9,369 research outputs found
Symbolic computation of moments of sampling distributions
By means of the notion of umbrae indexed by multisets, a general method to
express estimators and their products in terms of power sums is derived. A
connection between the notion of multiset and integer partition leads
immediately to a way to speed up the procedures. Comparisons of computational
times with known procedures show how this approach turns out to be more
efficient in eliminating much unnecessary computation.Comment: 21 pages, 7 table
On a symbolic representation of non-central Wishart random matrices with applications
By using a symbolic method, known in the literature as the classical umbral
calculus, the trace of a non-central Wishart random matrix is represented as
the convolution of the trace of its central component and of a formal variable
involving traces of its non-centrality matrix. Thanks to this representation,
the moments of this random matrix are proved to be a Sheffer polynomial
sequence, allowing us to recover several properties. The multivariate symbolic
method generalizes the employment of Sheffer representation and a closed form
formula for computing joint moments and cumulants (also normalized) is given.
By using this closed form formula and a combinatorial device, known in the
literature as necklace, an efficient algorithm for their computations is set
up. Applications are given to the computation of permanents as well as to the
characterization of inherited estimators of cumulants, which turn useful in
dealing with minors of non-central Wishart random matrices. An asymptotic
approximation of generalized moments involving free probability is proposed.Comment: Journal of Multivariate Analysis (2014
On photon statistics parametrized by a non-central Wishart random matrix
In order to tackle parameter estimation of photocounting distributions,
polykays of acting intensities are proposed as a new tool for computing photon
statistics. As unbiased estimators of cumulants, polykays are computationally
feasible thanks to a symbolic method recently developed in dealing with
sequences of moments. This method includes the so-called method of moments for
random matrices and results to be particularly suited to deal with convolutions
or random summations of random vectors. The overall photocounting effect on a
deterministic number of pixels is introduced. A random number of pixels is also
considered. The role played by spectral statistics of random matrices is
highlighted in approximating the overall photocounting distribution when acting
intensities are modeled by a non-central Wishart random matrix. Generalized
complete Bell polynomials are used in order to compute joint moments and joint
cumulants of multivariate photocounters. Multivariate polykays can be
successfully employed in order to approximate the multivariate Mendel-Poisson
transform. Open problems are addressed at the end of the paper.Comment: 18 pages, in press in Journal of Statistical Planning and Inference,
201
On the computation of classical, boolean and free cumulants
This paper introduces a simple and computationally efficient algorithm for
conversion formulae between moments and cumulants. The algorithm provides just
one formula for classical, boolean and free cumulants. This is realized by
using a suitable polynomial representation of Abel polynomials. The algorithm
relies on the classical umbral calculus, a symbolic language introduced by Rota
and Taylor in 1994, that is particularly suited to be implemented by using
software for symbolic computations. Here we give a MAPLE procedure. Comparisons
with existing procedures, especially for conversions between moments and free
cumulants, as well as examples of applications to some well-known distributions
(classical and free) end the paper.Comment: 14 pages. in press, Applied Mathematics and Computatio
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