Testing for the Presence of Self-Similarity of Gaussian Time Series Having Stationary Increments

A method for testing for the presence of self-similarity of a Gaussian time series with stationary increments is presented. The test is based on estimation of the distance between the time series and a set of time series containing all the fractional Brownian motions. This distance is constructed from two estimations of multiscale generalized quadratic variations expectations. The second one requires regression estimates of the self-similarity index H . Two estimations of H are then introduced. They present good robustness and computing time properties compared with the Whittle approach, with nearly similar convergence rate. The test is applied on simulated and real data. The self-similarity assumption is notably accepted for the famous Nile River data.     

Testing Normality of Functional Time Series

We develop tests of normality for time series of functions. The tests are related to the commonly used Jarque–Bera test. The assumption of normality has played an important role in many methodological and theoretical developments in the field of functional data analysis. Yet, no inferential procedures to verify it have been proposed so far, even for i.i.d. functions. We propose several approaches which handle two paramount challenges: (i) the unknown temporal dependence structure and (ii) the estimation of the optimal finite‐dimensional projection space. We evaluate the tests via simulations and establish their large sample validity under general conditions. We obtain useful insights by applying them to pollution and intraday price curves. While the pollution curves can be treated as normal, the normality of high‐frequency price curves is rejected.     

Testing for the Presence of a Random Walk in Series with Structural Breaks

We consider tests for the presence of a random walk component in a stationary or trend stationary time series and extend them to series that contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution is obtained. Then a modified test statistic is proposed. The advantage of this statistic is that its asymptotic distribution is not dependent on the location of the break point and its form is that of the generalized Cramer–von Mises distribution, with degrees of freedom depending on the number of break points. The performance of this modified test is shown, via some simulation experiments, to be comparable with that of the LBI test. An unconditional test, based on the assumption that there is a single break at an unknown point, is also examined. The use of the tests is illustrated with data on the flow of the Nile and US gross national product.     

The Use of Aggregate Time Series in Testing for Gaussianity

Many time series encountered in practice are non-Gaussian. Because of the process of data collection or the practice or researchers, time series used in analysis and modelling are frequently temporal aggregates. In this paper, we study the effects of the use of aggregate time series on testing for Gaussianity. We analyse how the test statistic is affected by aggregation and how that affects the power of the test. The results show that the use of aggregate time series induces Gaussianity and that the degree of inducement increases with the order of aggregation. In fact, the use of aggregate time series reduces the power of the test, although the effect is not significant for low orders of aggregation.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

A Direct Test for Cointegration Between a Pair of Time Series

In this paper we introduce a new test of the null hypothesis of no cointegration between a pair of time series. For a very simple generating model, our test compares favourably with the Engle–Granger/Dickey–Fuller test and the Johansen trace test. Indeed, shortcomings of the former motivated the development of our test. The applicability of our test is extended to series generated by low-order vector autoregressions. Again, we find evidence that this general version of our test is more powerful than the Johansen test. The paper concludes with an empirical example in which the new test finds strong evidence of cointegration, but the Johansen test does not.     

Data Driven Order Selection for Projection Estimator of the Spectral Density of Time Series with Long Range Dependence

Fractional exponential (FEXP) models have been introduced by Robinson (1991) and Beran (1993) to model the spectral density of a covariance stationary long-range dependent process. In this class of models, the spectral density f ( x ) of the process is decomposed as f ( x ) = |1 − exp( ix )|^{−2 d} f _*( x ), where f _*( x ) accounts for the short-memory component. In this contribution, FEXP models are used to construct semi-parametric estimates of the fractional differencing coefficient and of the spectral density, by considering an infinite Fourier series expansion of log f _*( x ). A data-driven order selection procedure, adapted from the Mallows' C _p procedure, is proposed to determine the order of truncation. The optimality of the data-driven procedure is established, under mild assumptions on the short-memory component f _*( x ). A limited Monte-Carlo experiment is presented to support our claims.     

An Improvement of the Portmanteau Statistic

Abstract. The portmanteau statistic is based on the first m‐residual autocorrelations, and is used for diagnostic checks on the adequacy of fit of a model. In this article, we propose a modified portmanteau statistic with a correction term that allows for the use of small values of m for the chi‐squared test. For this modification, we take a different approach to that suggested by Ljung [Biometrika (1986), Vol. 73, pp. 725–30]. Their empirical behaviour is clarified using asymptotic theory.     

