A note on the multivariate linear model with constraints on the dependent vector

N. I. Fisher; H. M. Hudson

doi:10.1111/j.1467-842X.1980.tb01156.x

A note on the multivariate linear model with constraints on the dependent vector

N. I. Fisher^*, H. M. Hudson

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

It is shown that the least squares estimators of B and Σ in the multivariate linear model {EY_i=X₁B, D(Y_i) =Σ, 1 ≤i≤n, Y₁Y_n uncorrelated} subject to the constraints Y_iM=X_iN are just the usual least squares estimators B̂= (X'X)‐¹X'Y and ΣC = 1/n(Y‐XB̂)(Y‐XB̂) in the unconstrained model where Σ has full rank. Tests of hypotheses concerning B are discussed for situations in which each Y_i has a multivariate normal distribution, and examples of the applicability of the model reviewed.

Original language	English
Pages (from-to)	75-78
Number of pages	4
Journal	Australian Journal of Statistics
Volume	22
Issue number	1
DOIs	https://doi.org/10.1111/j.1467-842X.1980.tb01156.x
Publication status	Published - 1980

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10.1111/j.1467-842X.1980.tb01156.x

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abstract = "It is shown that the least squares estimators of B and Σ in the multivariate linear model {EYi=X1B, D(Yi) =Σ, 1 ≤i≤n, Y1Yn uncorrelated} subject to the constraints YiM=XiN are just the usual least squares estimators {\^B}= (X'X)‐1X'Y and ΣC = 1/n(Y‐X{\^B})(Y‐X{\^B}) in the unconstrained model where Σ has full rank. Tests of hypotheses concerning B are discussed for situations in which each Yi has a multivariate normal distribution, and examples of the applicability of the model reviewed.",

author = "Fisher, {N. I.} and Hudson, {H. M.}",

year = "1980",

doi = "10.1111/j.1467-842X.1980.tb01156.x",

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AU - Hudson, H. M.

PY - 1980

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AB - It is shown that the least squares estimators of B and Σ in the multivariate linear model {EYi=X1B, D(Yi) =Σ, 1 ≤i≤n, Y1Yn uncorrelated} subject to the constraints YiM=XiN are just the usual least squares estimators B̂= (X'X)‐1X'Y and ΣC = 1/n(Y‐XB̂)(Y‐XB̂) in the unconstrained model where Σ has full rank. Tests of hypotheses concerning B are discussed for situations in which each Yi has a multivariate normal distribution, and examples of the applicability of the model reviewed.

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JO - Australian Journal of Statistics

JF - Australian Journal of Statistics

IS - 1

ER -

A note on the multivariate linear model with constraints on the dependent vector

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