491 research outputs found
Ratio tests for change point detection
We propose new tests to detect a change in the mean of a time series. Like
many existing tests, the new ones are based on the CUSUM process. Existing
CUSUM tests require an estimator of a scale parameter to make them
asymptotically distribution free under the no change null hypothesis. Even if
the observations are independent, the estimation of the scale parameter is not
simple since the estimator for the scale parameter should be at least
consistent under the null as well as under the alternative. The situation is
much more complicated in case of dependent data, where the empirical spectral
density at 0 is used to scale the CUSUM process. To circumvent these
difficulties, new tests are proposed which are ratios of CUSUM functionals. We
demonstrate the applicability of our method to detect a change in the mean when
the errors are AR(1) and GARCH(1,1) sequences.Comment: Published in at http://dx.doi.org/10.1214/193940307000000220 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
A test of significance in functional quadratic regression
We consider a quadratic functional regression model in which a scalar
response depends on a functional predictor; the common functional linear model
is a special case. We wish to test the significance of the nonlinear term in
the model. We develop a testing method which is based on projecting the
observations onto a suitably chosen finite dimensional space using functional
principal component analysis. The asymptotic behavior of our testing procedure
is established. A simulation study shows that the testing procedure has good
size and power with finite sample sizes. We then apply our test to a data set
provided by Tecator, which consists of near-infrared absorbance spectra and fat
content of meat.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ446 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Asymptotics of trimmed CUSUM statistics
There is a wide literature on change point tests, but the case of variables
with infinite variances is essentially unexplored. In this paper we address
this problem by studying the asymptotic behavior of trimmed CUSUM statistics.
We show that in a location model with i.i.d. errors in the domain of attraction
of a stable law of parameter , the appropriately trimmed CUSUM
process converges weakly to a Brownian bridge. Thus, after moderate trimming,
the classical method for detecting change points remains valid also for
populations with infinite variance. We note that according to the classical
theory, the partial sums of trimmed variables are generally not asymptotically
normal and using random centering in the test statistics is crucial in the
infinite variance case. We also show that the partial sums of truncated and
trimmed random variables have different asymptotic behavior. Finally, we
discuss resampling procedures which enable one to determine critical values in
the case of small and moderate sample sizes.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ318 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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
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