Discriminant Analysis of Time Series in the Presence of Within‐Group Spectral Variability

Many studies record replicated time series epochs from different groups with the goal of using frequency domain properties to discriminate between the groups. In many applications, there exists variation in cyclical patterns from time series in the same group. Although a number of frequency domain methods for the discriminant analysis of time series have been explored, there is a dearth of models and methods that account for within‐group spectral variability. This article proposes a model for groups of time series in which transfer functions are modelled as stochastic variables that can account for both between‐group and within‐group differences in spectra that are identified from individual replicates. An ensuing discriminant analysis of stochastic cepstra under this model is developed to obtain parsimonious measures of relative power that optimally separate groups in the presence of within‐group spectral variability. The approach possesses favourable properties in classifying new observations and can be consistently estimated through a simple discriminant analysis of a finite number of estimated cepstral coefficients. Benefits in accounting for within‐group spectral variability are empirically illustrated in a simulation study and through an analysis of gait variability.     

Consistency of the Hill Estimator for Time Series Observed with Measurement Errors

We investigate the asymptotic and finite sample behavior of the Hill estimator applied to time series contaminated by measurement or other errors. We show that for all discrete time models used in practice, whose non-contaminated marginal distributions are regularly varying, the Hill estimator is consistent. Essentially, the only assumption on the errors is that they have lighter tails than the underlying unobservable process. The asymptotic justification however depends on the specific class of models assumed for the underlying unobservable process. We show by means of a simulation study that the asymptotic robustness of the Hill estimator is clearly manifested in finite samples. We further illustrate this robustness by a numerical study of the interarrival times of anomalies in a backbone internet network, the Internet2 in the United States; the anomalies arrival times are computed with a roundoff error.     

Adaptive Fourier Series and the Analysis of Periodicities in Time Series Data

A Fourier series decomposes a function x ( t ) into a sum of periodic components that have sinusoidal shapes. This paper describes an adaptive Fourier series where the periodic components of x ( t ) may have a variety of differing shapes. The periodic shapes are adaptive since they depend on the function x ( t ) and the period. The results, which extend both Fourier analysis and Walsh–Fourier analysis, are applied to investigate the shapes of periodic components in time series data sets.     

A New Test of Linearity of Time Series Based on the Bispectrum

The linearity of a time series is tested by use of the bispectrum. We define the time series to be linear if the best predictor is linear. The bispectrum is estimated by stretching the data and smoothing by the Subba Rao–Gabr optimal window. The null hypothesis tested here is that the best predictor is linear against the alternative that the best predictor is quadratic. It turns out that the test statistic is asymptotically χ²-distributed under the hypothesis that the time series is linear. The results are demonstrated using simulated and real data.     

An Asymptotic Second-Order Lower Bound for the Bayes Risk of a Sequential Procedure

Abstract

Consider the problem of estimating the difference of the means of two populations, where each population distribution is a member of the one-parameter exponential family of probability distributions. A Bayesian approach is adopted in which the mean difference is estimated under the squared error loss and the prior distributions are of the form proposed by Diaconis and Ylivisaker [Diaconis, P.; Ylivisaker, D. Conjugate priors for exponential families. Ann. of Statist. 1979, 6, 269–281]. The main result determines an asymptotic second-order lower bound for the Bayes risk of a sequential procedure that takes N observations from one population and t ? N from the other population, and estimates the mean difference by the Bayes estimator, where N is determined according to a sequential design and t denotes the total number of observations sampled from both populations.     

The Effect of Linear Time Trends on the KPSS Test for Cointegration

The so-called KPSS test for the null hypothesis of cointegration builds on residuals from single equation regressions. Critical values have been provided for regressions with and without detrending. Here it is shown that the latter are not appropriate if the series display linear trends, although this does not mean that detrending is required. In this paper adequate percentiles are suggested for series that follow linear time trends, and tests are based on regressions without detrending. These percentiles are readily available from the literature.     

