Tailen Hsing

On the exceedance point process for a stationary sequence

SummaryIt is known that the exceedance points of a high level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate long range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. More generally the possible random measure limits for normalized exceedance point processes are characterized. Sufficient conditions are also given for the existence of a point process limit. The limiting distributions of extreme order statistics are derived as corollaries.

Theoretical foundations of functional data analysis, with an introduction to linear operators

Description: Functional data is data in the form of curves that is becoming a popular method for interpreting scientific data. Statistical Analysis of Functional Data provides an authoritative account of function data analysis covering its foundations, theory, methodology, and practical implementation. It also contains examples taken from a wide range of disciplines, including finance, medicine, and psychology. The book includes a supporting Web site hosting the real data sets analyzed in the book and related software. Statistical researchers or practitioners analyzing functional data will find this book useful.

Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data

We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.

On weighted U-statistics for stationary processes

A weighted U-statistic based on a random sample X1;::: ;Xn has the form Un = P 1•i;jn wiijK(Xi;Xj) where K is fixed symmetric measurable function and the wi are symmetric weights. A large class of statistics can be expressed as weighted U-statistics or variations thereof. This paper establishes the asymptotic normality of Un when the sample observations come from a non-linear time series and linear processes. MSC 2000 subject classifications. Primary 60F05; secondary 60G10.

Extreme value theory for multivariate stationary sequences

A distributional mixing condition is introduced for stationary sequences of random vectors to study their extremes. For a sequence satisfying the condition, the following topics which concern the weak limit F of properly normalized partial maxima are studied: (1) To obtain characterizations of F. (2) To study a condition under which the partial maxima behave as they would if the sequence were i.i.d. (3) To consider problems in connection with the independence of the margins of F.

ON THE ASYMPTOTIC JOINT DISTRIBUTION OF THE SUM AND MAXIMUM OF STATIONARY NORMAL RANDOM VARIABLES

Let X 1 , X 2 , ·· ·be stationary normal random variables with ρ n = cov( X 0 , X n ). The asymptotic joint distribution of and is derived under the condition ρ n log n → γ [0,∞). It is seen that the two statistics are asymptotically independent only if γ = 0.

On the characterization of certain point processes

This paper consists of two parts. First, a characterization is obtained for a class of infinitely divisible point processes on . Second, the result is applied to identify the weak limit of the point process Nn with points (j/n, un-1 ([xi]j)), j = 0, ±1, ±2, ..., where {[xi]j} is a stationary sequence satisfying a certain mixed conditio [Delta], and {un} is a sequence of non-increasing functions on (0, [infinity]) such that This application extends a result of Mori [14], which assumes that {[xi]j} is [alpha]-mixing, and that the distribution of max1[less-than-or-equals, slant]j[less-than-or-equals, slant]j [xi]j can be linearly normalized to converge to a maximum stable distribution.

On the extreme order statistics for a stationary sequence

Suppose that {[xi]j} is a strictly stationary sequence which satisfies the strong mixing condition. Denote by M(k)n the kth largest value of [xi]1,[xi]2,...,[xi]n, and {[upsilon]n(·)} a sequence of normalizing functions for which P[M(1)n[less-than-or-equals, slant][upsilon]n(x)]converges weakly to a continuous distribution G(x). It is shown that if for some k=2,3,...,P[M(k)n[less-than-or-equals, slant][upsilon]n(x)] converges for each x, then there exist probabilities p1,...,pk-1 such that P[M(j)n[less-than-or-equals, slant][upsilon]n(x)] converges weakly to for j=2,...,k, where natural interpretations can be given for the pj. This generalizes certain results due to Dziubdziela (1984) and Hsing, Husler and Leadbetter (1986). It is further demonstrated that, with minor modification, the technique can be extended to study the joint limiting distribution of the order statistics. In particular, Theorem 1 of Welsch (1972) is generalized, and some links between the convergence of the order statistics and that of certain point processes are established.

Relation Between Permutation-Test P Values and Classifier Error Estimates

Gene-expression-based classifiers suffer from the small number of microarrays usually available for classifier design. Hence, one is confronted with the dual problem of designing a classifier and estimating its error with only a small sample. Permutation testing has been recommended to assess the dependency of a designed classifier on the specific data set. This involves randomly permuting the labels of the data points, estimating the error of the designed classifiers for each permutation, and then finding the p value of the error for the actual labeling relative to the population of errors for the random labelings. This paper addresses the issue of whether or not this p value is informative. It provides both analytic and simulation results to show that the permutation p value is, up to very small deviation, a function of the error estimate. Moreover, even though the p value is a monotonically increasing function of the error estimate, in the range of the error where the majority of the p values lie, the function is very slowly increasing, so that inversion is problematic. Hence, the conclusion is that the p value is less informative than the error estimate. This result demonstrates that random labeling does not provide any further insight into the accuracy of the classifier or the precision of the error estimate. We have no knowledge beyond the error estimate itself and the various distribution-free, classifier-specific bounds developed for this estimate.

DECIDING THE DIMENSION OF EFFECTIVE DIMENSION REDUCTION SPACE FOR FUNCTIONAL AND HIGH-DIMENSIONAL DATA

In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is called the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential χ 2 testing procedures under the assumption that the predictor has an elliptically contoured distribution. We further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores. The proposed procedures are supported by theory, validated by simulation studies, and illustrated by a real-data example. Our methods and theory are applicable to functional data and high-dimensional multivariate data.