5,239 research outputs found
On Schauder Bases Properties of Multiply Generated Gabor Systems
Let be a finite subset of and . We
characterize the Schauder basis properties in of the Gabor
system
with a specific ordering on . The
characterization is given in terms of a Muckenhoupt matrix condition on
an associated Zibulski-Zeevi type matrix.Comment: 14 page
A Powerful Tuning Parameter Free Test of the Autoregressive Unit Root Hypothesis
This paper presents a family of simple nonparametric unit root tests indexed by one parameter, d, and containing Breitung's (2002) test as the special case d = 1. It is shown that (i) each member of the family with d > 0 is consistent, (ii) the asymptotic distribution depends on d, and thus reflects the parameter chosen to implement the test, and (iii) since the asymptotic distribution depends on d and the test remains consistent for all d > 0, it is possible to analyze the power of the test for different values of d. The usual Phillips-Perron or Dickey-Fuller type tests are characterized by tuning parameters (bandwidth, lag length, etc.), i.e. parameters which change the test statistic but are not reflected in the asymptotic distribution, and thus have none of these three properties. It is shown that members of the family with daugmented Dickey-Fuller test, fractional integration, GLS detrending, nonparametric, nuisance parameter, tuning parameter, power envelope, unit root test, variance ratio
A Powerful Test of the Autoregressive Unit Root Hypothesis Based on a Tuning Parameter Free Statistic
This paper presents a family of simple nonparametric unit root tests indexed by one parameter, d, and containing Breitung's (2002) test as the special case d=1. It is shown that (i) each member of the family with d>0 is consistent, (ii) the asymptotic distribution depends on d, and thus reflects the parameter chosen to implement the test, and (iii) since the asymptotic distribution depends on d and the test remains consistent for all d>0, it is possible to analyze the power of the test for different values of d. The usual Phillips-Perron or Dickey-Fuller type tests are indexed by bandwidth, lag length, etc., but have none of these three properties. It is shown that members of the family with dAugmented Dickey-Fuller test, fractional integration, GLS detrending, nonparametric, nuisance parameter, tuning parameter, power envelope, unit root test, variance ratio
Nonlinear approximation with nonstationary Gabor frames
We consider sparseness properties of adaptive time-frequency representations
obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical
Gabor frames by allowing for adaptivity in either time or frequency. It is
known that the concept of painless nonorthogonal expansions generalizes to the
nonstationary case, providing perfect reconstruction and an FFT based
implementation for compactly supported window functions sampled at a certain
density. It is also known that for some signal classes, NSGFs with flexible
time resolution tend to provide sparser expansions than can be obtained with
classical Gabor frames. In this article we show, for the continuous case, that
sparseness of a nonstationary Gabor expansion is equivalent to smoothness in an
associated decomposition space. In this way we characterize signals with sparse
expansions relative to NSGFs with flexible time resolution. Based on this
characterization we prove an upper bound on the approximation error occurring
when thresholding the coefficients of the corresponding frame expansions. We
complement the theoretical results with numerical experiments, estimating the
rate of approximation obtained from thresholding the coefficients of both
stationary and nonstationary Gabor expansions.Comment: 19 pages, 2 figure
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