On Approximating the Distribution of Quadratic Forms in Gamma Random Variables and Exponential Order Statistics

Mohsenipour, A. Akbar; Provost, Serge B.

doi:10.2991/jsta.2013.12.2.4

Cited by 3 publications

(2 citation statements)

References 10 publications

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“…One challenge is to extend the volume formula to the intersection with a ellipsoid. In [28], they propose a method to approximate the distribution f of quadratic forms in gamma random variables which is a similar problem to that in [27] (see Sect. 3).…”

Section: Discussionmentioning

confidence: 99%

Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises

Cales,

Chalkis,

Emiris

et al. 2018

Preprint

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We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be constructed by computing volumes of convex bodies.We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions, which are the method of choice for intersections with a single hyperplane; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple with additional structure; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed in the given body, such as a second-order cone program; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. * The views expressed are those of the authors and do not necessarily reflect official positions of the European Commission. 1 Calès acknowledges the financial support of the European Commission through its Proof-of-Concept program. 2 Emiris is partially supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No 734242 (Project LAMBDA).

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Section: Discussionmentioning

confidence: 99%

Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises

Cales,

Chalkis,

Emiris

et al. 2018

Preprint

View full text Add to dashboard Cite

show abstract

“…8 , 9 , 10 and also that the covariances of and are zero. If the random vector were sampled from distributions other than the multivariate normal such as the Gamma distribution, one can still derive approximations for the moments of quadratic forms (Mohsenipour and Provost 2013 ), but more detailed derivations for this case are beyond the scope of this study. For the derivation that we use (i) , which implies , and , (see Eq.…”

Section: Appendicesmentioning

confidence: 99%

Influence of the mating design on the additive genetic variance in plant breeding populations

Lanzl,

Melchinger,

Schön

2023

Theor Appl Genet

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Key message Mating designs determine the realized additive genetic variance in a population sample. Deflated or inflated variances can lead to reduced or overly optimistic assessment of future selection gains. Abstract The additive genetic variance $${V}_{A}$$ V A inherent to a breeding population is a major determinant of short- and long-term genetic gain. When estimated from experimental data, it is not only the additive variances at individual loci (QTL) but also covariances between QTL pairs that contribute to estimates of $${V}_{A}$$ V A . Thus, estimates of $${V}_{A}$$ V A depend on the genetic structure of the data source and vary between population samples. Here, we provide a theoretical framework for calculating the expectation and variance of $${V}_{A}$$ V A from genotypic data of a given population sample. In addition, we simulated breeding populations derived from different numbers of parents (P = 2, 4, 8, 16) and crossed according to three different mating designs (disjoint, factorial and half-diallel crosses). We calculated the variance of $${V}_{A}$$ V A and of the parameter b reflecting the covariance component in $${V}_{A},$$ V A , standardized by the genic variance. Our results show that mating designs resulting in large biparental families derived from few disjoint crosses carry a high risk of generating progenies exhibiting strong covariances between QTL pairs on different chromosomes. We discuss the consequences of the resulting deflated or inflated $${V}_{A}$$ V A estimates for phenotypic and genome-based selection as well as for applying the usefulness criterion in selection. We show that already one round of recombination can effectively break negative and positive covariances between QTL pairs induced by the mating design. We suggest to obtain reliable estimates of $${V}_{A}$$ V A and its components in a population sample by applying statistical methods differing in their treatment of QTL covariances.

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Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crises

Calès

Chalkis

Emiris

et al. 2023

Computational Geometry

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On Approximating the Distribution of Quadratic Forms in Gamma Random Variables and Exponential Order Statistics

Cited by 3 publications

References 10 publications

Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises

Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises

Influence of the mating design on the additive genetic variance in plant breeding populations

Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crises

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