Calculations of level probabilities for normal random variables with unequal variances with applications to Bartholomew's test in unbalanced one-way models

The evaluation of the cumulative distribution function of a multivariate normal distribution is considered. The multivariate normal distribution can have any positive definite correlation matrix and any mean vector. The approach taken has two stages. In the first stage, it is shown how non-centred orthoscheme probabilities can be evaluated by using a recursive integration method. In the second stage, some ideas of Schläfli and Abrahamson are extended to show that any non-centred orthant probability can be expressed as differences between at most ("m" - 1)! non-centred orthoscheme probabilities. This approach allows an accurate evaluation of many multivariate normal probabilities which have important applications in statistical practice. Copyright 2003 Royal Statistical Society.

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“…This recursive computational approach has been employed in Hayter and Liu (1996) and Miwa et al . (2000).…”

Section: 1 Recursive Integration Formulaementioning

confidence: 91%

“…In this section attention is directed towards an m × m tridiagonal correlation matrix R ={ ρ ij } for which ρ ij =0 for | i − j | > 1. Miwa et al . (2000) gave a procedure to calculate centred orthoscheme probabilities P m ( 0 , R ).…”

Section: The Evaluation Of Non‐centred Orthoscheme Probabilitiesmentioning

confidence: 99%

The Evaluation of General Non-Centred Orthant Probabilities

Miwa

Hayter

Kuriki

2003

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…Derivatives of the integrands of (1) and (16) for three or more dimensional problems can be computed easily because the functions R(·) and R J ∪{i} (·) in the integrands were evaluated by integrating the results of lower dimensional problems as pointed out by Miwa et al (2000). The derivatives of integrands for the two-dimensional problem can be evaluated explicitly because the integrand is the product of two density functions of one-dimensional normal distribution.…”

Section: A Numerical Examplementioning

confidence: 99%

“…The new procedure proposed here requires computational time in the order of p 2 2 p . The procedures proposed by Miwa et al (2000Miwa et al ( , 2003 were essentially based on this property of the conditional distribution of normal distributions. I will consider the relationship between the two procedures briefly in Sect.…”

Section: Introductionmentioning

confidence: 99%

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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“…, n k ) for k 4 are given in Barlow et al [2,Section 3.3]. For general k, Miwa et al [24] pointed out that P (l, k; n 1 , . .…”

Section: Remark 22mentioning

confidence: 99%

Asymptotic distribution of inequality-restricted canonical correlation with application to tests for independence in ordered contingency tables

Kuriki

2005

Journal of Multivariate Analysis

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For two-way ordered categorical data, correspondence analysis and the RC association model (the row-column-effect association model) with order-restricted scores have been proposed mainly for descriptive purposes. In this paper, tests for independence in two-way ordered contingency tables based on these models are developed in a general framework of inequality-restricted canonical correlation analysis. The limiting null distributions are characterized as the maxima of Gaussian random fields and asymptotic expansions of their tail probabilities are derived by the tube method, an integral geometric approach. Some numerical techniques for fitting order-restricted models are discussed. An example of data analysis is given to demonstrate the practical usefulness of the proposed method. Correspondence analysis and the RC association model in two-way ordered categorical dataThe method of canonical correlation or correspondence analysis (CA) is one of the most popular tools for analyzing two-way contingency tables. Suppose that an a × b table

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Calculations of level probabilities for normal random variables with unequal variances with applications to Bartholomew's test in unbalanced one-way models

Cited by 20 publications

References 15 publications

The Evaluation of General Non-Centred Orthant Probabilities

The Evaluation of General Non-Centred Orthant Probabilities

Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces

Asymptotic distribution of inequality-restricted canonical correlation with application to tests for independence in ordered contingency tables

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