Gregory Rice

Test of independence for functional data

We wish to test the null hypothesis that a collection of functional observations are independent and identically distributed. Our procedure is based on the sum of the L^2 norms of the empirical correlation functions. The limit distribution of the proposed test statistic is established under the null hypothesis. Under the alternative the sample exhibits serial correlation, and consistency is shown when the sample size as well as the number of lags used in the test statistic tend to ~. A Monte Carlo study illustrates the small sample behavior of the test and the procedure is applied to data sets, Eurodollar futures and magnetogram records.

Testing Equality of Means When the Observations are from Functional Time Series

There are numerous examples of functional data in areas ranging from earth science to finance where the problem of interest is to compare several functional populations. In many instances, the observations are obtained consecutively in time, and thus, the classical assumption of independence within each population may not be valid. In this article, we derive a new, asymptotically justified method to test the hypothesis that the mean curves of multiple functional populations are the same. The test statistic is constructed from the coefficient vectors obtained by projecting the functional observations into a finite dimensional space. Asymptotics are established when the observations are considered to be from stationary functional time series. Although the limit results hold for projections into arbitrary finite dimensional spaces, we show that higher power is achieved by projecting onto the principle components of empirical covariance operators that diverge under the alternative. Our method is further illustrated by a simulation study as well as an application to electricity demand data.

On the asymptotic normality of kernel estimators of the long run covariance of functional time series

We consider the asymptotic normality in

A Plug-in Bandwidth Selection Procedure for Long-Run Covariance Estimation with Stationary Functional Time Series

L^2

Inference for the autocovariance of a functional time series under conditional heteroscedasticity

of kernel estimators of the long run covariance kernel of stationary functional time series. Our results are established assuming a weakly dependent Bernoulli shift structure for the underlying observations, which contains most stationary functional time series models, under mild conditions. As a corollary, we obtain joint asymptotics for functional principal components computed from empirical long run covariance operators, showing that they have the favorable property of being asymptotically independent.

ASYMPTOTIC PROPERTIES OF THE CUSUM ESTIMATOR FOR THE TIME OF CHANGE IN LINEAR PANEL DATA MODELS

In arenas of application including environmental science, economics, and medicine, it is increasingly common to consider time series of curves or functions. Many inferential procedures employed in the analysis of such data involve the long run covariance function or operator, which is analogous to the long run covariance matrix familiar to finite dimensional time series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies crucially on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite dimensional setting, we propose a bandwidth selection method that aims to minimize the estimators asymptotic mean squared normed error (AMSNE) in

Asymptotics for empirical eigenvalue processes in high-dimensional linear factor models

L^2[0,1]^2

Testing stationarity of functional time series

. As the AMSNE depends on unknown population quantities including the long run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.

Extensions of some classical methods in change point analysis

Abstract Most methods for analyzing functional time series rely on the estimation of lagged autocovariance operators or surfaces. As in univariate time series analysis, testing whether or not such operators are zero is an important diagnostic step that is well understood when the data, or model residuals, form a strong white noise. When functional data are constructed from dense records of, for example, asset prices or returns, a weak white noise model allowing for conditional heteroscedasticity is often more realistic. Applying inferential procedures for the autocovariance based on a strong white noise to such data often leads to the erroneous conclusion that the data exhibit significant autocorrelation. We develop methods for performing inference for the lagged autocovariance operators of stationary functional time series that are valid under general conditional heteroscedasticity conditions. These include a portmanteau test to assess the cumulative significance of empirical autocovariance operators up to a user selected maximum lag, as well as methods for obtaining confidence bands for a functional version of the autocorrelation that are useful in model selection/validation. We analyze the efficacy of these methods through a simulation study, and apply them to functional time series derived from asset price data of several representative assets. In this application, we found that strong white noise tests often suggest that such series exhibit significant autocorrelation, whereas our tests, which account for functional conditional heteroscedasticity, show that these data are in fact uncorrelated in a function space.

Weak invariance principles for sums of dependent random functions

We consider the problem of estimating the common time of a change in the mean parameters of panel data when dependence is allowed between the cross-sectional units in the form of a common factor. A CUSUM type estimator is proposed, and we establish first and second order asymptotics that can be used to derive consistent confidence intervals for the time of change. Our results improve upon existing theory in two primary directions. Firstly, the conditions we impose on the model errors only pertain to the order of their long run moments, and hence our results hold for nearly all stationary time series models of interest, including nonlinear time series like the ARCH and GARCH processes. Secondly, we study how the asymptotic distribution and norming sequences of the estimator depend on the magnitude of the changes in each cross-section and the common factor loadings. The performance of our results in finite samples is demonstrated with a Monte Carlo simulation study, and we consider applications to two real data sets: the exchange rates of 23 currencies with respect to the US dollar, and the GDP per capita in 113 countries.