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recursive integration methodologies with statistical applications

JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2006年第7期136卷 2284-2296页

作者： Hayter, AJ Georgia Tech Ind & Syst Engn Atlanta GA 30332 USA

This paper shows how recursive integration methodologies can be used to evaluate high-dimensional integral expressions. This has applications to many areas of statistical inference where probability calculations and critical point evaluations often require such high-dimensional integral evaluations. recursive integration can allow an integral expression of a given dimension to be evaluated by a series of calculations of a smaller dimension. This significantly reduces the computation time. The application of the recursive integration methodology is illustrated with several examples. (c) 2005 Elsevier B.V. All rights reserved.

关键词： recursive integration numerical integration computational intensity tree structure Markov property critical point calculation multiple comparisons sequential analysis

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recursive integration Methodologies with Applications to the Evaluation of Multivariate Normal Probabilities

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JOURNAL OF STATISTICAL THEORY AND PRACTICE 2011年第4期5卷 563-589页

作者： Hayter, A. J. Univ Denver Dept Business Informat & Analyt Denver CO 80208 USA

This paper considers the evaluation of probabilities defined in terms of the multivariate normal distribution. The multivariate normal distribution can have any covariance matrix and any mean vector. Probabilities defined by a set of inequalities of linear combinations of the multivariate normal random variables are considered. It is shown how these probabilities can be evaluated from a series of one-dimensional integrations. This approach affords a practical algorithm for the evaluation of these probabilities which is considerably more efficient than more direct numerical integration approaches. Consequently, it enlarges the class of probabilities of this kind which are computationally feasible.

关键词： Multivariate normal distribution recursive integration Numerical integration Computational intensity Tree structure Tri-diagonal matrix Orthant probability

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A flexible true plurigaussian code for spatial facies simulations

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COMPUTERS & GEOSCIENCES 2006年第10期32卷 1629-1645页

作者： Xu, Chaoshui Dowd, Peter Alan Mardia, Kanti V. Fowell, Robert J. Univ Adelaide Fac Engn Comp & Math Sci Adelaide SA 5005 Australia Univ Leeds Dept Min & Mineral Engn Leeds LS2 9JT W Yorkshire England Univ Leeds Dept Stat Leeds LS2 9JT W Yorkshire England

The current forms of plurigaussian simulation have serious limitations for applications to large numbers of geological facies, or units, which have complex contact relations. In this paper the authors present a true plurigaussian simulation (PGS) method, which can be applied in a simple way to any number of geological facies by using any number of Gaussians. A recursive technique is used for multi-dimensional integration of the Gaussian functions, which forms the major part of the PGS computation. A binary, dynamic contact matrix (DCM) is used to specify the contact relations among the facies;this method has proved to be simple, flexible and capable of dealing with general, complex contact relations. A method for incorporating into PGS multivariate correlations among any number of random variables is also included. A simulated example is used to demonstrate the application of the generalised PGS. This example shows that PGS is more robust to under-sampling than traditional direct indicator simulation. (c) 2006 Elsevier Ltd. All rights reserved.

关键词： geostatistical simulation plurigaussian simulation spatial facieses simulation spatial modelling multivariate correlation contact relations dynamic contact matrix recursive integration

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Rank constrained distribution and moment computations

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2017年 105卷 229-242页

作者： Kiatsupaibul, Seksan Hayter, Anthony J. Liu, Wei Chulalongkorn Univ Dept Stat Bangkok 10330 Thailand Univ Denver Dept Business Informat & Analyt Denver CO USA Univ Southampton Sch Math Southampton Hants England Univ Southampton S3RI Southampton Hants England

