Functional Data Analysis with Increasing Number of Projections

Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d tends to infinity with the sample size. We propose a specific framework in which we let d tend to infinity by deriving a normal approximation for the two-parameter partial sum process of the scores \xi_{i,j} of the i-th function with respect to the j-th FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example

Кілька зауважень стосовно статті “Про одне O-зворотне твердження” Воїслава Авакумовича

Some comments concerning the origin of the (R–O) notion for real functions are given, which has been used in the paper above, but was first introduced by Avakumović (1935). Moreover, some later extensions and generalizations of such functions are briefly discussed.Статтю присвячено розвитку ідеї правильно змінної зміни функцій дійсного аргументу. Особливу увагу приділено узагальненню цього поняття, яке вперше з’явилось у статті В. Авакумовіча в 1936 році сербсько-хорватською мовою. Зараз ця властивість позначається ORV або OR, хоча іноді використовується й оригінальне позначення R–O. Функції, які мають цю властивість, називаються функціями Авакумовича–Карамати. У статті просліджено розвиток властивості ORV в статтях інших авторів, починаючи з роботи Й. Карамати, також опублікованій у 1936 році. У XX сторіччі ця властивість досліджувалась здебільшого у зв’язку з конкретними застосуваннями у математичному аналізі або теорії ймовірностей (див., наприклад, Bari та Stechkin (1956) або Feller (1969), а також інші роботи у списку літератури). Пізніше з’явилися роботи, у яких властивість ORV використовувалась у теорії звичайних диференціальних рівнянь, теорії чисел, комплексному аналізі, функціональному аналізі тощо. Разом з цим з’явилося розуміння, що ця властивість є надто загальною, а її часткові випадки також мають широке коло змістовних застосувань. Особливу увагу у другій частині статті приділено властивостям PRV та невиродженості групи регулярних точок, які автори досліджували разом з В. В. Булдигіним та К.-Х. Індлекофером, починаючи з 1999 року. Властивість невиродженості групи регулярних точок вирізняє з класу ORV ті функції, у яких границя Карамати існує для принаймні двох точок. Виявляється, що кожну з таких функцій можна зобразити як добуток певної функції Карамати на іншу логарифмічно періодичну функцію, тобто такі функції утоворюють більш широкий клас, ніж правильно змінні функції Карамати. Клас RV складається з тих функцій, які зберігають асимптотичну еквівалентність як послідовностей, так і функцій. Неявно таку властивість використовували ранішо багато інших авторів, проте вони не помічали, що за нею прихована теорія, багата на результати внутрішнього характеру та на застосування. Детально властивості невиродженості групи регулярних точок та збереження асимптотичної еквівалентності викладено в монографії Buldygin, Indlekofer, Klesov та Steinebach (2018)

SFB 823 Reaction times of monitoring schemes for ARMA time series Discussion Paper Reaction times of monitoring schemes for ARMA time series *

Abstract This paper is concerned with deriving the limit distributions of stopping times devised to sequentially uncover structural breaks in the parameters of an autoregressive moving average, ARMA, time series. The stopping rules are defined as the first time lag for which detectors, based on CUSUMs and Page's CUSUMs for residuals, exceed the value of a prescribed threshold function. It is shown that the limit distributions crucially depend on a drift term induced by the underlying ARMA parameters. The precise form of the asymptotic is determined by an interplay between the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications to electroencephalography, EEG, and IBM data. The empirical results indicate a satisfactory behavior in finite samples

Estimating a gradual parameter change in an AR(1)-process

We discuss the estimation of a change-point t(0) at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations X-1,..., X-n, we shall study the least squares estimator (t(0)) over cap for t(0), which is obtained by minimizing the sum of squares of residuals with respect to the given parameters. As a first result it can be shown that, under certain regularity and moment assumptions, (t(0)) over cap /n is a consistent estimator for t(0), where t(0) = left perpendicularn tau(0)right perpendicular, with 0 (P) tau(0) (n ->infinity). Based on the rates obtained in the proof of the consistency result, a first, but rough, convergence rate statement can immediately be given. Under somewhat stronger assumptions, a precise rate can be derived via the asymptotic normality of our estimator. Some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator

Robust monitoring of CAPM portfolio betas II

In this work, we extend our study in Chochola et al. [7] and propose some robust sequential procedure for the detection of structural breaks in a Functional Capital Asset Pricing Model (FCAPM). The procedure is again based on M-estimates and partial weighted sums of M-residuals and robustifies the approach of Aue et al. [3], in which ordinary least squares (OLS) estimates have been used. Similar to Aue et al. [3], and in contrast to Chochola et al. [7], high-frequency data can now also be taken into account. The main results prove some null asymptotics for the suggested test as well as its consistency under local alternatives. In addition to the theoretical results, some conclusions from a small simulation study together with an application to a real data set are presented in order to illustrate the finite sample performance of our monitoring procedure. (C) 2014 Elsevier Inc. All rights reserved

Pseudo-regularly varying functions and generalized renewal processes

One of the main aims of this book is to exhibit some fruitful links between renewal theory and regular variation of functions. Applications of renewal processes play a key role in actuarial and financial mathematics as well as in engineering, operations research and other fields of applied mathematics. On the other hand, regular variation of functions is a property that features prominently in many fields of mathematics. The structure of the book reflects the historical development of the authors’ research work and approach – first some applications are discussed, after which a basic theory is created, and finally further applications are provided. The authors present a generalized and unified approach to the asymptotic behavior of renewal processes, involving cases of dependent inter-arrival times. This method works for other important functionals as well, such as first and last exit times or sojourn times (also under dependencies), and it can be used to solve several other problems. For example, various applications in function analysis concerning Abelian and Tauberian theorems can be studied as well as those in studies of the asymptotic behavior of solutions of stochastic differential equations. The classes of functions that are investigated and used in a probabilistic context extend the well-known Karamata theory of regularly varying functions and thus are also of interest in the theory of functions. The book provides a rigorous treatment of the subject and may serve as an introduction to the field. It is aimed at researchers and students working in probability, the theory of stochastic processes, operations research, mathematical statistics, the theory of functions, analytic number theory and complex analysis, as well as economists with a mathematical background. Readers should have completed introductory courses in analysis and probability theory.

Monitoring risk in a ruin model perturbed by diffusion

Perturbed risk model, Ruin probability, Adjustment coefficient, Nonparametric sequential test, Asymptotic size, Power 1, Change-point test, Empirical moment-generating function, Strong approximation, Wiener process,