Tests for Harmonic Components in the Spectra of Categorical Time Series

A test for categorical time series is developed which is based on Fisher's test for continuous-parameter time series. Instead of using a test based on the Fourier periodog ram for spectral analysis, we utilize the Walsh–Fourier periodogram for testing purposes. We briefly explain the theory behind Walsh–Fourier analysis and some of its recent applications. Asymptotic results for the distribution of the new test statistic for Walsh–Fourier spectra are presented and compared with a simulated distribution. We also perform power studies in order to assess the detection capability of the test. In the presence of multiple peaks in the spectrum, this test tends to lose power. Therefore, we also explore several alternatives to the test for Walsh–Fourier spectra and apply all of the alternative methods to a realization of geomagnetic reversals     

Optimal Rate of Convergence for Empirical Quantiles and Distribution Functions for Time Series

Given a stationary sequence , we are interested in the rate of convergence in the central limit theorem of the empirical quantiles and the empirical distribution function. Under a general notion of weak dependence, we show a Berry–Esseen result with optimal rate n^?1/2. The setup includes many prominent time series models, such as functions of ARMA or (augmented) GARCH processes. In this context, optimal Berry–Esseen rates for empirical quantiles appear to be novel.     

Zero‐Modified Geometric INAR(1) Process for Modelling Count Time Series with Deflation or Inflation of Zeros

In this article, we propose a first‐order integer‐valued autoregressive [INAR(1)] process for dealing with count time series with deflation or inflation of zeros. The proposed process has zero‐modified geometric marginals and contains the geometric INAR(1) process as a particular case. The proposed model is also capable of capturing underdispersion and overdispersion, which sometimes are caused by deflation or inflation of zeros. We explore several statistical and mathematical properties of the process, discuss point estimation of the parameters and find the asymptotic distribution of the proposed estimators. We also propose a test based on our model for checking if the count time series considered is deflated or inflated of zeros. Two empirical illustrations are presented in order to show the potential for practice of our zero‐modified geometric INAR(1) process. This article contains a Supporting Information.     

A Frequency‐Domain Test to Check Equality in Spectral Densities of Multiple Time Series With Unequal Lengths

In this paper, a new frequency‐domain test is proposed to check the equality of spectral densities of two or more stationary time series. The proposed test is able to deal with multiple independent time series of different lengths naturally, based on some regression models of log periodograms. The asymptotic null distribution of the proposed test is obtained. The consistency is shown under any fixed alternative and a sequence of local alternatives. A simulation study is conducted to examine the finite sample performance of the test. By jointly modeling all log periodograms, the test is empirically robust when multiple time series are mutually dependent to some extent. It also works well for non‐Gaussian time series. The proposed test is applied to compare several vibrational signals for damage detection of a mechanical system.     

Partition Coefficients of Alkaloids in Biphasic Ionic-Liquid-Aqueous Systems and their Dependence on the Hofmeister Series

The aim of this work is to test the potential of hydrophobic phosphonium-based ionic liquids for the extraction of caffeine and nicotine from aqueous phases through the determination of the alkaloids' partition coefficients. It was found that the caffeine partitioning for the ionic-liquid-rich phase increases with the ionic liquid hydrogen-bonding accepting capability (or water content), while for nicotine a nearly opposite behavior was observed. In addition, both the influence of the ionic concentration of the aqueous solution (ranging from 0.0 mol · kg^?1 to 3.0 mol · kg^?1), and the salt nature (with K- and Na-based salts), in the partitioning of caffeine for the ionic-liquid-rich phase were investigated. The influence of the inorganic salt nature in the alkaloid partitioning for the ionic-liquid-rich phase closely follows the Hofmeister series.     

Prediction Interval for Autoregressive Time Series via Oracally Efficient Estimation of Multi‐Step‐Ahead Innovation Distribution Function

A kernel distribution estimator (KDE) is proposed for multi‐step‐ahead prediction error distribution of autoregressive time series, based on prediction residuals. Under general assumptions, the KDE is proved to be oracally efficient as the infeasible KDE and the empirical cumulative distribution function (cdf) based on unobserved prediction errors. Quantile estimator is obtained from the oracally efficient KDE, and prediction interval for multi‐step‐ahead future observation is constructed using the estimated quantiles and shown to achieve asymptotically the nominal confidence levels. Simulation examples corroborate the asymptotic theory.