2,755 research outputs found
On image segmentation using information theoretic criteria
Image segmentation is a long-studied and important problem in image
processing. Different solutions have been proposed, many of which follow the
information theoretic paradigm. While these information theoretic segmentation
methods often produce excellent empirical results, their theoretical properties
are still largely unknown. The main goal of this paper is to conduct a rigorous
theoretical study into the statistical consistency properties of such methods.
To be more specific, this paper investigates if these methods can accurately
recover the true number of segments together with their true boundaries in the
image as the number of pixels tends to infinity. Our theoretical results show
that both the Bayesian information criterion (BIC) and the minimum description
length (MDL) principle can be applied to derive statistically consistent
segmentation methods, while the same is not true for the Akaike information
criterion (AIC). Numerical experiments were conducted to illustrate and support
our theoretical findings.Comment: Published in at http://dx.doi.org/10.1214/11-AOS925 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Functional generalized autoregressive conditional heteroskedasticity
Heteroskedasticity is a common feature of financial time series and is
commonly addressed in the model building process through the use of ARCH and
GARCH processes. More recently multivariate variants of these processes have
been in the focus of research with attention given to methods seeking an
efficient and economic estimation of a large number of model parameters. Due to
the need for estimation of many parameters, however, these models may not be
suitable for modeling now prevalent high-frequency volatility data. One
potentially useful way to bypass these issues is to take a functional approach.
In this paper, theory is developed for a new functional version of the
generalized autoregressive conditionally heteroskedastic process, termed
fGARCH. The main results are concerned with the structure of the fGARCH(1,1)
process, providing criteria for the existence of a strictly stationary
solutions both in the space of square-integrable and continuous functions. An
estimation procedure is introduced and its consistency verified. A small
empirical study highlights potential applications to intraday volatility
estimation
Selection from a stable box
Let be independent, identically distributed random variables. It is
well known that the functional CUSUM statistic and its randomly permuted
version both converge weakly to a Brownian bridge if second moments exist.
Surprisingly, an infinite-variance counterpart does not hold true. In the
present paper, we let be in the domain of attraction of a strictly
-stable law, . While the functional CUSUM statistics
itself converges to an -stable bridge and so does the permuted version,
provided both the and the permutation are random, the situation turns
out to be more delicate if a realization of the is fixed and
randomness is restricted to the permutation. Here, the conditional distribution
function of the permuted CUSUM statistics converges in probability to a random
and nondegenerate limit.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6014 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Bootstrapping spectral statistics in high dimensions
Statistics derived from the eigenvalues of sample covariance matrices are
called spectral statistics, and they play a central role in multivariate
testing. Although bootstrap methods are an established approach to
approximating the laws of spectral statistics in low-dimensional problems,
these methods are relatively unexplored in the high-dimensional setting. The
aim of this paper is to focus on linear spectral statistics as a class of
prototypes for developing a new bootstrap in high-dimensions --- and we refer
to this method as the Spectral Bootstrap. In essence, the method originates
from the parametric bootstrap, and is motivated by the notion that, in high
dimensions, it is difficult to obtain a non-parametric approximation to the
full data-generating distribution. From a practical standpoint, the method is
easy to use, and allows the user to circumvent the difficulties of complex
asymptotic formulas for linear spectral statistics. In addition to proving the
consistency of the proposed method, we provide encouraging empirical results in
a variety of settings. Lastly, and perhaps most interestingly, we show through
simulations that the method can be applied successfully to statistics outside
the class of linear spectral statistics, such as the largest sample eigenvalue
and others.Comment: 42 page
Spectral analysis of linear time series in moderately high dimensions
This article is concerned with the spectral behavior of -dimensional
linear processes in the moderately high-dimensional case when both
dimensionality and sample size tend to infinity so that . It
is shown that, under an appropriate set of assumptions, the empirical spectral
distributions of the renormalized and symmetrized sample autocovariance
matrices converge almost surely to a nonrandom limit distribution supported on
the real line. The key assumption is that the linear process is driven by a
sequence of -dimensional real or complex random vectors with i.i.d. entries
possessing zero mean, unit variance and finite fourth moments, and that the
linear process coefficient matrices are Hermitian and
simultaneously diagonalizable. Several relaxations of these assumptions are
discussed. The results put forth in this paper can help facilitate inference on
model parameters, model diagnostics and prediction of future values of the
linear process
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
Reaction times of monitoring schemes for ARMA time series
This paper is concerned with deriving the limit distributions of stopping
times devised to sequentially uncover structural breaks in the parameters of an
autoregressive moving average, ARMA, time series. The stopping rules are
defined as the first time lag for which detectors, based on CUSUMs and Page's
CUSUMs for residuals, exceed the value of a prescribed threshold function. It
is shown that the limit distributions crucially depend on a drift term induced
by the underlying ARMA parameters. The precise form of the asymptotic is
determined by an interplay between the location of the break point and the size
of the change implied by the drift. The theoretical results are accompanied by
a simulation study and applications to electroencephalography, EEG, and IBM
data. The empirical results indicate a satisfactory behavior in finite samples.Comment: Published at http://dx.doi.org/10.3150/14-BEJ604 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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