zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Mean squared error matrix comparisons between biased estimators - an overview of recent results. (English) Zbl 0703.62066
Summary: We give a systematic report on mean squared error marix comparisons of competing biased estimators. Our approach is quite general: The parameter vector to be estimated is assumed to belong to a subset of the p- dimensional Euclidean space. However, to illustrate our results, we shall pay attention to the linear regression model where biased estimation is very popular. Especially we are interested in generalized ridge and restricted least squares estimation.
MSC:
62H12Multivariate estimation
62J05Linear regression
62J07Ridge regression; shrinkage estimators
References:
[1]BAKSALARY, J.K. and KALA, R. (1983): ”Partial orderings between matrices one of which is of rank one”, Bulletin of the Polish Academy of Sciences, Mathematics 31, 5–7.
[2]BAKSALARY, J.K., LISKI, E.P. and TRENKLER, G. (1989): ”Mean square error matrix improvements and admissibility of linear estimators”, Journal of Statistical Planning and Inference 23, 313–325. · Zbl 0685.62052 · doi:10.1016/0378-3758(89)90075-X
[3]BEKKER, P.A. and NEUDECKER, H. (1989): ”Albert’s theorem applied to problems of efficiency and MSE superiority”, Statistica Neerlandica 43, 157–167. · Zbl 0731.62105 · doi:10.1111/j.1467-9574.1989.tb01256.x
[4]BEN-ISRAEL, A. and GREVILLE, T.N.E. (1974): ”Generalized inverses: Theory and applications”, John Wiley, New York.
[5]CAMPBELL, S.L. and MEYER, C.D. (1979): ”Generalized inverses of linear transformations”, Pitman, London.
[6]CHAWLA, J.S. (1988): ”A note on general ridge estimator”, Communications in Statistics, A 17, 739–744. · Zbl 0664.62069 · doi:10.1080/03610928808829652
[7]CHAWLA, J.S. (1988): ”On necessary and sufficient conditions for superiority of ridge estimator over least squares estimator”, Statistical Papers 29, 227–230. · Zbl 0649.62062 · doi:10.1007/BF02924527
[8]FAREBROTHER, R.W. (1976): ”Further results on the mean square error of ridge regression”, Journal of the Royal Statistica Society B 38, 248–250.
[9]OBENCHAIN, R.I. (1975): ”Ridge analysis following a preliminary test of the shrunken hypothesis”, Technometrics 17, 431–445, (with discussion) · Zbl 0324.62050 · doi:10.2307/1268429
[10]PERLMAN, M.D. (1972): ”Reduced mean square error estimation for several parameters”, Sankhya B 34, 89–92.
[11]RAO, C.R. (1973): ”Linear statistical inference and its applications”, John Wiley, New York.
[12]TERÄSVIRTA, T. (1982): ”Superiority comparisons of homogeneous linear estimators”, Communications in Statistics A 11, 1595–1601. · Zbl 0506.62052 · doi:10.1080/03610928208828333
[13]TERÄSVIRTA, T. (1983): ”Strong superiority of heterogeneous estimators”, ASA Proceedings of Business and Economic Statistics Section, 135–139.
[14]TOUTENBURG, H. (1982): ”Prior information in linear models”, John Wiley, New York.
[15]TOUTENBURG, H. (1986): ”Weighted mixed regression with applications to regressor’s nonresponse. I: Theoretical results”, Preprint IMath., 29/86, Berlin.
[16]TOUTENBURG, H. and STAHLECKER, P. (1989): ”Report on MSE-comparisons between biased restricted least squares estimators”, Universität Dortmund, Fachbereich Statistik, Forschungsbericht 89/15.
[17]TOUTENBURG, H. (1989): ”Mean-square-error-comparisons between restricted least squares, mixed and weighted mixed estimators”, Forschungsbericht Nr. 89/12, Universität Dortmund.
[18]TOUTENBURG, H. and SCHAFFRIN, B. (1990): ”Weighted mixed regression”, Proceedings of the GAMM-Conference at Karlsruhe, ZAMM, 70, 4–6.
[19]TRENKLER, D. (1986): ”Superiority comparisons of generalized ridge estimators”, Mathematica Japonica 31, 301–307.
[20]TRENKLER, D. (1986): ”Verallgemeinerte Ridge-Regression”, Mathematical Systems in Economics, Vol. 104, Anton Hain, Meisenheim.
[21]TRENKLER, G. (1981): ”Biased estimators in the linear regression model”, Mathematical Systems in Economics, Vol. 58, Gunn & Hain, Cambridge Massachusetts.
[22]TRENKLER, G. (1985): ”Mean square error matrix comparisons of estimators in linear regression”, Communications in Statistics A 14, 2495–2509. · Zbl 0594.62075 · doi:10.1080/03610928508829058
[23]TRENKLER, G. (1987): ”Mean square error matrix comparisons between biased restricted least squares estimators”, Sankhya A 49, 96–104.
[24]TRENKLER, G. and PORDZIK, P. (1988): ”Pre-Test Estimation in the Linear Regression Model Based on Competing Restrictions”, Submitted to publication.
[25]TRENKLER, G. and TRENKLER, D. (1988): ”A note on superiority comparisons of homogenous linear estimators”, Communications in Statistics A 12, 799–808. · Zbl 0523.62066 · doi:10.1080/03610928308828496