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 $1994.$ 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 kk-statistics, polykays and their multivariate generalizations

Through the classical umbral calculus, we provide a unifying syntax for single and multivariate $k$-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 $k$-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