Weak convergence of the tail empirical process for dependent sequences

This paper proves weak convergence in D$D$ of the tail empirical process–the renormalized extreme tail of the empirical process–for a large class of stationary sequences. The conditions needed for convergence are (i) moment restrictions on the amount of clustering of extremes, (ii) restrictions on long range dependence (absolute regularity or strong mixing), and (iii) convergence of the covariance function. We further show how the limit process is changed if exceedances of a nonrandom level are replaced by exceedances of a high quantile of the observations. Weak convergence of the tail empirical process is one key to asymptotics for extreme value statistics and its wide range of applications, from geoscience to finance.     

Limit theorems for runs based on ‘small spacings’

A new concept of runs was proposed in the work of Eryilmaz and Stepanov (2008). A sequence of spacings forms a run if the lengths of these spacings do not exceed ε>0. In that paper, asymptotic properties of such spacings were investigated and statistical criteria proposed. In our present study, we maintain research on runs associated with these spacings. We derive limit theorems for the total number of runs, longest run and propose a statistical criterion.     

Estimation of two ordered bivariate mean residual life functions

Situations occur frequently in which the mean residual life (mrl) functions of two populations must be ordered. For example, if a mechanical device is improved, the mrl function for the improved device should not be less than that of the original device. Also, mrl functions for medical patients should often be ordered depending on the status of concomitant variables. This paper proposes nonparametric estimators of the bivariate mrl function under a mrl ordering. The estimators are shown to be asymptotically unbiased, strongly uniformly consistent and weakly convergent to a bivariate Gaussian process. The estimators are shown to be the projections, in a sense to be made precise, of the empirical mrl function onto an appropriate convex set of mrl functions. In the one-sample problem, the new estimators dominate the empirical mrl function in terms of risk with respect to a wide class of loss functions.     

Spatial autoregression and related spatio-temporal models

We propose a spatial autoregressive random field of order p on the spatial domain for p?2 in this paper, whose univariate margins are the continuous-time autoregression of order p on the real line, and introduce a class of semiparametric spatio-temporal covariance models stationary in space with the spatial autoregressive margin.     

Some notes on extremal discriminant analysis

Classical discriminant analysis focusses on Gaussian and nonparametric models where in the second case the unknown densities are replaced by kernel densities based on the training sample. In the present article we assume that it suffices to base the classification on exceedances above higher thresholds, which can be interpreted as observations in a conditional framework. Therefore, the statistical modeling of truncated distributions is merely required. In this context, a nonparametric modeling is not adequate because the kernel method is inaccurate in the upper tail region. Yet one may deal with truncated parametric distributions like the Gaussian ones. Our primary aim is to replace truncated Gaussian distributions by appropriate generalized Pareto distributions and to explore properties and the relationship of discriminant functions in both models.     

Limit distribution of a roundoff error

Let [X] and {X} be the integer and the fractional parts of a random variable X. The conditional distribution function F_n(x)=P({X}≤x|[X]=n) for an integer n is investigated. F_n for a large n is regarded as the distribution of a roundoff error in an extremal event. For most well-known continuous distributions, it is shown that F_n converges as n→∞ and three types of limit distributions appear as the limit distribution according to the tail behavior of F.     

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Summary Consider the stationary sequenceX ₁=G(Z ₁),X ₂=G(Z ₂),..., whereG(·) is an arbitrary Borel function andZ ₁,Z ₂,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z ₁ Z _k+1) satisfyingr(0)=1 and _k=1 |r(k)| ^m < , where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X _n }, the integerm is the Hermite rank of the family {I{G(·) x} –F(x):xR}. LetF _n (·) be the empirical distribution function ofX ₁,...,X _n . We prove that, asn, the empirical processn ^1/2{F _n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[–,].Partially supported by NSF Grant DMS-9208067     

On some extremal problems in H p and the prediction ofL p -harmonizable stochastic processes

Summary We providesimple andsuccinct solutions to two dual extremal problems in the Hardy spacesH ^p , and to an aspect of the linear prediction problem for a certain class of discrete and continuous parameter L ^p -harmonizable stochastic processes, for all 1p<. Two of the results presented appear new. The methods of proof of the rest of the results provide alternatesimpler andshorter proofs for some earlier known theorems.This research is partially supported by AFSOR Grant No. 90-016 8 and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence     

Properties of spatial cross-periodograms using fixed-domain asymptotics

Cross-periodograms can be used to study a multivariate spatial process observed on a lattice. For spatial data, it is often appropriate to study asymptotic properties of statistical procedures under fixed-domain asymptotics in which the number of observations increases in a fixed region while shrinking distances between neighboring observations. Using fixed-domain asymptotics, we prove relative asymptotic unbiasedness and relative consistency of a smoothed cross-periodogram after appropriate filtering of the data. In addition, we show that smoothed cross-periodograms are asymptotically normal when the process is stationary multivariate Gaussian with appropriate assumptions on high-frequency behavior of the spectral density.     

Peaks-over-threshold stability of multivariate generalized Pareto distributions

It is well-known that the univariate generalized Pareto distributions (GPD) are characterized by their peaks-over-threshold (POT) stability. We extend this result to multivariate GPDs.It is also shown that this POT stability is asymptotically shared by distributions which are in a certain neighborhood of a multivariate GPD. A multivariate extreme value distribution is a typical example.The usefulness of the results is demonstrated by various applications. We immediately obtain, for example, that the excess distribution of a linear portfolio with positive weights a_i, i≤d, is independent of the weights, if (U₁,…,U_d) follows a multivariate GPD with identical univariate polynomial or Pareto margins, which was established by Macke [On the distribution of linear combinations of multivariate EVD and GPD distributed random vectors with an application to the expected shortfall of portfolios, Diploma Thesis, University of Würzburg, 2004, (in German)] and Falk and Michel [Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58 (2006) 261-290]. This implies, for instance, that the expected shortfall as a measure of risk fails in this case.     

Sharp probability estimates for generalized Smirnov statistics

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line. Author’s address: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds

Summary Here we discuss the regularity of solutions of SDE's and obtain conditions under which a SDE on a complete Riemannian manifoldM has a global smooth solution flow, in particular improving the usual global Lipschitz hypothesis whenM=R ⁿ . There are also results on non-explosion of diffusions.Research supported by SERC grant GR/H67263     

The point process approach for fractionally differentiated random walks under heavy traffic

We prove some heavy-traffic limit theorems for some nonstationary linear processes which encompass the fractionally differentiated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a non-Gaussian stable distribution. The results are based on an extension of the point process methodology to linear processes with nonsummable coefficients and make use of a new maximal type inequality.     

On homogeneous stochastic processes on compact Abelian groups

This article discusses the extension to general compact Abelian groups of some results previously established by R. Roy for the case of the circle and the sphere. Estimators of the covariance function and spectral parameters for a homogeneous stochastic process defined on a compact Abelian group are considered and their properties are derived.     

Local asymptotic normality in a stationary model for spatial extremes

De Haan and Pereira (2006) [6] provided models for spatial extremes in the case of stationarity, which depend on just one parameter β>0 measuring tail dependence, and they proposed different estimators for this parameter. We supplement this framework by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold. Standard arguments from LAN theory then provide the asymptotic minimum variance within the class of regular estimators of β. It turns out that the relative frequency of exceedances is a regular estimator sequence with asymptotic minimum variance, if the underlying observations follow a multivariate extreme value distribution or a multivariate generalized Pareto distribution.