By James D. Malley

This monograph brings jointly my paintings in mathematical information as i've got considered it throughout the lens of Jordan algebras. 3 technical domain names are to be noticeable: purposes to random quadratic types (sums of squares), the research of algebraic simplifications of maxi mum probability estimation of patterned covariance matrices, and a extra broad open mathematical exploration of the algebraic area from which i've got drawn the consequences utilized in the statistical difficulties simply pointed out. Chapters 1, 2, and four current the statistical results i've got built utilizing the algebraic effects that seem, for the main half, in bankruptcy three. As a much less daunting, but fairly effective, element of access into this fabric, one heading off many of the summary algebraic concerns, the reader could use the 1st 1/2 bankruptcy four. the following I current a streamlined, yet nonetheless absolutely rigorous, definition of a Jordan algebra (as it's utilized in that bankruptcy) and its crucial homes. those proof are then instantly utilized to simplifying the M:-step of the EM set of rules for multivariate general covariance matrix estimation, within the presence of linear constraints, and knowledge lacking thoroughly at random. the implications awarded basically get to the bottom of a pragmatic statistical quest all started by way of Rubin and Szatrowski [1982], and persisted, occasionally implicitly, through many others. After this, one can then go back to Chapters 1 and a couple of to determine how i've got tried to generalize the paintings of Cochran, Rao, Mitra, and others, on vital and valuable houses of sums of squares.