By Nel D. G., Groenewald P. C. N.
Self sustaining random samples of sizesN 1 andN 2 from multivariate general populationsN p (θ1,∑1) andN p (θ2,∑2) are thought of. lower than the null hypothesisH zero: θ1=θ2, a unmarried θ is generated from aN p(μ, Σ) earlier distribution, whereas underH 1: θ1≠θ2 skill are generated from the exchangeable priorN p(μ,σ). In either circumstances Σ can be assumed to have a obscure previous distribution. For an easy covariance constitution, the Bayes factorB and minimal Bayes think about favour of the null hypotheses is derived. The Bayes threat for every speculation is derived and a technique is mentioned for utilizing the Bayes issue and Bayes hazards to check the speculation.
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Extra info for A Bayesian Approach to the Multivariate Behrens-Fisher Problem Under the Assumption of Proportional Covariance Matrices
These two ratios are drawn from the third category of ratios described on pp. 33-36. The same passage cited above on p. , III. ” Once again, Oresme can offer no ex amples, but one fitting the specifications would be and(s/j)V^ which are unrelatable as a number to a number, that is they are unrelatable by any rational exponentpjq since The reader of the De proportionibus proportionum upon arriving almost at the end of Chapter III, the last of the strictly mathematical chapters, is met with an unexpected burst of enthusiasm which serves as introduction to the tenth proposition.
571-71), it follows that V c jV ^ = ^/2- Without furnishing any additional informa tion, Oresme states categorically that the ratio which produces V c must be irrational. 525-32), Oresme draws upon other propositions in the third chapter. By Chapter III, Proposition III, all velocities resulting from multiple ratios nji, where n is any integer greater than i, are incommensurable to all velocities arising from ratios of the form where p > q > i and pjq is in its lowest terms. This is obvious because nji ^ where m/r is a ratio of integers.
It is extremely improbable that Oresme knew of, or could have represented, such concepts. 49293). In the present example he needed to know only that five exponential parts were required, which was obvious from the fact that the given ratio is From the denominator of the exponent he knew that each of the five parts had to be He then simply assumes that the first two mean proportionals are appropriate, so that A , C, D , and B are in geometric proportional ity, from which it follows that AjC-CjDD/B constitute three of the five required exponential parts.
A Bayesian Approach to the Multivariate Behrens-Fisher Problem Under the Assumption of Proportional Covariance Matrices by Nel D. G., Groenewald P. C. N.
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