944 research outputs found
Minimax hypothesis testing for curve registration
This paper is concerned with the problem of goodness-of-fit for curve
registration, and more precisely for the shifted curve model, whose application
field reaches from computer vision and road traffic prediction to medicine. We
give bounds for the asymptotic minimax separation rate, when the functions in
the alternative lie in Sobolev balls and the separation from the null
hypothesis is measured by the l2-norm. We use the generalized likelihood ratio
to build a nonadaptive procedure depending on a tuning parameter, which we
choose in an optimal way according to the smoothness of the ambient space.
Then, a Bonferroni procedure is applied to give an adaptive test over a range
of Sobolev balls. Both achieve the asymptotic minimax separation rates, up to
possible logarithmic factors
Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression
We consider the problem of testing a particular type of composite null
hypothesis under a nonparametric multivariate regression model. For a given
quadratic functional , the null hypothesis states that the regression
function satisfies the constraint , while the alternative
corresponds to the functions for which is bounded away from zero. On the
one hand, we provide minimax rates of testing and the exact separation
constants, along with a sharp-optimal testing procedure, for diagonal and
nonnegative quadratic functionals. We consider smoothness classes of
ellipsoidal form and check that our conditions are fulfilled in the particular
case of ellipsoids corresponding to anisotropic Sobolev classes. In this case,
we present a closed form of the minimax rate and the separation constant. On
the other hand, minimax rates for quadratic functionals which are neither
positive nor negative makes appear two different regimes: "regular" and
"irregular". In the "regular" case, the minimax rate is equal to
while in the "irregular" case, the rate depends on the smoothness class and is
slower than in the "regular" case. We apply this to the issue of testing the
equality of norms of two functions observed in noisy environments
Landmark-Based Registration of Curves via the Continuous Wavelet Transform
This paper is concerned with the problem of the alignment of multiple sets of curves. We analyze two real examples arising from the biomedical area for which we need to test whether there are any statistically significant differences between two subsets of subjects. To synchronize a set of curves, we propose a new nonparametric landmark-based registration method based on the alignment of the structural intensity of the zero-crossings of a wavelet transform. The structural intensity is a multiscale technique recently proposed by Bigot (2003, 2005) which highlights the main features of a signal observed with noise. We conduct a simulation study to compare our landmark-based registration approach with some existing methods for curve alignment. For the two real examples, we compare the registered curves with FANOVA techniques, and a detailed analysis of the warping functions is provided
A scale-space approach with wavelets to singularity estimation
This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, "the structural intensity", that computes the "density" of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed
Statistical inferences for functional data
With modern technology development, functional data are being observed
frequently in many scientific fields. A popular method for analyzing such
functional data is ``smoothing first, then estimation.'' That is, statistical
inference such as estimation and hypothesis testing about functional data is
conducted based on the substitution of the underlying individual functions by
their reconstructions obtained by one smoothing technique or another. However,
little is known about this substitution effect on functional data analysis. In
this paper this problem is investigated when the local polynomial kernel (LPK)
smoothing technique is used for individual function reconstructions. We find
that under some mild conditions, the substitution effect can be ignored
asymptotically. Based on this, we construct LPK reconstruction-based estimators
for the mean, covariance and noise variance functions of a functional data set
and derive their asymptotics. We also propose a GCV rule for selecting good
bandwidths for the LPK reconstructions. When the mean function also depends on
some time-independent covariates, we consider a functional linear model where
the mean function is linearly related to the covariates but the covariate
effects are functions of time. The LPK reconstruction-based estimators for the
covariate effects and the covariance function are also constructed and their
asymptotics are derived. Moreover, we propose a -norm-based global test
statistic for a general hypothesis testing problem about the covariate effects
and derive its asymptotic random expression. The effect of the bandwidths
selected by the proposed GCV rule on the accuracy of the LPK reconstructions
and the mean function estimator is investigated via a simulation study. The
proposed methodologies are illustrated via an application to a real functional
data set collected in climatology.Comment: Published at http://dx.doi.org/10.1214/009053606000001505 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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