9,217 research outputs found
On generalized Cauchy-Stieltjes transforms of some Beta distributions
We express generalized Cauchy-Stieltjes transforms of some particular Beta
distributions (of ultraspherical type generating functions for orthogonal
polynomials) as a powered Cauchy-Stieltjes transform of some measure. For
suitable values of the power parameter, the latter measure turns out to be a
probability measure and its density is written down using Markov transforms.
The discarded values give a negative answer to a deformed free probability
unless a restriction on the power parameter is made. A particular symmetric
distribution interpolating between Wigner and arcsine distributions is
obtained. Its moments are expressed through a terminating hypergeometric series
interpolating between Catalan and shifed Catalan numbers. for small values of
the power parameter, the free cumulants are computed. Interesting opne problems
related to a deformed representation theory of the infinite symmetric group and
to a deformed Bozejko's convolution are discussed
Laguerre and Meixner orthogonal bases in the algebra of symmetric functions
Analogs of Laguerre and Meixner orthogonal polynomials in the algebra of
symmetric functions are studied. This is a detailed exposition of part of the
results announced in arXiv:1009.2037. The work is motivated by a connection
with a model of infinite-dimensional Markov dynamics.Comment: Latex, 52p
Non-equispaced B-spline wavelets
This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.Comment: 42 pages, 2 figure
Wavelet Analysis and Denoising: New Tools for Economists
This paper surveys the techniques of wavelets analysis and the associated methods of denoising. The Discrete Wavelet Transform and its undecimated version, the Maximum Overlapping Discrete Wavelet Transform, are described. The methods of wavelets analysis can be used to show how the frequency content of the data varies with time. This allows us to pinpoint in time such events as major structural breaks. The sparse nature of the wavelets representation also facilitates the process of noise reduction by nonlinear wavelet shrinkage , which can be used to reveal the underlying trends in economic data. An application of these techniques to the UK real GDP (1873-2001) is described. The purpose of the analysis is to reveal the true structure of the data - including its local irregularities and abrupt changes - and the results are surprising.Wavelets, Denoising, Structural breaks, Trend estimation
Gibbs Sampling, Exponential Families and Orthogonal Polynomials
We give families of examples where sharp rates of convergence to stationarity
of the widely used Gibbs sampler are available. The examples involve standard
exponential families and their conjugate priors. In each case, the transition
operator is explicitly diagonalizable with classical orthogonal polynomials as
eigenfunctions.Comment: This paper commented in: [arXiv:0808.3855], [arXiv:0808.3856],
[arXiv:0808.3859], [arXiv:0808.3861]. Rejoinder in [arXiv:0808.3864].
Published in at http://dx.doi.org/10.1214/07-STS252 the Statistical Science
(http://www.imstat.org/sts/) by the Institute of Mathematical Statistics
(http://www.imstat.org
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