Alexander Aue

University of California, Davis | UCD · Department of Statistics

Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing stationarity tests for functional time series based on frequency domain methods. Setting up the tests requires a delicate...

Functional time series have become an integral part of both functional data and time series analysis. Important contributions to methodology, theory and application for the prediction of future trajectories and the estimation of functional time series parameters have been made in the recent past. This paper continues this line of research by propos...

An estimator for the time of a break in the mean of stationary functional data is proposed that is fully functional in the sense that it does not rely on dimension reduction techniques such as functional principal component analysis (fPCA). A thorough asymptotic theory is developed for the estimator of the break date for fixed break size and shrink...

In this paper, we propose a test for the two-sample problem of testing equality of mean vectors in the high-dimensional regime. The proposed test is based on a ridge-regularized Hotelling's $T^2$ statistic. We derive the cut-off values for the test through asymptotic analysis and suggest several finite sample modifications. We also propose a data d...

Spectral statistics play a central role in many multivariate testing problems. It is therefore of interest to approximate the distribution of functions of the eigenvalues of sample covariance matrices. Although bootstrap methods are an established approach to approximating the laws of spectral statistics in low-dimensional problems, these methods a...

The M6 Competition assessed the performance of competitors using a ranked probability score and an information ratio (IR). While these metrics do well at picking the winners in the competition, crucial questions remain for investors with longer-term incentives. To address these questions, we compare the competitors' performance to a number of conve...

Invertible processes naturally arise in many aspects of functional time series analysis, and consistent estimation of the infinite dimensional operators that define them are of interest. Asymptotic upper bounds for the estimation error of such operators for processes in the Hilbert space $L^2[0, 1]$ have been considered in recent years. This articl...

Quality control charts aim at raising an alarm as soon as sequentially obtained observations of an underlying random process no longer seem to be within stochastic fluctuations prescribed by an ‘in-control’ scenario. Such random processes can often be modelled using the concept of stationarity, or even independence as in most classical works. An im...

The accurate estimation of prediction errors in time series is an important problem, which has immediate implications for the accuracy of prediction intervals as well as the quality of a number of widely used time series model selection criteria such as the Akaike information criterion. Except for simple cases, however, it is difficult or even impo...

This paper tackles one of the most fundamental goals in functional time series analysis which is to provide reliable predictions for future functions. Existing methods for predicting a complete future functional observation use only completely observed trajectories. We develop a new method, called partial functional prediction (PFP), which uses bot...

In many complex systems, networks and graphs arise in a natural manner. Often, time evolving behavior can be easily found and modeled using time-series methodology. Amongst others, two common research problems in network analysis are community detection and change-point detection. Community detection aims at finding specific sub-structures within t...

This paper deals with two-sample tests for functional time series data, which have become widely available in conjunction with the advent of modern complex observation systems. Here, particular interest is in evaluating whether two sets of functional time series observations share the shape of their primary modes of variation as encoded by the eige...

Providing reliable predictions is one of the fundamental topics in functional time series analysis. Existing functional time series methodology seeks to predict a complete future functional observation based on a set of observed functions. The problem of interest discussed here is how to advance prediction methodology to cases where partial informa...

Networks and graphs arise naturally in many complex systems, often exhibiting dynamic behavior that can be modeled using dynamic networks. Two major research problems in dynamic networks are (1) community detection, which aims to find specific sub-structures within the networks, and (2) change point detection, which tries to find the time points at...

We are interested in testing general linear hypotheses in a high-dimensional multivariate linear regression model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. A family of rotation-invariant tests is proposed that involves a flexible spectral shrinka...

This paper deals with analyzing structural breaks in the covariance operator of sequentially observed functional data. For this purpose, procedures are developed to segment an observed stretch of curves into periods for which second-order stationarity may be reasonably assumed. The proposed methods are based on measuring the fluctuations of sample...

This paper deals with analyzing structural breaks in the covariance operator of sequentially observed functional data. For this purpose, procedures are developed to segment an observed stretch of curves into periods for which second-order stationarity may be reasonably assumed. The proposed methods are based on measuring the fluctuations of sample...

This article is concerned with the spectral behavior of p-dimensional linear processes in the moderately high-dimensional case when both dimensionality p and sample size n tend to infinity so that p/n→0. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocov...

Functional data analysis is typically conducted within the $L^2$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same ti...

