115 research outputs found
Augmented GARCH sequences: Dependence structure and asymptotics
The augmented GARCH model is a unification of numerous extensions of the
popular and widely used ARCH process. It was introduced by Duan and besides
ordinary (linear) GARCH processes, it contains exponential GARCH, power GARCH,
threshold GARCH, asymmetric GARCH, etc. In this paper, we study the
probabilistic structure of augmented sequences and the
asymptotic distribution of various functionals of the process occurring in
problems of statistical inference. Instead of using the Markov structure of the
model and implied mixing properties, we utilize independence properties of
perturbed GARCH sequences to directly reduce their asymptotic behavior to the
case of independent random variables. This method applies for a very large
class of functionals and eliminates the fairly restrictive moment and
smoothness conditions assumed in the earlier theory. In particular, we derive
functional CLTs for powers of the augmented GARCH variables, derive the error
rate in the CLT and obtain asymptotic results for their empirical processes
under nearly optimal conditions.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ120 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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
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
Split invariance principles for stationary processes
The results of Koml\'{o}s, Major and Tusn\'{a}dy give optimal Wiener
approximation of partial sums of i.i.d. random variables and provide an
extremely powerful tool in probability and statistical inference. Recently Wu
[Ann. Probab. 35 (2007) 2294--2320] obtained Wiener approximation of a class of
dependent stationary processes with finite th moments, , with error
term , , and Liu and Lin [Stochastic
Process. Appl. 119 (2009) 249--280] removed the logarithmic factor, reaching
the Koml\'{o}s--Major--Tusn\'{a}dy bound . No similar results exist
for , and in fact, no existing method for dependent approximation yields
an a.s. rate better than . In this paper we show that allowing a
second Wiener component in the approximation, we can get rates near to
for arbitrary . This extends the scope of applications of the
results essentially, as we illustrate it by proving new limit theorems for
increments of stochastic processes and statistical tests for short term
(epidemic) changes in stationary processes. Our method works under a general
weak dependence condition covering wide classes of linear and nonlinear time
series models and classical dynamical systems.Comment: Published in at http://dx.doi.org/10.1214/10-AOP603 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the prediction of stationary functional time series
This paper addresses the prediction of stationary 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 prediction techniques
can be utilized in this context. The connection between functional and
multivariate predictions is made precise for the important case of vector and
functional autoregressions. The proposed method is easy to implement, making
use of existing statistical software packages, and may therefore be attractive
to a broader, possibly non-academic, audience. Its practical applicability is
enhanced through the introduction of a novel functional final prediction error
model selection criterion that allows for an automatic determination of the lag
structure and the dimensionality of the model. The usefulness of the proposed
methodology is demonstrated in a simulation study and an application to
environmental data, namely the prediction of daily pollution curves describing
the concentration of particulate matter in ambient air. It is found that the
proposed prediction method often significantly outperforms existing methods
Break detection in the covariance structure of multivariate time series models
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 dimensionality of the
underlying data. Since it is nonparametric, the procedure avoids the
difficulties associated with parametric model selection, model fitting and
parameter estimation. We provide the theoretical foundation for the test and
demonstrate its applicability via a simulation study and an analysis of
financial data. Extensions to multiple changes and the case of infinite fourth
moments are also discussed.Comment: Published in at http://dx.doi.org/10.1214/09-AOS707 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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