Antieigenvalue techniques in statistics. (English) Zbl 1168.15006

Seddighin, Morteza

Linear Algebra Appl. 430, No. 10, 2566-2580 (2009).

Given an operator \(T\) on a Hilbert space \(H\), the first antieigenvalue of \(T\), also called the (real) cosine of \(T\), denoted by \(\mu_1(T)\) or \(\cos T\), was first defined by K. Gustafson [Bull. Am. Math. Soc. 74, 488–492 (1968; Zbl 0172.40702)] to be
\[ \mu_1(T)= \inf_{Tf\neq 0}\frac{\text{Re}(Tf,f)}{\| Tf\|\, \| f\|}\,. \]
The author continues to explore two techniques used in antieigenvalue computations in previous papers. The first is basically a two nonzero component property for certain class of functionals. The second consists on converting the matrix optimization problems in statistics to a convex programming problem, finding the minimum of a convex function on the numerical range of an operator. The two techniques will permit generalizing some of the matrix optimization problems arising in statistics to strongly accretive operators on finite or infinite dimensional Hilbert spaces.

Reviewer: C. M. da Fonseca (Coimbra)

MSC:

15A18	Eigenvalues, singular values, and eigenvectors
65F15	Numerical computation of eigenvalues and eigenvectors of matrices
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
62C20	Minimax procedures in statistical decision theory
90C25	Convex programming

Keywords:

antieigenvalue; antieigenvector; statistical efficiency; strongly accretive operator; numerical range; convex set; Hilbert space; convex programming

Citations:

Zbl 0172.40702

References:

[1]	Gustafson, K., The angle of an operator and positive operator products, Bull. amer. math. soc., 74, 492-499, (1968) · Zbl 0172.40702
[2]	Gustafson, K., Operator trigonometry of statistics and econometrics, Linear algebra appl., 354, 141-158, (2002) · Zbl 1015.62054
[3]	Gustafson, K., The trigonometry of matrix statistics, Int. stat. rev., 74, 2, 187-202, (2006)
[4]	Gustafson, K.; Rao, D., Numerical range, (1997), Springer
[5]	Gustafson, K.; Seddighin, M., Antieigenvalue bounds, J. math. anal. appl., 143, 327-340, (1989) · Zbl 0696.47004
[6]	Liu, S., Efficiency comparisons between two estimators based on matrix determinant kantrovich-type inequalities, Metrika, 51, 145-155, (2000) · Zbl 1093.62555
[7]	Liu, S.; King, M., Two Kantorovich-type inequalities and efficiency comparisons between the OLSE and BLUE, J. inequal. appl., 7, 169-177, (2002) · Zbl 1009.15008
[8]	Magnus, R.; Neudecker, H., Matrix differential calculus with applications in statistics and econometrics, (1999), Wiley Chichester · Zbl 0912.15003
[9]	Rao, C.R.; Rao, M.B., Stationary values of the product of two Rayleigh quotients homologous canonical correlations, Sankhya: Indian stat., 49B, 113-125, (1987) · Zbl 0639.62051
[10]	Rao, C.R.; Rao, M.B., Matrix algebra and its applications to statistics and econometrics, (1998), World Scientific Singapore · Zbl 0915.15001
[11]	Seddighin, M., Antieigenvalues and total antieigenvalues of normal operators, J. math. anal. appl., 274, 239-254, (2002) · Zbl 1020.47020
[12]	Seddighin, M., On the joint antieigenvalue of operators on normal subalgebras, J. math. anal. appl., 312, 61-71, (2005) · Zbl 1096.47007
[13]	Seddighin, M.; Gustafson, K., On the eigenvalues which express antieigenvalues, Int. J. math. math. sci., 10, 1543-1554, (2005) · Zbl 1094.47030
[14]	Wang, S.G.; Chow, S.C., Advanced linear models, (1997), Marcel-Dekker New York