Long-range Dependent Time Series Research Articles

Rank-based change-point analysis for long-range dependent time series

We consider change-point tests based on rank statistics to test for structural changes in long-range dependent observations. Under the hypothesis of stationary time series and under the assumption of a change with decreasing change-point height, the asymptotic distributions of corresponding test statistics are derived. For this, a uniform reduction principle for the sequential empirical process in a two-parameter Skorohod space equipped with a weighted supremum norm is proved. Moreover, we compare the efficiency of rank tests resulting from the consideration of different score functions. Under Gaussianity, the asymptotic relative efficiency of rank-based tests with respect to the CuSum test is 1, irrespective of the score function. Regarding the practical implementation of rank-based change-point tests, we suggest to combine self-normalized rank statistics with subsampling. The theoretical results are accompanied by simulation studies that, in particular, allow for a comparison of rank tests resulting from different score functions. With respect to the finite sample performance of rank-based change-point tests, the Van der Waerden rank test proves to be favorable in a broad range of situations. Finally, we analyze data sets from economy, hydrology, and network traffic monitoring in view of structural changes and compare our results to previous analysis of the data.

Covariance-engaged Classification of Sets via Linear Programming

Set classification aims to classify a set of observations as a whole, as opposed to classifying individual observations separately. To formally understand the unfamiliar concept of binary set classification, we first investigate the optimal decision rule under the normal distribution, which utilizes the empirical covariance of the set to be classified. We show that the number of observations in the set plays a critical role in bounding the Bayes risk. Under this framework, we further propose new methods of set classification. For the case where only a few parameters of the model drive the difference between two classes, we propose a computationally-efficient approach to parameter estimation using linear programming, leading to the Covariance-engaged LInear Programming Set (CLIPS) classifier. Its theoretical properties are investigated for both independent case and various (short-range and long-range dependent) time series structures among observations within each set. The convergence rates of estimation errors and risk of the CLIPS classifier are established to show that having multiple observations in a set leads to faster convergence rates, compared to the standard classification situation in which there is only one observation in the set. The applicable domains in which the CLIPS performs better than competitors are highlighted in a comprehensive simulation study. Finally, we illustrate the usefulness of the proposed methods in classification of real image data in histopathology.

Ordinal Pattern Dependence in the Context of Long-Range Dependence.

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for a long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so, we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

A Comparison of Hurst Exponent Estimators in Long-range Dependent Curve Time Series

Abstract The Hurst exponent is the simplest numerical summary of self-similar long-range dependent stochastic processes. We consider the estimation of Hurst exponent in long-range dependent curve time series. Our estimation method begins by constructing an estimate of the long-run covariance function, which we use, via dynamic functional principal component analysis, in estimating the orthonormal functions spanning the dominant sub-space of functional time series. Within the context of functional autoregressive fractionally integrated moving average (ARFIMA) models, we compare finite-sample bias, variance and mean square error among some time- and frequency-domain Hurst exponent estimators and make our recommendations.

A modified Wilcoxon test for change points in long-range dependent time series

This paper considers testing for structural change in long-memory time series. We modify the Wilcoxon two-sample rank test by standardizing it with a kernel-based fixed bandwidth long-run variance estimator. The corresponding test statistic converges to a well-defined distribution under the null hypothesis. In a Monte Carlo simulation we confirm that the test provides good finite sample size and power results and compare it with an existing approach.

Improved Estimation of the Memory Parameter

In this paper, it is proposed to estimate the memory parameter of a potentially long-range dependent time series by applying goodness-of-fit tests to the cumulative normalized periodogram in the neighborhood of frequency zero. The results of an extensive simulation study show that this new estimator performs well compared to conventional frequency-domain estimators which are based on the Whittle likelihood or are obtained from the popular log periodogram estimator by trimming, smoothing, and utilizing non-Fourier frequencies, respectively. In an empirical investigation of log absolute daily index returns, we find evidence of long-range dependence with values of the memory parameter in the range between 0.2 and 0.3 both in developed and developing stock markets. There are no indications of long-range dependence in the case of the original index returns.

Beta autoregressive fractionally integrated moving average models

In this work we introduce the class of beta autoregressive fractionally integrated moving average models for continuous random variables taking values in the continuous unit interval (0,1). The proposed model accommodates a set of regressors and a long-range dependent time series structure. We derive the partial likelihood estimator for the parameters of the proposed model, obtain the associated score vector and Fisher information matrix. We also prove the consistency and asymptotic normality of the estimator under mild conditions. Hypotheses testing, diagnostic tools and forecasting are also proposed. A Monte Carlo simulation is considered to evaluate the finite sample performance of the partial likelihood estimators and to study some of the proposed tests. An empirical application is also presented and discussed.