Consider a set of independent random variables with specified distributions or a set of multivariate normal random variables with a product correlation structure. This paper shows how the distributions and moments of these random variables can be calculated conditional on a specified ranking of their values. This can be useful when the ordering of the variables can be determined without observing the actual values of the variables, as in ranked set sampling, for example. Thus, prior information on the distributions and moments from their individual specified distributions can be updated to provide improved posterior information using the known ranking. While these calculations ostensibly involve high dimensional integral expressions, it is shown how the previously developed general recursive integration methodology can be applied to this problem so that they can be evaluated in a straightforward manner as a series of one-dimensional or two-dimensional integral calculations. Furthermore, the proposed methodology possesses a self-correction mechanism in the computation that prevents any serious growth of the errors. Examples illustrate how different kinds of ranking information affect the distributions, expectations, variances, and covariances of the variables, and how they can be employed to solve a decision making problem. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Conditional distribution Moments Normal distribution Order restriction Ranked set sampling recursive integration

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ELIAS: An Accurate and Extensible Lithography Aerial Image Simulator With Improved Numerical Algorithms

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IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING 2009年第2期22卷 276-289页

作者： Yu, Peng Pan, David Z. Univ Texas Austin Dept Elect & Comp Engn Austin TX 78712 USA

Lithography simulators have been playing an indispensable role in process optimization and design for manufacturability (DFM). The ever smaller feature sizes demand higher numerical accuracy and faster runtime on these lithography simulators. Aerial image simulation is the first key step in lithography simulation, and the method using transmission cross coefficient (TCC), which is a two-dimensional integral, is the most commonly used technique for full-chip aerial image simulation. In this paper, we present a very accurate, yet efficient and extensible aerial image simulator, ELIAS. We find that the majority of the numerical error during the TCC computation is due to the discontinuous boundaries of the support of the TCC integrand. We reduce the error dramatically by using a recursive integration algorithm. Because TCC is usually computed on uniform grids, we further speed up the algorithm without increasing the errors. Given the same accuracy, our new algorithm can speed up the runtime by 100 x -1000 X. Our algorithm also provides smooth tradeoff between accuracy and runtime. It can be used to benchmark other lithography aerial simulators. In addition, ELIAS provides an open-source, flexible software framework to incorporate different lithography settings.

关键词： Accuracy aerial image simulation C plus ELIAS fast Fourier transform (FFT) Hopkins equation lithography simulation numerical algorithm recursive integration runtime transmission cross coefficient (TCC)

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Simultaneous confidence interval estimation for successive comparisons of ordered treatment effects

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2000年第1期88卷 75-86页

作者： Liu, W Miwa, T Hayter, AJ Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA Univ Southampton Dept Math Southampton SO9 5NH Hants England Natl Inst Agroenvironm Sci Stat Lab Tsukuba Ibaraki 3058604 Japan

Lee and Spurrier (J. Statist. Plann. Inference 43 (1995) 323-330) present a new procedure for making successive comparisons between ordered treatments. Their procedure has important applications for problems where the treatments can be assumed to satisfy a simple ordering, such as for a sequence of increasing dose-levels of a drug. In this paper we provide some additional work on their procedure. Specifically, we illustrate how a recursive computational approach can be used to calculate the critical points required by the procedure for any number of treatment levels. We also show how the procedure can be extended to provide information on larger sets of contrasts of the treatment effects. The application of the procedure is illustrated with an example. (C) 2000 Elsevier Science B.V. All rights reserved. MSG: 62F03;62F07;62F25;62J10;62J15.

关键词： contrasts critical points multivariate-t distribution ordered inference pairwise comparisons recursive integration simultaneous confidence intervals

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Distribution of ratio of extreme of isotonic estimators of chi-square variables with applications

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2009年第9期139卷 3262-3269页

作者： Singh, Parminder Gill, Amar Nath Kumar, Narendra Guru Nanak Dev Univ Dept Math Amritsar Punjab India Panjab Univ Dept Stat Chandigarh 160014 India

