Construction, properties and statistical applications of positive definite intraclass matrix. (English) Zbl 1223.15041
Given a real positive definite matrix \(S\) with real entries, the author considers the intraclass matrix \(S_0\) whose diagonal and off-diagonal elements are chosen as the averages of the diagonal elements and off-diagonal elements of the former matrix, respectively. Various algebraic properties of this intraclass matrix are given. For instance, \(S_0\) is shown to be positive definite, and many inequalities on the trace, norm, determinant and spectral condition number are presented. Finally, the paper concludes with a discussion of statistical applications of such matrices.
Reviewer: Igor Klep (Auckland)
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 15A45 | Miscellaneous inequalities involving matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15A12 | Conditioning of matrices |
Keywords:
positive definite matrix; intraclass matrix; trace; norm; eigenvalue; ill-conditioned matrix; determinant; spectral condition numberSoftware:
itsmrReferences:
| [1] | DOI: 10.1093/biomet/55.2.327 · Zbl 0157.48501 · doi:10.1093/biomet/55.2.327 |
| [2] | Bellman R, Introduction to Matrix Analysis,, 2. ed. (1974) |
| [3] | DOI: 10.1007/b97391 · Zbl 0994.62085 · doi:10.1007/b97391 |
| [4] | DOI: 10.1007/BF02310555 · Zbl 1367.62314 · doi:10.1007/BF02310555 |
| [5] | DOI: 10.1214/aos/1176342465 · Zbl 0261.62037 · doi:10.1214/aos/1176342465 |
| [6] | DOI: 10.2307/2282714 · Zbl 0116.37002 · doi:10.2307/2282714 |
| [7] | DOI: 10.2307/2528472 · doi:10.2307/2528472 |
| [8] | Graybill FA, Introduction to Matrices with Applications in Statistics (1969) |
| [9] | DOI: 10.1037/13240-000 · doi:10.1037/13240-000 |
| [10] | DOI: 10.1016/0024-3795(85)90244-7 · Zbl 0578.15013 · doi:10.1016/0024-3795(85)90244-7 |
| [11] | Jolicoeur P, Growth 24 pp 339– (1960) |
| [12] | Kshirsagar AM, A Course in Linear Models (1983) |
| [13] | Marcus M, A Survey of Matrix Theory & Matrix Inequalities (1964) |
| [14] | Morrison DF, Multivariate Statistical Methods,, 2. ed. (1978) |
| [15] | DOI: 10.1111/j.2044-8317.1970.tb00438.x · Zbl 0215.26901 · doi:10.1111/j.2044-8317.1970.tb00438.x |
| [16] | DOI: 10.1111/j.1467-842X.1976.tb01288.x · Zbl 0404.62044 · doi:10.1111/j.1467-842X.1976.tb01288.x |
| [17] | B.N. Mukherjee,A Simple Approach to Testing of Hypothesis Regarding a Class of Covariance Structures, Proceedings of the Indian Statistical Institute Golden Jubilee International Conference on Statistics: Applications and New Directions, Calcutta, 1981, pp. 171–192 |
| [18] | Mukherjee BN, Calcutta Stat. Assoc. Bull. 37 pp 171– (1988) |
| [19] | Rao CR, Linear Statistical Inference and its Applications,, 2. ed. (2002) |
| [20] | Rogers GS, Commun. Stat. Theory Methods 6 pp 121– (1977) · Zbl 0352.62028 · doi:10.1080/03610927708827477 |
| [21] | DOI: 10.1111/j.1469-185X.1965.tb00806.x · doi:10.1111/j.1469-185X.1965.tb00806.x |
| [22] | Takeuchi K, The Foundations of the Multivariate Analysis (1982) |
| [23] | DOI: 10.1214/aoms/1177730940 · Zbl 0063.08259 · doi:10.1214/aoms/1177730940 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
