Operator trigonometry of statistics and econometrics. (English) Zbl 1015.62054

Gustafson, Karl

Linear Algebra Appl. 354, No. 1-3, 141-158 (2002).

Summary: A new and useful geometric point of view for the understanding and analysis of certain matrix methods as they are used in statistics and econometrics is presented. Applications to statistical efficiency, parameter estimation, and correlation theory are given. In particular we show that worst case relative least squares efficiency, although achieved by maximally inefficient regressors, is also achieved by maximal covariance matrix turning vectors. Also we elaborate geometrically a commutator trace efficiency result of P. Bloomfield and G.S. Watson [Biometrika 62, 121-128 (1975; Zbl 0308.62056)]. Well-established Lagrange multiplier methods for constrained optimization are compared to the use of Euler equations from the new geometric theory.

MSC:

62H12	Estimation in multivariate analysis
15A99	Basic linear algebra
62J05	Linear regression; mixed models
62A01	Foundations and philosophical topics in statistics

Keywords:

operator trigonometry; statistical efficiency; Lagrange multiplier

Citations:

Zbl 0308.62056

References:

[1]	Aitken, A.C., On least squares and linear combination of observations, Proc. royal soc. Edinburgh A, 55, 42-48, (1934) · Zbl 0011.26603
[2]	G. Alpargu, G.P.H. Styan, A third bibliography on the Frucht-Kantorovich Inequality and on some related inequalities, in: S. Jensen, G.P.H. Styan (Eds.), The Seventh International Workshop on Matrices and Statistics, Ft. Lauderdale, December 11-14, 1998, McGill University, Montreal, pp. 17-26
[3]	Bloomfield, P.; Watson, G.S., The inefficiency of least squares, Biometrika, 62, 121-128, (1975) · Zbl 0308.62056
[4]	Durbin, J.; Kendall, M.G., The geometry of estimation, Biometrika, 38, 150-158, (1951) · Zbl 0045.40904
[5]	C.F. Gauss (1821). See J. Bertrand (1888) Calcul des probabilités, or A. Borsh, P. Simon Abhandlangen zur Methode der kleinsten Quadratic von Carl Friedrich Gauss, Würzburg, 1887
[6]	Gustafson, K.E., The angle of an operator and positive operator products, Bull. AMS, 74, 488-492, (1968) · Zbl 0172.40702
[7]	Gustafson, K.E., A min – max theorem, Notices AMS, 15, 799, (1968)
[8]	Gustafson, K.E., Matrix trigonometry, Linear algebra appl., 217, 117-140, (1995) · Zbl 0826.15022
[9]	Gustafson, K.E., Lectures on computational fluid dynamics, mathematical physics, and linear algebra, (1997), World Scientific Singapore
[10]	Gustafson, K.E., Operator trigonometry of iterative methods, Numer. linear algebra appl., 4, 333-347, (1997) · Zbl 0889.65030
[11]	Gustafson, K.E., The geometrical meaning of the kantorovich – wielandt inequalities, Linear algebra appl., 296, 143-151, (1999) · Zbl 0938.47026
[12]	K.E. Gustafson, A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin, Comput. Technol. 4 (Novosibirsk), (1999) pp. 73-83 · Zbl 0928.65028
[13]	Gustafson, K.E., Parallel computing forty years ago, Math. comput. simulation, 51, 47-62, (1999)
[14]	Gustafson, K.E., An extended operator trigonometry, Linear algebra appl., 319, 117-135, (2000) · Zbl 0969.15013
[15]	Gustafson, K.E.; Rao, D.K.M., Numerical range: the field of values of linear operators and matrices, (1997), Springer Berlin
[16]	Hannan, E.J., Multiple time series, (1970), Wiley New York · Zbl 0211.49804
[17]	Horn, R.A.; Johnson, C.R., Matrix analysis, (1990), Cambridge UK · Zbl 0704.15002
[18]	Knott, M., On the minimum efficiency of least squares, Biometrika, 62, 129-132, (1975) · Zbl 0308.62057
[19]	Liu, S., Efficiency comparisons between two estimators based on matrix determinant Kantorovich-type inequalities, Metrika, 51, 145-155, (2000) · Zbl 1093.62555
[20]	Magnus, J.R.; Neudecker, H., Matrix differential calculus with applications in statistics and econometrics, (1999), Wiley Chichester · Zbl 0912.15003
[21]	Plackett, R.I., A historical note on the method of least squares, Biometrika, 36, 458-460, (1949) · Zbl 0041.46802
[22]	Puntanen, S.; Styan, G., The equality of the ordinary least squares estimator and the best linear unbiased estimator, Amer. statist., 43, 153-161, (1989)
[23]	Rao, C.R., Estimation of variance covariance components-MINQUE theory, J. multivariate anal., 1, 257-275, (1971)
[24]	Rao, C.R.; Rao, M.B., Stationary values of the product of two Rayleigh quotients: homologous canonical correlations, Sankhya: indianj.statist., 49B, 113-125, (1987) · Zbl 0639.62051
[25]	Rao, C.R.; Rao, M.B., Matrix algebra and its applications to statistics and econometrics, (1998), World Scientific Singapore · Zbl 0915.15001
[26]	Searle, S.R., The infusion of matrices into statistics, IMAGE: bull. internat. linear algebra soc., 24, 25-32, (2000)
[27]	Wang, S.G.; Chow, S.C., Advanced linear models, (1994), Marcel-Dekker New York
[28]	Wang, S.G.; Ip, W.C., A matrix verson of the Wielandt inequality and its applications to statistics, Linear algebra appl., 296, 171-181, (2000)
[29]	Watson, G.S., Serial correlation in regression analysis I, Biometrika, 42, 327-341, (1955) · Zbl 0068.33201
[30]	Watson, G.S.; Alpargu, G.; Styan, G.P.H., Some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor, Linear algebra appl., 264, 13-54, (1997) · Zbl 0948.62046