Locally and uniformly best estimators in replicated regression model

Júlia Volaufová; Lubomír Kubáček

Aplikace matematiky (1983)

  • Volume: 28, Issue: 5, page 386-390
  • ISSN: 0862-7940

Abstract

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The aim of the paper is to estimate a function γ = t r ( D β β ' ) + t r ( C ∑ ) (with d , C known matrices) in a regression model ( Y , X β , ∑ ) with an unknown parameter β and covariance matrix ∑ . Stochastically independent replications Y 1 , ... , Y m of the stochastic vector Y are considered, where the estimators of X β and ∑ are Y ¯ = 1 m ∑ i = 1 m Y i and ∑ ^ = ( m - 1 ) - 1 ∑ i = 1 m ( Y i - Y ¯ ) ( Y i - Y ¯ ) ' , respectively. Locally and uniformly best inbiased estimators of the function γ , based on Y ¯ and ∑ ^ , are given.

How to cite

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Volaufová, Júlia, and Kubáček, Lubomír. "Locally and uniformly best estimators in replicated regression model." Aplikace matematiky 28.5 (1983): 386-390. <http://eudml.org/doc/15317>.

@article{Volaufová1983,
abstract = {The aim of the paper is to estimate a function $\gamma =tr(D\beta \beta ^\{\prime \})+tr(C\sum )$ (with $d, C$ known matrices) in a regression model $(Y, X\beta ,\sum )$ with an unknown parameter $\beta $ and covariance matrix $\sum $. Stochastically independent replications $Y_1,\ldots , Y_m$ of the stochastic vector $Y$ are considered, where the estimators of $X\beta $ and $\sum $ are $\bar\{Y\}=\frac\{1\}\{m\} \sum ^m _\{i=1\} Y_i$ and $\hat\{\sum \}=(m-1)^\{-1\} \sum ^m_\{i=1\}(Y_i-\bar\{Y\})(Y_i-\bar\{Y\})^\{\prime \}$, respectively. Locally and uniformly best inbiased estimators of the function $\gamma $, based on $\bar\{Y\}$ and $\hat\{\sum \}$, are given.},
author = {Volaufová, Júlia, Kubáček, Lubomír},
journal = {Aplikace matematiky},
keywords = {replicated regression model; best unbiased estimators; replicated regression model; best unbiased estimators},
language = {eng},
number = {5},
pages = {386-390},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Locally and uniformly best estimators in replicated regression model},
url = {http://eudml.org/doc/15317},
volume = {28},
year = {1983},
}

TY - JOUR
AU - Volaufová, Júlia
AU - Kubáček, Lubomír
TI - Locally and uniformly best estimators in replicated regression model
JO - Aplikace matematiky
PY - 1983
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 28
IS - 5
SP - 386
EP - 390
AB - The aim of the paper is to estimate a function $\gamma =tr(D\beta \beta ^{\prime })+tr(C\sum )$ (with $d, C$ known matrices) in a regression model $(Y, X\beta ,\sum )$ with an unknown parameter $\beta $ and covariance matrix $\sum $. Stochastically independent replications $Y_1,\ldots , Y_m$ of the stochastic vector $Y$ are considered, where the estimators of $X\beta $ and $\sum $ are $\bar{Y}=\frac{1}{m} \sum ^m _{i=1} Y_i$ and $\hat{\sum }=(m-1)^{-1} \sum ^m_{i=1}(Y_i-\bar{Y})(Y_i-\bar{Y})^{\prime }$, respectively. Locally and uniformly best inbiased estimators of the function $\gamma $, based on $\bar{Y}$ and $\hat{\sum }$, are given.
LA - eng
KW - replicated regression model; best unbiased estimators; replicated regression model; best unbiased estimators
UR - http://eudml.org/doc/15317
ER -

References

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  1. Jürgen Kleffe, C. R. Rao's MINQUE for replicated and multivariate observations, Lecture Notes in Statistics 2. Mathematical Statistics and Probability Theory. Proceedings Sixth International Conference. Wisla (Poland) 1978. Springer N. York, Heidelberg, Berlin 1979, 188-200. (1978) 
  2. Jürgen Kleffe, Júlia Volaufová, Optimality of the sample variance-covariance matrix in repeated measurement designs, (Submitted to Sankhyā). 
  3. C. R. Rao, Linear Statistical Inference and Its Applications, J. Wiley, N. York 1965. (1965) Zbl0137.36203MR0221616
  4. C. R. Rao S. K. Mitra, Generalized Inverse of Matrices and Its Applications, J. Wiley, N. York 1971. (1971) MR0338013
  5. R. Thrum J. Kleffe, Inequalities for moments of quadratic forms with applications to a.s. convergence, Math. Operationsforsch. Statistics Ser. Statistics (in print). MR0704788

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