578 research outputs found
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 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
Symbolic Calculus in Mathematical Statistics: A Review
In the last ten years, the employment of symbolic methods has substantially
extended both the theory and the applications of statistics and probability.
This survey reviews the development of a symbolic technique arising from
classical umbral calculus, as introduced by Rota and Taylor in The
usefulness of this symbolic technique is twofold. The first is to show how new
algebraic identities drive in discovering insights among topics apparently very
far from each other and related to probability and statistics. One of the main
tools is a formal generalization of the convolution of identical probability
distributions, which allows us to employ compound Poisson random variables in
various topics that are only somewhat interrelated. Having got a different and
deeper viewpoint, the second goal is to show how to set up algorithmic
processes performing efficiently algebraic calculations. In particular, the
challenge of finding these symbolic procedures should lead to a new method, and
it poses new problems involving both computational and conceptual issues.
Evidence of efficiency in applying this symbolic method will be shown within
statistical inference, parameter estimation, L\'evy processes, and, more
generally, problems involving multivariate functions. The symbolic
representation of Sheffer polynomial sequences allows us to carry out a
unifying theory of classical, Boolean and free cumulants. Recent connections
within random matrices have extended the applications of the symbolic method.Comment: 72 page
A unifying framework for -statistics, polykays and their multivariate generalizations
Through the classical umbral calculus, we provide a unifying syntax for
single and multivariate -statistics, polykays and multivariate polykays.
From a combinatorial point of view, we revisit the theory as exposed by
Stuart and Ord, taking into account the Doubilet approach to symmetric
functions. Moreover, by using exponential polynomials rather than set
partitions, we provide a new formula for -statistics that results in a very
fast algorithm to generate such estimators.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6163 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A cumulant approach for the first-passage-time problem of the Feller square-root process
The paper focuses on an approximation of the first passage time probability
density function of a Feller stochastic process by using cumulants and a
Laguerre-Gamma polynomial approximation. The feasibility of the method relies
on closed form formulae for cumulants and moments recovered from the Laplace
transform of the probability density function and using the algebra of formal
power series. To improve the approximation, sufficient conditions on cumulants
are stated. The resulting procedure is made easier to implement by the symbolic
calculus and a rational choice of the polynomial degree depending on skewness,
kurtosis and hyperskewness. Some case-studies coming from neuronal and
financial fields show the goodness of the approximation even for a low number
of terms. Open problems are addressed at the end of the paper
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