Change-in-mean tests in long-memory time series: a review of recent developments

It is well known that standard tests for a mean shift are invalid in long-range dependent time series. Therefore, several long-memory robust extensions of standard testing principles for a change-in-mean have been proposed in the literature. These can be divided into two groups: those that utilize consistent estimates of the long-run variance and self-normalized test statistics. Here, we review this literature and complement it by deriving a new long-memory robust version of the sup-Wald test. Apart from giving a systematic review, we conduct an extensive Monte Carlo study to compare the relative performance of these methods. Special attention is paid to the interaction of the test results with the estimation of the long-memory parameter. Furthermore, we show that the power of self-normalized test statistics can be improved considerably by using an estimator that is robust to mean shifts.

A simple test on structural change in long-memory time series

We propose a simple test on structural change in long-range dependent time series. It is based on the idea that the test statistic of the standard CUSUM test retains its asymptotic distribution if it is applied to fractionally differenced data. We prove that our approach is asymptotically valid, if the memory is estimated consistently under the null hypothesis. Therefore, the well-known CUSUM test can be used on the differenced data without any further modification. In a simulation study, we compare our test with a CUSUM test on structural change that is specifically constructed for long-memory time series and show that our approach performs reasonably well.

Bootstrap methods for long-memory processes: a review

This manuscript summarized advances in bootstrap methods for long-range dependent time series data. The stationary linear long-memory process is briefly described, which is a target process for bootstrap methodologies on time-domain and frequency-domain in this review. We illustrate time-domain bootstrap under long-range dependence, moving or non-overlapping block bootstraps, and the autoregressive-sieve bootstrap. In particular, block bootstrap methodologies need an adjustment factor for the distribution estimation of the sample mean in contrast to applications to weak dependent time processes. However, the autoregressive-sieve bootstrap does not need any other modification for application to long-memory. The frequency domain bootstrap for Whittle estimation is provided using parametric spectral density estimates because there is no current nonparametric spectral density estimation method using a kernel function for the linear long-range dependent time process.

Regularized Estimation of Linear Functionals of Precision Matrices for High-Dimensional Time Series

This paper studies a Dantzig-selector type regularized estimator for linear functionals of high-dimensional linear processes. Explicit rates of convergence of the proposed estimator are obtained and they cover the broad regime from i.i.d. samples to long-range dependent time series and from sub-Gaussian innovations to those with mild polynomial moments. It is shown that the convergence rates depend on the degree of temporal dependence and the moment conditions of the underlying linear processes. The Dantzig-selector estimator is applied to the sparse Markowitz portfolio allocation and the optimal linear prediction for time series, in which the ratio consistency when compared with an oracle estimator is established. The effect of dependence and innovation moment conditions is further illustrated in the simulation study. Finally, the regularized estimator is applied to classify the cognitive states on a real fMRI dataset and to portfolio optimization on a financial dataset.

How Much Information Does Dependence Between Wavelet Coefficients Contain?

ABSTRACTThis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short- as well as long-range dependent time series is assumed. We focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic. To this end, we derive asymptotic distributional properties of the sample variance for a time series that is synthesized, ignoring some or all dependence between wavelet coefficients. We show that the second-order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent Gaussian case. This holds true even if the dependency is correct within each level and only the dependence between levels is ignored. In the case of sample autocovariances and sample autocorrelations at lag one, we indicate that first-order properties are erroneous. In a second step, we investigate several nonparametric bootstrap schemes in the wavelet domain, which take more and more dependence into account until finally the full dependency is mimicked. We obtain very similar results, where only a bootstrap mimicking the full covariance structure correctly can be valid asymptotically. A simulation study supports our theoretical findings for the wavelet domain bootstraps. For long-range-dependent time series with long-memory parameter d > 1/4, we show that some additional problems occur, which cannot be solved easily without using additional information for the bootstrap. Supplementary materials for this article are available online.

A generalization of a Gaussian semiparametric estimator on multivariate long-range dependent processes

In this paper, we propose and study a general class of Gaussian semiparametric estimators (GSE) of the fractional differencing parameter in the context of long-range dependent multivariate time series. We establish large sample properties of the estimator without assuming Gaussianity. The class of models considered here satisfies simple conditions on the spectral density function, restricted to a small neighbourhood of the zero frequency and includes important class of vector autoregressive fractionally integrated moving average processes. We also present a simulation study to assess the finite sample properties of the proposed estimator based on a smoothed version of the GSE which supports its competitiveness.

