A Bayesian Approach to the Multivariate Behrens-Fisher by Nel D. G., Groenewald P. C. N. PDF

By Nel D. G., Groenewald P. C. N.

Autonomous random samples of sizesN 1 andN 2 from multivariate general populationsN p (θ1,∑1) andN p (θ2,∑2) are thought of. less than the null hypothesisH zero: θ1=θ2, a unmarried θ is generated from aN p(μ, Σ) past distribution, whereas underH 1: θ1≠θ2 capacity are generated from the exchangeable priorN p(μ,σ). In either instances Σ can be assumed to have a obscure earlier distribution. For an easy covariance constitution, the Bayes factorB and minimal Bayes think about favour of the null hypotheses is derived. The Bayes danger for every speculation is derived and a method is mentioned for utilizing the Bayes issue and Bayes dangers to check the speculation.

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Finally, take the case of two rotations, s — s(0,a), s' = s(0',a'). To compute the product u = ss' when O ^ 0 ' , define three lines in E 2 as follows: • / is the line through O and 0 ' , • /' is the line through O such that the angle from /' to / is a / 2 , • I" is the line through O' such that the angle from I to I" is Q ' / 2 . 1, s = r(l')r(l), 8'=r(l)r(l"), whence u = ss' = r ( / > ( / " ) , as above. There are now two cases according as /' and /" are parallel or not, and these are illustrated in Fig.

B + b). 23 2. +(-l)ea, c = ar6s(f3) + b. 3. 1 Let r be reflection in a line I, s — s{fi) rotation through fi G Emod 2n about a point O £ I, and t(a) translation by a vector a G E. Then *(a) r = *(ar), *(a) 5 = t(as), s(p)r = s(-/3). 14) • Some comments are appropriate. First, in view of the normal form theorem and the fact that translations commute with each other, the first two of these equations show that the translation subgroup T is closed under conjugation in E. ) The third equation says that when you look at a clock through a mirror, the hands appear to move anti-clockwise.

Prove that any rotation of E 3 can be written as the product of two reflections. Deduce that the product of two rotations of E 3 with intersecting axes is again a rotation. What if the axes are parallel? 8. Give another proof, along the lines of that of the Normal Form Theorem, that any isometry of E 2 is the product of at most three reflections. 9. Extend the result of the previous exercise by showing that any isometry of E 3 is the product of at most four reflections. 10. Prove that any OR isometry u of E 2 is of the form r£, where t is a translation and r is reflection in some line through a prescribed point O.

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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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