zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Improvement of the Liu estimator in linear regression model. (English) Zbl 1125.62055

Summary: In the presence of stochastic prior information, in addition to the sample, H. Thiel and A. S. Goldberger [Int. Econ. Rev. 2, 65–77 (1961)] introduced a mixed estimator β ^ for the parameter vector β in the standard multiple linear regression model (Y,Xβ,σ 2 I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by K. Liu [Commun. Stat., Theory Methods 22, No. 2, 393–402 (1993; Zbl 0784.62065)].

We introduce another new Liu type biased estimator, called stochastic restricted Liu estimator, β ^ srd for β, and discuss its efficiency. Necessary and sufficient conditions for the mean squared error matrix of the stochastic restricted Liu estimator β ^ srd to exceed the mean squared error matrix of the mixed estimator β ^ m will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the mixed estimator β ^ m and to the biased estimator introduced by Liu.

62H12Multivariate estimation
62J05Linear regression
[1]Akdeniz, F. and Kaçiranlar, S. (2001) More on the new biased estimator in linear regression. Sankhya: The Indian Journal of Statistics, 63B, 321–325.
[2]Baksalary, J. K. and Trenkler, G. (1991) Nonnegative and positive definiteness of matrices modified by two matrices of rank one, Linear algebra and its Applications, 151, 169–184. · Zbl 0728.15011 · doi:10.1016/0024-3795(91)90362-Z
[3]Liu, K. (1993) A new class of biased estimate in linear regression. Communication in Statistics–Theory and Methods, 22(2), 393–402. · Zbl 0784.62065 · doi:10.1080/03610929308831027
[4]Rao, C.R. and Toutenburg, H. (1995) Linear models, Least squares and Alternatives. Springer Verlag.
[5]Kaçiranlar, S., Sakallioğlu, S., Akdeniz, F., Styan, G.P.H. and Werner, H.J. (1999) A new biased estimator in linear regression and a detailed analysis of the widely analyzed dataset on Portland Cement. Sankhya: The Indian Journal of Statistics. 61B, 443–459.
[6]Thiel, H. and Goldberger, A.S. (1961) On pure and Mixed estimation in Economics. International Economic review, 2, 65–77. · doi:10.2307/2525589