Let V(1),...,V(k) be k independent chi-square variables each with v degrees of freedom (d.f.) and let W(1) <= ... <= W(k) be the corresponding isotonic estimators computed by the pool adjacent violator algorithm under simple ordering among Vs. In this paper, a recursive integration algorithm for computing the distribution function of statistic T = W(k)/W(1) is presented. The test statistic F(max)* proposed by Fujino (Biometrika 66 (1979) 133-139) to test the homogeneity of variances against ordered alternative follows same null distribution as of the statistic T. Fujino discussed the null F(max)* distribution of and provided critical constant only for k = 3,...,6, which limited its applications. The recursive procedure given in this article can be implemented efficiently for sufficient higher value of k. Test procedures based on statistic T for testing homogeneity against simple ordered alternative of k normal variances and equality of scale parameters of k exponential distributions are discussed separately with tables of critical constants for their implementation. Correctness of the critical constants is verified through simulation study. The proposed methodology is also extended to address the problem of testing exponentiality (constant failure rate) versus increasing failure rate, using isotonic estimators of normalized spaces between consecutive order statistics, along with tables of relevant critical constants. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Isotonic estimator Scale parameter Simple ordered alternative Critical constants recursive integration Constant failure rate Normalized spacing between successive order statistics Increasing failure rate

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Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces

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STATISTICS AND COMPUTING 2016年第1-2期26卷 187-197页

作者： Nomura, Noboru Natl Inst Adv Ind Sci & Technol Tsukuba Ibaraki 3058569 Japan

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 non-centered 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.

关键词： Conditional distribution Cumulative distribution function Orthant probability Polyhedral cone recursive integration

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On the isotonic range statistic for testing against an ordered alternative

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2002年第2期105卷 347-362页

作者： Kuriki, S Shimodaira, H Hayter, T Inst Stat Math Minato Ku Tokyo 1068569 Japan Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

Consider the balanced one-way layout for comparing k treatment effects mu(i), 1 less than or equal to i less than or equal to k. Marcus (Biometrika 63 (1976) 177) considers a test procedure for testing the null hypothesis H-0: mu(1) = mu(2) = (...) = mu(k) against the simple ordered alternative H-A: mu(1) less than or equal to mu(2) less than or equal to (...) less than or equal to mu(k) with at least one strict inequality based upon the range (μ) over cap (k) - (μ) over cap (1) of the isotonic estimates of the treatment effects. This test statistic was originally proposed in Williams (Biometrics 27 (1971) 103) and is sometimes referred to as the modified Williams statistic. It has many useful applications in biostatistical studies. In this article, we present an algorithm for calculating the distribution function of this isotonic range statistic for a general number of treatments k using recursive integration techniques and we provide tables of critical points for this test procedure. This allows the full implementation of this test procedure. We also show how this procedure can be used to construct a set of simultaneous confidence intervals for a set of monotone contrasts of the treatment effects and compare it with other multiple comparisons procedures. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： analysis of variance order restricted inference isotonic estimates multiple comparisons recursive integration modified Williams statistic simultaneous inference monotone contrasts

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Orthant probabilities of elliptical distributions from orthogonal projections to subspaces

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STATISTICS AND COMPUTING 2019年第2期29卷 289-300页

作者： Nomura, Noboru Kochi Univ Fac Sci & Technol Akebono Cho Kochi 7808520 Japan

A new procedure is proposed for evaluating non-centred orthant probabilities of elliptical distributed vectors, which is the probabilities that all elements of a vector are non-negative. The definition of orthant probabilities is simple, formulated as a multiple integral of the density function;however, applying direct numerical integration is not practical, except in low-dimensional cases, and methods for evaluating orthant probabilities are not trivial. This probability arises frequently in statistics;in particular, the normal distribution and Student's t-distribution are in the family of elliptical distribution. In the procedure proposed in this paper, an orthant probability is approximated by the probability that the vector falls in a simplex. In the process, the problem is decomposed into sub-problems of lower dimension based on the symmetry of elliptical distributions. Intermediate sub-problems can be generated by projection onto subspaces, and the sub-problems form a lattice structure. Considering this structure, intermediate computations are shared between the evaluations of higher-dimensional problems, and computational time is reduced. The procedure can be applied not only to normal distributions, but also to general elliptical distributions, especially t-distributions, which are used in the multiple comparison procedure.

关键词： Cumulative distribution function Elliptical distribution Orthant probability Polyhedral cone recursive integration

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