Asymptotic independence of multiple Wiener–Itô integrals and the resulting limit laws

We characterize the asymptotic independence between blocks consisting of multiple Wiener–Itô integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension and other related results on the multivariate convergence of multiple Wiener–Itô integrals, that involve Gaussian and non Gaussian limits. We give applications to the study of the asymptotic behavior of functions of short and long-range dependent stationary Gaussian time series and establish the asymptotic independence for discrete non-Gaussian chaoses.

Recursive Nonparametric Estimation for Time Series

This paper considers online kernel estimation for both short- and long-range dependent time series data. Utilizing the predictive dependence measure of Wu, we carefully study the asymptotic properties of recursive kernel density and regression estimators for a general class of stationary processes. In particular, we prove that the proposed estimators have the asymptotic normality and the corresponding central limit theorems are provided. In addition, we establish the sharp laws of the iterated logarithms that precisely characterize the asymptotic almost sure behavior of the proposed estimators.

Long-range dependent time series specification

In this paper we propose using a nonparametric model specification test for parametric time series with long-range dependence (LRD). To establish asymptotic distributions of the proposed test statistic, we develop new central limit theorems for certain weighted quadratic forms of stationary time series with LRD. To implement our proposed test in practice, we develop a computer-intensive parametric bootstrap simulation procedure for finding simulated critical values. As a result, our finite-sample studies demonstrate that both the proposed theory and the simulation procedure work well, and that the proposed test has little size distortion and reasonable power.

Discriminating between long-range dependence and non-stationarity

This paper is devoted to the discrimination between a stationary long-range dependent model and a non stationary process. We develop a nonparametric test for stationarity in the framework of locally stationary long memory processes which is based on a Kolmogorov-Smirnov type distance between the time varying spectral density and its best approximation through a stationary spectral density. We show that the test statistic converges to the same limit as in the short memory case if the (possibly time varying) long memory parameter is smaller than $1/4$ and justify why the limiting distribution is different if the long memory parameter exceeds this boundary. Concerning the latter case the novel FARI($\infty$) bootstrap is introduced which provides a bootstrap-based test for stationarity which shows good empirical properties if the long memory parameter is smaller than $1/2$ which is the usual restriction in the framework of long-range dependent time series. We investigate the finite sample properties of our approach in a comprehensive simulation study and employ the new test in an analysis of two data sets.

Sequential Anomaly Detection in Wireless Sensor Networks and Effects of Long-Range Dependent Data

Anomaly detection is important for the correct functioning of wireless sensor networks. Recent studies have shown that node mobility along with spatial correlation of the monitored phenomenon in sensor networks can lead to observation data that have long range dependency, which could significantly increase the difficulty of anomaly detection. In this article, we develop an anomaly detection scheme based on multiscale analysis of the long-range dependent traffic to address this challenge. In this proposed detection scheme, the discrete wavelet transform is used to approximately de-correlate the traffic data and capture data characteristics in different timescales. The remaining dependencies are then captured by a multilevel hidden Markov model in the wavelet domain. To estimate the model parameters, we develop an online discounting expectation maximization (EM) algorithm, which also tracks variations of the estimated models over time. Network anomalies are detected as abrupt changes in the tracked model variation scores. Statistical properties of our detection scheme are evaluated numerically using long-range dependent time series. We also evaluate our detection scheme in malicious scenarios simulated using the NS-2 network simulator.

Tail index estimation in the presence of long-memory dynamics

Most tail index estimators are formulated under assumptions of weak serial dependence, but nevertheless are applied in practice to long-range dependent time series data. This issue arises because for many time series found in teletraffic and financial econometric applications, both heavy tails and long memory are prevalent features. For a certain class of Heavy-Tail Long-Memory (HTLM) processes, McElroy and Politis (2007a) and Jach et al. (2011) found that the probabilistic behavior of the sample mean depends delicately on the interplay of the tail index and the long memory parameter. In contrast, results in Kulik and Soulier (2011) indicate that the sample quantiles for a related HTLM process are unaffected by long-range dependence. Motivated by these results, we undertake an extensive numerical study to compare the finite-sample performance of several tail index estimators–both those based on sample quantiles, such as the Hill and DEdH ( Hill (1975) and Dekkers et al. (1989)) as well as those based on moments, e.g. Meerschaert and Scheffler (1998)–in the HTLM context. Our results largely confirm and expand those of Kulik and Soulier (2011), in that the Hill and DEdH estimators perform well despite the presence of long memory.

Invariance principles for linear processes with application to isotonic regression

In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range dependence and the limiting distribution is a fractional Brownian motion. The proofs are based on new approximations by a linear process with martingale difference innovations. The results are then applied to study an estimator of the isotonic regression when the error process is a (possibly long-range dependent) time series.