Symbolic computation of moments of sampling distributions

Nardo, E. Di; Guarino, Giuseppe; Senato, D.

doi:10.1016/j.csda.2008.04.017

Cited by 10 publications

(10 citation statements)

References 23 publications

(59 reference statements)

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“…By virtue of the fundamental theorem of symmetric polynomials, U-statistics can be expressed in terms of elementary symmetric polynomials. The symbolic method proposed here provides a way to find this expression [22]. The starting point is to replace R by R[x 1 , x 2 , .…”

Section: Is Umbrally Represented Bymentioning

confidence: 99%

Symbolic Calculus in Mathematical Statistics: A Review

Nardo¹

2015

Preprint

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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, Lévy 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.

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Section: Is Umbrally Represented Bymentioning

confidence: 99%

Symbolic Calculus in Mathematical Statistics: A Review

Nardo¹

2015

Preprint

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“…In [1], it is shown how procedures for computing moments and cumulants may themselves be derived from a few elementary identities. An efficient method for symbolic computation of moments and cumulants of sampling distributions is presented in [8]. For most excellent accounts of the literature, we refer the readers to [2,14].…”

Section: Introductionmentioning

confidence: 99%

Estimation in additive models and ANOVA-like applications

Antunes

Ferreira

et al. 2020

Journal of Applied Statistics

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A well-known property of cumulant generating function is used to estimate the first four order cumulants, using least-squares estimators. In the case of additive models, empirical best linear unbiased predictors are also obtained. Pairs of independent and identically distributed models associated with the treatments of a base design are used to obtain unbiased estimators for the fourth-order cumulants. An application to real data is presented, showing the good behaviour of the least-squares estimators and the great flexibility of our approach.

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“…with no reference to any probability space, someway getting closer to statistical methods. Compared with previous symbolic methods employed in statistics, see for example [1] and [20], by a theoretical point of view it has the advantage to reduce the combinatorics of symmetric functions, commonly used by statisticians, to few relations which cover a great variety of calculations [11]. By a computational point of view, the efficiency of umbral calculus in manipulating expressions involving r.v.…”

Section: Introductionmentioning

confidence: 99%

“…By a computational point of view, the efficiency of umbral calculus in manipulating expressions involving r.v. 's has been tested on the theory of k-statistics [10] and their generalizations [12] as well as in manipulating U -statistics and product moments of sample moments [11]. Recently, also the free cumulant theory has been approached by means of this new syntax [13], showing promises for future developments [14].…”

Section: Introductionmentioning

confidence: 99%