The Evaluation of General Non-Centred Orthant Probabilities

Miwa, Tetsuhisa; Hayter, Anthony J.; Kuriki, Shinya

doi:10.1111/1467-9868.00382

Cited by 80 publications

(82 citation statements)

References 13 publications

(24 reference statements)

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“…As shown on p. 227 in Miwa et al (2003), the corresponding polyhedral cone Q can be represented as a set of points that can be expressed as a linear combination of edge vectorsṽ i of the cone Q with non-negative coefficients ξ i , z = ξ 1ṽ1 + · · · + ξ pṽ p , where the vectorsṽ i are the column vectors of the matrix V. Craig (2008) considered a procedure to evaluate orthant probabilities by splitting the cone Q into edge orthoscheme cones, where two edgesṽ i andṽ j of the cone are orthogonal if |i − j| > 1. He showed that the polyhedral cone Q can be split into a linear combination of edge orthoscheme cones.…”

Section: Introductionmentioning

confidence: 89%

“…In this case, the polyhedral cone Q is represented by b j ( j−1) z j−1 + b j j z j ≥ 0. Miwa et al (2003) showed a method for evaluating this probability, which they called the orthoscheme probability, using recursive integration. They showed that non-centered orthant probability can be expressed as a linear combination of at most ( p − 1)!…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces

Nomura

2014

Stat Comput

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In this paper, a new procedure is described for evaluating the probability that all elements of a normally distributed vector are non-negative, which is called the noncentered orthant probability. This probability is defined by a multivariate integral of the density function. The definition is simple, and this probability arises frequently in statistics because the normal distribution is prevalent. The method for evaluating this probability, however, is not obvious, because applying direct numerical integration is not practical except in low dimensional cases. In the procedure proposed in this paper, the problem is decomposed into sub-problems of lower dimension. Considering the projection onto subspaces, the solutions of the sub-problems can be shared in the evaluation of higher dimensional problems. Thus the sub-problems form a lattice structure. This reduces the computational time from a factorial order, where the interim results are not shared, to order p 2 2 p , which is faster than the procedures that have been reported in the literature.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces

Nomura

2014

Stat Comput

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“…Given the central role of the normal distribution, computation of multivariate normal probability functions has been extensively studied. For low dimensions, various deterministic numerical integration rules are available (Joe, 1995;Miwa et al, 2003). However, these rules are subject to the curse of dimensionality and cannot be employed for moderate to large dimensions, as those occurring in the type of animal behavior experiments that motivated this paper.…”

Section: Likelihood Inferencementioning

confidence: 99%

Hybrid Pairwise Likelihood Analysis of Animal Behavior Experiments

Cattelan

Varin

2013

Biometrics

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The study of the determinants of contests between animals is an important issue in understanding animal behavior. Tournament experiments among a set of animals are used by zoologists for this purpose. From a statistical point of view, the results of these tournament experiments are naturally analyzed by paired comparison models such as the Bradley-Terry and the Thurstone models. A major complication is the presence of dependence between the outcomes of couples of contests with an animal in common. Likelihood analysis of this type of animal behavior experiments in presence of interdependence between contests is computationally demanding. An alternative fitting method that mixes optimal estimation equations and pairwise likelihood inference is then suggested. The performance of the proposed methodology is investigated by simulation studies and then applied to a real data set about adult male Cape Dwarf Chameleons.

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“…It employs a quasi-Monte Carlo algorithm, for which the computation time increases linearly with the dimension n and the computing time of ΔG(y) is proportional to n 2 (n ∈ N and n < 1000) (Genz and Bretz 1999). With a recursive linear integration procedure, mvtnorm calculates the MVN integral with even higher accuracy without compromising the computation time when n < 20 (Miwa et al 2003;Mi et al 2009).…”

Section: Introductionmentioning

confidence: 99%

Selectiongain: an R package for optimizing multi-stage selection

Utz

Melchinger

2015

Comput Stat

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Multi-stage selection is practised in numerous fields of the life sciences and particularly in breeding. A special characteristic of multi-stage selection is that candidates are evaluated in successive stages with increasing intensity and efforts, and only a fraction of the superior candidates is selected and promoted to the next stage. For the optimum design of such selection programs, the selection gain ΔG(y) plays a central role. It can be calculated by integration of a truncated multivariate normal distribution. While mathematical formulas for calculating ΔG(y) and ψ(y), the variance among the selected candidates, were developed a long time ago, solutions and software for numerical calculations were not available. We developed the R package selectiongain for efficient and precise calculation of ΔG(y) and ψ(y) for (i) a given matrix Σ * of correlations among the unobservable target character and the selection criteria and (ii) given coordinates Q of the truncation point or the selected fractions α in each stage. In addition, our software can be used for optimizing multi-stage selection programs under a given total budget and different costs of evaluating the candidates in each stage. Besides a detailed description of the functions of the software, the package is illustrated with two examples.

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The Evaluation of General Non-Centred Orthant Probabilities

Cited by 80 publications

References 13 publications

Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces

Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces

Hybrid Pairwise Likelihood Analysis of Animal Behavior Experiments

Selectiongain: an R package for optimizing multi-stage selection

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