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Further identities for the Wishart distribution with applications in regression
Author(s) -
Haff L. R.
Publication year - 1981
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314615
Subject(s) - wishart distribution , identity matrix , mathematics , scalar (mathematics) , inverse wishart distribution , matrix (chemical analysis) , multivariate statistics , identity (music) , scatter matrix , matrix t distribution , multivariate normal distribution , regression analysis , statistics , pure mathematics , eigenvalues and eigenvectors , physics , geometry , materials science , quantum mechanics , acoustics , composite material
Matrix analogues are given for a known scalar identity which relates certain expectations with respect to the Wishart distribution. (The scalar identity was independently derived by C. Stein and L. Haff.) The matrix analogues are more aptly called “matrix extensions.” They can be derived by using the scalar identity; nevertheless, they are seen (in quite elementary terms) to be more general than the latter. A method of doing multivariate calculations is developed from the identities, and several examples are worked in detail. We compute the first two moments of the regression coefficients and another matrix arising in regression analysis. Also, we give a new result for the matrix analogue of squared multiple correlation: the bias correction of Ezekiel (1930), a result often used in model building, is extended to the case of two or more dependent variables.