http://library.um.ac.id/free-contents/downloadpdf.php/buku/laws-of-small-numbers-extremes-and-rare-events-michael-falk-jurg-husler-rolf-dieter-reiss-42718.pdf

Laws of small numbers: extremes and rare events / Michael Falk, Jurg Husler, Rolf-Dieter Reiss

: The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes. The theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts. Emphasis is placed on stationary cases but other important classes (e.g. Mark of sequences) are included. Significant ideas and methods are described rather than details, and in particular the nature and role of important underlying point processes (such as exceedances and upcrossings) are emphasized. Applications are given to particular classes of process (e.g. normal, moving average) and connections with related theory (such as convergence of sums) are indicated.

/pdf/external-theory-for-stochastic-processes-ojd30t0xp1.pdf

External Theory for Stochastic Processes.

We consider density estimates of the usual type generated by a weight function. Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity. Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved.

/pdf/on-some-global-measures-of-the-deviations-of-density-2k83ayeluy.pdf

On Some Global Measures of the Deviations of Density Function Estimates

Extremes of Gaussian processes over an infinite horizon

In the present paper we construct asymptotic confidence bands in non-parametric regression. Our assumptions cover unequal variances of the observations and nonuni-form, possibly considerably clustered design. The confidence band is based on an undersmoothed local polynomial estimator. An appropriate quantile is obtained via the wild bootstrap. We derive certain rates (in the sample size n) for the error in coverage probability, which improves on existing results for methods that rely on the asymptotic distribution of the maximum of some Gaussian process. We propose a practicable rule for a data-dependent choice of the band-width. A small simulation study illustrates the possible gains by our approach over alternative frequently used methods.

Simultaneous bootstrap confidence bands in nonparametric regression

: The authors consider two problems for separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho(t,s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or <2. In showing the existence of such (stationary) processes for 0 = or < alpha < 2, the authors relate the magnitude of the tails of the spectral distributionsto the behavior of the covariance function at the origin. For 0 < alpha = or < 2, the authors obtain the asymptotic distribution of the maximum of X(t). This second result is used to obtain a result for X(t) as t approaches infinity similar to the 'so called' law of the iterated logarithm. (Author)

/pdf/asymptotic-properties-of-gaussian-processes-2vji8ixt44.pdf

Asymptotic Properties of Gaussian Processes

: The paper presents a result which generalizes the paper of Watanabe (Trans. AMS 148, 233-248). The result is extended also to the nonstationary process treated by Watanabe.

/pdf/an-asymptotic-0-1-behavior-of-gaussian-processes-q182w25f6n.pdf

Hisao Watanabe

Papers

Asymptotic Properties of Gaussian Processes

An Asymptotic 0-1 Behavior of Gaussian Processes