线性混合模型 Linear Mixed Models STATS5054_1/STATS4045_1

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The methods discussed above for the full parameter vector now directly carry over to calculate local influences on the geometric surface defined by $\mathrm{LD}{1}(\omega)$. We now partition $\ddot{L}$ as $$\ddot{L}=\left(\begin{array}{ll} \ddot{L}{11} & \ddot{L}{12} \ \ddot{L}{21} & \ddot{L}{22} \end{array}\right),$$ according to the dimensions of $\theta{1}$ and $\theta_{2}$. Cook (1986) has then shown that the local influence on the estimation of $\theta_{1}$, of perturbing the model in the direction of a normalized vector $\boldsymbol{h}$, is given by
$$C_{h}\left(\boldsymbol{\theta}{1}\right)=2\left|\boldsymbol{h}^{\prime} \Delta^{\prime}\left[\ddot{L}^{-1}-\left(\begin{array}{cc} 0 & 0 \ 0 & \ddot{L}{22}^{-1} \end{array}\right)\right] \Delta \boldsymbol{h}\right|$$
Because all eigenvalues of the malrix
$$\left(\begin{array}{ll} \ddot{L}{11} & \ddot{L}{12} \ \ddot{L}{21} & \ddot{L}{22} \end{array}\right)\left(\begin{array}{cc} 0 & 0 \ 0 & \ddot{L}{22}^{-1} \end{array}\right)=\left(\begin{array}{cc} 0 & \ddot{L}{12} \ddot{L}_{22}^{-1} \ 0 & I \end{array}\right)$$

MATHS5077_1COURSE NOTES ：

A measure of influence, proposed , is then
$$\rho_{i}=-\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)^{\prime} \ddot{L}\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)$$$$\boldsymbol{\Delta}{i}=-\sum{j \neq i} \boldsymbol{\Delta}{j}=\ddot{L}{(i)}(\widehat{\boldsymbol{\theta}})\left(\hat{\boldsymbol{\theta}}{(i)}^{1}-\widehat{\boldsymbol{\theta}}\right)$$ such that expression (11.4) becomes $$C{i}=-2\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)^{\prime} \ddot{L}{(i)} \ddot{L}^{-1} \ddot{L}{(i)}\left(\hat{\boldsymbol{\theta}}-\hat{\boldsymbol{\theta}}{(i)}^{1}\right)$$