New PDF release: Analysis of variance for random models, vol.2: Unbalanced

By Sahai H., Ojeda M.M.

ISBN-10: 0817632298

ISBN-13: 9780817632298

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P. 13) j = 1, . . 14) and E(Lσ 2 σ 2 ) = − = − ∂ 2 V −1 1 ∂ 2 n|V | 1 tr E(Y − Xα)(Y − Xα) − 2 ∂σi2 ∂σj2 2 ∂σi2 ∂σj2 1 ∂ 2 n|V | 1 V ∂ 2 V −1 tr − 2 ∂σi2 ∂σj2 2 ∂σi2 ∂σj2 i, j = 1, . . , p. 16) 32 Chapter 10. 17) . 17), we obtain 2 ∂ 2 { n|V |} ∂V ∂V −1 ∂ V = tr V − V −1 2 V −1 2 2 2 2 2 ∂σi ∂σj ∂σi ∂σj ∂σj ∂σi . 18) with respect to σi2 , we obtain ∂ 2 V −1 ∂V ∂V ∂ 2V = V −1 2 V −1 2 V −1 − V −1 2 2 V −1 2 2 ∂σi ∂σj ∂σj ∂σi ∂σi ∂σj + V −1 ∂V −1 ∂V −1 V V . 20) by V and taking the trace yields tr V ∂ 2 V −1 ∂σi2 ∂σj2 = tr ∂ 2V ∂V −1 ∂V −1 V V − V −1 ∂σj2 ∂σi2 ∂σi2 ∂σj2 + ∂V −1 ∂V −1 V V ∂σi2 ∂σj2 = tr 2V −1 2 ∂V −1 ∂V −1 ∂ V .

Restricted Maximum Likelihood Estimation Hence, letting αˆ and σˆ 2 denote the ML estimators of α and σ 2 , their variance-covariance matrix is given by ⎡ ⎤ ⎡ . 2 ˆ σˆ ) ⎥ ⎢ −E(Lαα ) .. −E(Lασ 2 ) ˆ . Cov(α, ⎢ Var(α) ⎢ ⎥=⎢ ··· ··· ··· ··· ⎣ ⎦ ⎣ . . 2 2 ˆ σˆ ) . Var(σˆ ) Cov(α, −E(Lασ 2 ) . −E(Lσ 2 σ 2 ) ⎡ ⎤−1 . −1 X .. 0 V X ⎢ ⎥ ⎢ ⎥ ··· ··· =⎢ ⎥ . ⎣ ⎦ .. 1 ∂V ∂V −1 −1 . 22) and Var(σˆ ) = 2 tr V 2 −1 ∂V −1 ∂V , i, j = 1, . . , p V ∂σi2 ∂σj2 −1 . 22) represents the lower bound for the variance-covariance matrix of unbiased estimators.

5 that the SSP method involved the construction of an unbiased estimate of the square of the population mean. As proposed by Koch (1967b), this procedure can be used to obtain an unbiased estimate of the mean itself by proceeding as follows. 1) where E(µˆ 2 ) = E{Q(Y )} = µ2 . Now, consider the set of transformations on the data obtained by adding a constant θ to each observation. After making such a transformation, the SSP method is used to obtain the unbiased estimator of the square of the population mean of the transformed data, which will have the form Q(Y + θ1).

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Analysis of variance for random models, vol.2: Unbalanced data by Sahai H., Ojeda M.M.


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