Functional data analysis is typically conducted within the $L^2$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same ti...

Statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics, and they play a central role in multivariate testing. Although bootstrap methods are an established approach to approximating the laws of spectral statistics in low-dimensional problems, these methods are relatively unexplored in the high-dimension...

This contribution discusses the estimation of an invertible functional time series through fitting of functional moving average processes. The method uses a functional version of the innovations algorithm and dimension reduction onto a number of principal directions. Several methods are suggested to automate the procedures. Empirical evidence is pr...

Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is bas...

We develop a new methodology for the fitting of nonstationary time series that exhibit nonlinearity, asymmetry, local persistence and changes in location scale and shape of the underlying distribution. In order to achieve this goal, we perform model selection in the class of piecewise stationary quantile autoregressive processes. The best model is...

We develop a new methodology for the fitting of nonstationary time series that exhibit nonlinearity, asymmetry, local persistence and changes in location scale and shape of the underlying distribution. In order to achieve this goal, we perform model selection in the class of piecewise stationary quantile autoregressive processes. The best model is...

We propose a two-sample test for detecting the difference between mean vectors in a high-dimensional regime based on a ridge-regularized Hotelling's $T^2$. To choose the regularization parameter, a method is derived that aims at maximizing power within a class of local alternatives. We also propose a composite test that combines the optimal tests c...

Partition-wise models offer a flexible approach for modeling complex and multidimensional data that are capable of producing interpretable results. They are based on partitioning the observed data into regions, each of which is modeled with a simple submodel. The success of this approach highly depends on the quality of the partition, as too large...

Heteroskedasticity is a common feature of financial time series and is commonly addressed in the model building process through the use of ARCH and GARCH processes. More recently multivariate variants of these processes have been in the focus of research with attention given to methods seeking an efficient and economic estimation of a large number...

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 p...

This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/n\to0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized samp...

We give an overview of random matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the develo...

This article proposes new model-fitting techniques for quantiles of an observed data sequence, including methods for data segmentation and variable selection. The main contribution, however, is in providing a means to perform these two tasks simultaneously. This is achieved by matching the data with the best-fitting piecewise quantile regression mo...

We study sequential monitoring procedures that detect instabilities of the regression operator in an underlying (fully) functional regression model allowing for dependence. These open-end and closed-end procedures are built on a functional principal components analysis of both the predictor and response functions, thus giving rise to multivariate d...

In this paper, novel joint semiparametric spline-based modeling of conditional mean and volatility of financial time series is proposed and evaluated on daily stock return data. The modeling includes functions of lagged response variables and time as predictors. The latter can be viewed as a proxy for omitted economic variables contributing to the...

The concentration of aerosol particles, largely caused by traffic volume and often enhanced during temperature inversion episodes in the cold season, can be a concern for human health in the urban environment. This particulate matter is typically recorded as PM10, the total mass of particles below 10 μm in diameter. It is suspected that, within the...

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 p...

High-dimensional time series arise naturally in econometrics and finance, atmospheric and environmental science, genomics, experimental chemistry, and electrical engineering, among a multitude of disciplines. Recent developments in the statistical analysis of high-dimensional data have demonstrated the limitations of many existing statistical proce...

In astrophysics a common goal is to infer the flux distribution of populations of scientifically interesting objects such as pulsars or supernovae. In practice, inference for the flux distribution is often conducted using the cumulative distribution of the number of sources detected at a given sensitivity. The resulting "log(N>S)-log(S)" relationsh...

This paper gives an account of some of the recent work on structural breaks in time series models. In particular, we show how procedures based on the popular cumulative sum, CUSUM, statistics can be modified to work also for data exhibiting serial dependence. Both structural breaks in the unconditional and conditional mean as well as in the varianc...

This paper addresses the prediction of functional time series. Existing contributions to this problem have largely focused on the special case of first-order functional autoregressive processes because of their technical tractability and the current lack of advanced functional time series methodology. It is shown here how standard multivariate pred...

Despite substantial criticism, variants of the capital asset pricing model (CAPM) remain to this day the primary statistical tools for portfolio managers to assess the performance of financial assets. In the CAPM, the risk of an asset is expressed through its correlation with the market, widely known as the beta. There is now a general consensus am...

