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- Title
On the distribution of sample scale-free scatter matrices.
- Authors
Mathai, A. M.; Provost, Serge B.
- Abstract
This paper addresses certain distributional aspects of a scale-free scatter matrix denoted by R that is stemming from a matrix-variate gamma distribution having a positive definite scale parameter matrix B. Under the assumption that B is a diagonal matrix, a structural representation of the determinant of R is derived; the exact density functions of products and ratios of determinants of matrices possessing such a structure are obtained; a closed form expression is given for the density function of R. Moreover, a novel procedure is utilized to establish that certain functions of the determinant of the sample scatter matrix are asymptotically distributed as chi-square or normal random variables. Then, representations of the density function of R that respectively involve multiple integrals, multiple series and Gauss' hypergeometric function are provided for the general case of a positive definite scale parameter matrix, and an illustrative numerical example is presented. Cutting-edge mathematical techniques have been employed to derive the results. Naturally, they also apply to the conventional sample correlation matrix which is encountered in various multivariate inference contexts.
- Subjects
S-matrix theory; RANDOM variables; HYPERGEOMETRIC series; HYPERGEOMETRIC functions; GAMMA distributions; CHI-square distribution
- Publication
Statistical Papers, 2024, Vol 65, Issue 1, p121
- ISSN
0932-5026
- Publication type
Article
- DOI
10.1007/s00362-022-01388-8