurin series and then truncating to three terms. The spectrum factorization and inversion leads in a straightforward fashion to the heavy traffic approximation. If one carries two additional terms from the Maclaurin series, the characteristic equation is a cubic with exactly one real negative root. This root provides an easy way to extend the heavy traffic formula to cases where the traffic is not so heavy. This paper studies the quality of this approximation and includes some numerical evaluation based on data actually encountered.     

The influence of deterministic noise on empirical measures generated by stationary processes

We consider weak convergence of empirical measures generated by stationary random process perturbed by deterministic noise . We assume that the noise has asymptotic distribution. In particular, we demonstrate that if the process is ergodic, or satisfies some mixing assumptions, then the influence of deterministic noise on is the same as it would be if were stochastic. Such results are of importance when investigating fluctuations and convex rearrangements of stochastic processes.