In this paper, we quantify the reaction time of on-line monitoring schemes for changes in the mean based on moving sums. The corresponding sequential test procedure requires a historical sample of size m as a baseline, while decisions are made based on a window of size h=h(m)h=h(m) containing the h most recent observations. Perhaps surprisingly, th...

Image segmentation is a long-studied and important problem in image processing. Different solutions have been proposed, many of which follow the information theoretic paradigm. While these information theoretic segmentation methods often produce excellent empirical results, their theoretical properties are still largely unknown. The main goal of th...

This paper studies the problem of local bandwidth selection for local linear regression. It is known that the optimal local bandwidth for estimating the unknown curve f at design point x depends on the curve’s second derivative f''(x) at x. Therefore one could select the local bandwidth h(x) at x via estimating f''(x). However, as typically estimat...

Several tests for detecting mean shifts at an unknown time in stationary time series have been proposed, including cumulative sum (CUSUM), Gaussian likelihood ratio (LR), maximum of F(F) and extreme value statistics. This article reviews these tests, connects them with theoretical results, and compares their finite sample performance via simulation...

The blocking method offers a quick and simple procedure for obtaining a scatterplot smooth from a noisy data set. The idea is to partition the data into blocks of equal size and fit a low order polynomial to each block. An important component of this method is the choice of the number of blocks. This article develops an automatic selection method f...

We propose a unified quasi-likelihood procedure for the estimation of the unknown parameters of a first-order random coefficient autoregressive, RCA, model that works both for stationary and nonstationary processes. For this procedure, the weak consistency and the asymptotic normality are established under minimal assumptions on the noise sequences...

The talk concerns sequential procedures detection of changes in linear relationship Yk(t) = ò10Yk(t,s)Xk(S)ds+ek(t),1 £ k < ¥{\rm{Y}_{k}(t)}\, = \, {\int^{1}_{0}}\,{\Psi}_{k}{\rm(t,\,s)}\,{\rm{X}_{k}(S){ds}}\,+\,{\varepsilon_{k}}{\rm\,(t)},\,{1}\,\leq\,{\rm\,{k}}< \infty between random functions Yk and Xk on [0,1], where errors { εk} [0,1] and { Ψ...

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility, and non-trading periods that may be different for the two assets. We also allow for asy...

A long studied and important image processing problem is image segmentation. In this paper theoretical properties of some image segmentation methods are investigated. More precisely, we are interested if these methods are statistically consistent, that is, if they can accurately recover the number of segments together with their boundaries in the i...

In this paper, we introduce an asymptotic test procedure to assess the stability of volatilities and cross-volatilites of linear and nonlinear multivariate time series models. The test is very flexible as it can be applied, for example, to many of the multivariate GARCH models established in the literature, and also works well in the case of high d...

The paper develops a comprehensive asymptotic theory for the estimation of a change-point in the mean function of functional observations. We consider both the case of a constant change size, and the case of a change whose size approaches zero, as the sample size tends to infinity. We show how the limit distribution of a suitably defined change-poi...

In this paper, we provide a segmentation procedure for mean-nonstationary time series. The segmentation is obtained by casting the problem into the framework of detecting structural breaks in trending regression models in which the regressors are generated by suitably smooth functions. As test statistics we propose to use the maximally selected lik...

We study in this paper the extremal behavior of stochastic integrals of Legendre polynomial transforms with respect to Brownian motion. As the main results, we obtain the exact tail behavior of the supremum of these integrals taken over intervals [0,h] with h>0 fixed, and the limiting distribution of the supremum on intervals [0,T] as T→∞. We show...

We consider a multiple regression model in which the explanatory variables are specified by time series. To sequentially test for the stability of the regression parameters in time, we introduce a detector which is based on the first excess time of a CUSUM-type statistic over a suitably constructed threshold function. The aim of this paper is to st...

We study test procedures that detect structural breaks in underlying data sequences. In particular, we wish to discriminate between different reasons for these changes, such as (1) shifting means, (2) random walk behavior, and (3) constant means but innovations switching from stationary to difference stationary behavior. Almost all procedures prese...

We consider a nonlinear polynomial regression model in which we wish to test the null hypothesis of structural stability in the regression parameters against the alternative of a break at an unknown time. We derive the extreme value distribution of a maximum-type test statistic which is asymptotically equivalent to the maximally selected likelihood...

The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economist...

