1,258 research outputs found
Weakly dependent functional data
Functional data often arise from measurements on fine time grids and are
obtained by separating an almost continuous time record into natural
consecutive intervals, for example, days. The functions thus obtained form a
functional time series, and the central issue in the analysis of such data
consists in taking into account the temporal dependence of these functional
observations. Examples include daily curves of financial transaction data and
daily patterns of geophysical and environmental data. For scalar and vector
valued stochastic processes, a large number of dependence notions have been
proposed, mostly involving mixing type distances between -algebras. In
time series analysis, measures of dependence based on moments have proven most
useful (autocovariances and cumulants). We introduce a moment-based notion of
dependence for functional time series which involves -dependence. We show
that it is applicable to linear as well as nonlinear functional time series.
Then we investigate the impact of dependence thus quantified on several
important statistical procedures for functional data. We study the estimation
of the functional principal components, the long-run covariance matrix, change
point detection and the functional linear model. We explain when temporal
dependence affects the results obtained for i.i.d. functional observations and
when these results are robust to weak dependence.Comment: Published in at http://dx.doi.org/10.1214/09-AOS768 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Consistency of the mean and the principal components of spatially distributed functional data
This paper develops a framework for the estimation of the functional mean and
the functional principal components when the functions form a random field.
More specifically, the data we study consist of curves
, observed at spatial points
. We establish conditions for
the sample average (in space) of the to be a consistent
estimator of the population mean function, and for the usual empirical
covariance operator to be a consistent estimator of the population covariance
operator. These conditions involve an interplay of the assumptions on an
appropriately defined dependence between the functions and
the assumptions on the spatial distribution of the points . The
rates of convergence may be the same as for i.i.d. functional samples, but
generally depend on the strength of dependence and appropriately quantified
distances between the points . We also formulate conditions for
the lack of consistency.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ418 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Subsampling the mean of heavy-tailed dependent observations
We establish the validity of subsampling confidence intervals for the mean of a dependent series with heavy-tailed marginal distributions. Using point process theory, we study both linear and nonlinear GARCH-like time series models. We propose a data-dependent method for the optimal block size selection and investigate its performance by means of a simulation study.Heavy tails, linear time series, subsampling
Detection of periodicity in functional time series
We derive several tests for the presence of a periodic component in a time
series of functions. We consider both the traditional setting in which the
periodic functional signal is contaminated by functional white noise, and a
more general setting of a contaminating process which is weakly dependent.
Several forms of the periodic component are considered. Our tests are motivated
by the likelihood principle and fall into two broad categories, which we term
multivariate and fully functional. Overall, for the functional series that
motivate this research, the fully functional tests exhibit a superior balance
of size and power. Asymptotic null distributions of all tests are derived and
their consistency is established. Their finite sample performance is examined
and compared by numerical studies and application to pollution data
Near-integrated GARCH sequences
Motivated by regularities observed in time series of returns on speculative
assets, we develop an asymptotic theory of GARCH(1,1) processes {y_k} defined
by the equations y_k=\sigma_k\epsilon_k, \sigma_k^2=\omega +\alpha
y_{k-1}^2+\beta \sigma_{k-1}^2 for which the sum \alpha +\beta approaches unity
as the number of available observations tends to infinity. We call such
sequences near-integrated. We show that the asymptotic behavior of
near-integrated GARCH(1,1) processes critically depends on the sign of \gamma
:=\alpha +\beta -1. We find assumptions under which the solutions exhibit
increasing oscillations and show that these oscillations grow approximately
like a power function if \gamma \leq 0 and exponentially if \gamma >0. We
establish an additive representation for the near-integrated GARCH(1,1)
processes which is more convenient to use than the traditional multiplicative
Volterra series expansion.Comment: Published at http://dx.doi.org/10.1214/105051604000000783 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A randomness test for functional panels
Functional panels are collections of functional time series, and arise often
in the study of high frequency multivariate data. We develop a portmanteau
style test to determine if the cross-sections of such a panel are independent
and identically distributed. Our framework allows the number of functional
projections and/or the number of time series to grow with the sample size. A
large sample justification is based on a new central limit theorem for random
vectors of increasing dimension. With a proper normalization, the limit is
standard normal, potentially making this result easily applicable in other FDA
context in which projections on a subspace of increasing dimension are used.
The test is shown to have correct size and excellent power using simulated
panels whose random structure mimics the realistic dependence encountered in
real panel data. It is expected to find application in climatology, finance,
ecology, economics, and geophysics. We apply it to Southern Pacific sea surface
temperature data, precipitation patterns in the South-West United States, and
temperature curves in Germany.Comment: Supplemental material from the authors' homepage or upon reques
Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends
We develop methodology for the estimation of the functional mean and the
functional principal components when the functions form a spatial process. The
data consist of curves observed at spatial
locations . We propose several
methods, and evaluate them by means of a simulation study. Next, we develop a
significance test for the correlation of two such functional spatial fields.
After validating the finite sample performance of this test by means of a
simulation study, we apply it to determine if there is correlation between
long-term trends in the so-called critical ionospheric frequency and decadal
changes in the direction of the internal magnetic field of the Earth. The test
provides conclusive evidence for correlation, thus solving a long-standing
space physics conjecture. This conclusion is not apparent if the spatial
dependence of the curves is neglected.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS524 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On discriminating between long-range dependence and changes in mean
We develop a testing procedure for distinguishing between a long-range
dependent time series and a weakly dependent time series with change-points in
the mean. In the simplest case, under the null hypothesis the time series is
weakly dependent with one change in mean at an unknown point, and under the
alternative it is long-range dependent. We compute the CUSUM statistic ,
which allows us to construct an estimator of a change-point. We then
compute the statistic based on the observations up to time
and the statistic based on the observations after time . The
statistic converges to a well-known distribution
under the null, but diverges to infinity if the observations exhibit long-range
dependence. The theory is illustrated by examples and an application to the
returns of the Dow Jones index.Comment: Published at http://dx.doi.org/10.1214/009053606000000254 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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