We study a CUSUM–type monitoring scheme designed to sequentially detect changes in the regression parameter of an underlying linear model. The test statistic used is based on recursive residuals. Main aim of this paper is to derive the limiting extreme value distribution under the null hypothesis of structural stability. The model assumptions are f...

Let $\{X_j\}$ be independent, identically distributed random variables. It is well known that the functional CUSUM statistic and its randomly permuted version both converge weakly to a Brownian bridge if second moments exist. Surprisingly, an infinite-variance counterpart does not hold true. In the present paper, we let $\{X_j\}$ be in the domain o...

We determine the limiting behavior of near-integrated first-order random coefficient autoregressive RCA(1) time series. It is shown that the asymptotics of the finite-dimensional distributions crucially depends on how the critical value 1 is approached, which determines whether the process is near-stationary, has a unit root, or is mildly explosive...

We study the limiting behavior of the prominent R/S test statistic, aimed at detecting long-range dependence, if instead of long memory a stochastic trend given by cumulative random shocks is present. As the main result we derive the convergence rate of the R/S statistic to its limit.

We discuss the limiting behavior of the serial correlation coefficient in mildly explosive autoregression, where the error sequence is in the domain of attraction of an -stable law, (0,2 . Therein, the autoregressive coefficient = n 1 is assumed to satisfy the condition n 1 such that n( n 1) as n . In contrast to the vast majority of existing liter...

We provide a characterization of strictly stationary solutions to the stochastic recurrence equation zk = c("k 1)zk 1 + g("k 1) with Borel-measurable functions c and g, and independent, identically distributed random variables {"k}. Strictly stationary solutions that are functions of the past, respectively, of the future exist if and only if the ex...

We consider a linear regression model with errors modelled by martingale difference sequences, which include heteroskedastic augmented GARCH processes. We develop asymptotic theory for two monitoring schemes aimed at detecting a change in the regression parameters. The first method is based on the CUSUM of the residuals and was studied earlier in t...

We utilize strong invariance principles to construct tests for the stability of model parameters determining a random coefficient autoregressive time series of order one. The test statistics are based on (conditional) least squares estimators for the unknown parameters.

We study so-called augmented GARCH sequences, which include many submodels of considerable interest, such as polynomial and exponential GARCH. To model the returns of speculative assets, it is particularly important to understand the behaviour of the squares of the observations. The main aim of this paper is to present a strong approximation for th...

We propose the quasi-maximum likelihood method to estimate the parameters of an RCA(1) process, i.e. a random coefficient autoregressive time series of order 1. The strong consistency and the asymptotic normality of the estimators are derived under optimal conditions. Copyright 2006 Blackwell Publishing Ltd.

We study the limiting behaviour of the prominent V /S and R/S test statistics, aimed at detecting long–range dependence, under the presence of a stochastic trend, which is given by cumulative random shocks. Depending on the size of these shocks, the asymptotic distribution is characterized either solely by the stochastic trend, or solely by the err...

In this paper, we derive a strong invariance principle for the partial sums of RCA(1) random variables. An application yields asymptotic tests for a change in the mean of the observations both for sequential and a posteriori procedures based on CUSUMs.

We consider a sequential test procedure which detects possible changes in the mean of observations satisfying a weak invariance principle. Our test statistic is based on weighted CUSUMs of the underlying random variables. In this paper, we study the asymptotic behaviour of the delay time if a change has occurred in the sample after a training perio...

Giving a generalization of Berkes and Horváth (2003), we consider the Euclidean norm of vector-valued stochastic processes, which can be approximated with a vector-valued Wiener process having a linear drift. The suprema of the Euclidean norm of the processes are not far away from the norm of the processes at the right most point. We also obtain an...

Köln, University, Diss., 2004 (Nicht für den Austausch).

We consider an estimator of the change-point of a stochastic process satisfying some weak invariance principles. Making use of the known asymptotics of the approximating Wiener processes we derive various limiting distributions according to different orders of magnitude of the underlying change. The results take into account, but also extend those...

Consider a linear model setting in which the explanatory variables are specified by time series. To sequentially test for the stability of the regression parameters in time, we introduce a detector which is based on the first excess time of a CUSUM-type statistic over a suitably defined threshold function. The main aim of this paper is to derive th...

We derive the limit distribution of a stopping time used to sequentially detect a change from stationary to random walk behavior. Results are compared to known results in a level shift setting and are underlined by a simulation study.