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统计代写|广义线性模型代写Generalized linear model代考|STAT3030 Maximum Likelihood

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统计代写|广义线性模型代写Generalized linear model代考|Point Estimation

Under a Gaussian mixed model, the distribution of $y$ is given by (1.4), which has a joint pdf

$$f(y)=\frac{1}{(2 \pi)^{n / 2}|V|^{1 / 2}} \exp \left{-\frac{1}{2}(y-X \beta)^{\prime} V^{-1}(y-X \beta)\right},$$
where $n$ is the dimension of $y$. Thus, the log-likelihood function is given by
$$l(\beta, \theta)=c-\frac{1}{2} \log (|V|)-\frac{1}{2}(y-X \beta)^{\prime} V^{-1}(y-X \beta),$$
where $\theta$ represents the vector of all of the variance components (involved in $V$ ) and $c$ is a constant. By differentiating the log-likelihood with respect to the parameters (see Appendix A), we obtain the following:
\begin{aligned} \frac{\partial l}{\partial \beta}=X^{\prime} V^{-1} y-X^{\prime} V^{-1} X \beta, \ \frac{\partial l}{\partial \theta_r}=\frac{1}{2}\left{(y-X \beta)^{\prime} V^{-1} \frac{\partial V}{\partial \theta_r} V^{-1}(y-X \beta)-\operatorname{tr}\left(V^{-1} \frac{\partial V}{\partial \theta_r}\right)\right}, \ \quad r=1, \ldots, q, \end{aligned}
where $\theta_r$ is the $r$ th component of $\theta$, which has dimension $q$. The standard procedure of finding the ML estimator, or MLE, is to solve the ML equations $\partial l / \partial \beta=$ $0, \partial l / \partial \theta=0$. However, finding the solution may not be the end of the story. In other words, the solution to (1.7) and (1.8) is not necessarily the MLE. See Sect. $1.8$ for further discussion. Let $p$ be the dimension of $\beta$. For simplicity, we assume that $X$ is of full (column) rank; that is, $\operatorname{rank}(X)=p$ (see Sect. 1.8). Let $(\hat{\beta}, \hat{\theta})$ be the MLE. From (1.7) one obtains
$$\hat{\beta}=\left(X^{\prime} \hat{V}^{-1} X\right)^{-1} X^{\prime} \hat{V}^{-1} y,$$
where $\hat{V}=V(\hat{\theta})$, that is, $V$ with the variance components involved replaced by their MLE. Thus, once the MLE of $\theta$ is found, the MLE of $\beta$ can be calculated by the “closed-form” expression (1.9). As for the MLE of $\theta$, by (1.7) and (1.8), it is easy to show that it satisfies
$$y^{\prime} P \frac{\partial V}{\partial \theta_r} P y=\operatorname{tr}\left(V^{-1} \frac{\partial V}{\partial \theta_r}\right), \quad r=1, \ldots, q,$$
where
$$P=V^{-1}-V^{-1} X\left(X^{\prime} V^{-1} X\right)^{-1} X^{\prime} V^{-1} \text {. }$$

统计代写|广义线性模型代写Generalized linear model代考|Asymptotic Covariance Matrix

Under suitable conditions (see Sect. $1.8$ for discussion), the MLE is consistent and asymptotically normal with the asymptotic covariance matrix equal to the inverse of the Fisher information matrix. Let $\psi=\left(\beta^{\prime}, \theta^{\prime}\right)^{\prime}$. Then, under regularity conditions, the Fisher information matrix has the following expressions:
$$\operatorname{Var}\left(\frac{\partial l}{\partial \psi}\right)=-\mathrm{E}\left(\frac{\partial^2 l}{\partial \psi \partial \psi^{\prime}}\right)$$
By (1.7) and (1.8), further expressions can be obtained for the elements of (1.12). For example, assuming that $V$ is twice continuously differentiable (with respect to the components of $\theta$ ), then, using the results of Appendices $B$ and $C$, it can be shown (Exercise 1.6) that
\begin{aligned} \mathrm{E}\left(\frac{\partial^2 l}{\partial \beta \partial \beta^{\prime}}\right) &=-X^{\prime} V^{-1} X, \ \mathrm{E}\left(\frac{\partial^2 l}{\partial \beta \partial \theta_r}\right) &=0, \quad 1 \leq r \leq q \ \mathrm{E}\left(\frac{\partial^2 l}{\partial \theta_r \partial \theta_s}\right) &=-\frac{1}{2} \operatorname{tr}\left(V^{-1} \frac{\partial V}{\partial \theta_r} V^{-1} \frac{\partial V}{\partial \theta_s}\right), \quad 1 \leq r, s \leq q . \end{aligned}
It follows that (1.12) does not depend on $\beta$, and therefore may be denoted by $I(\theta)$, as we do in the sequel.

统计代写|广义线性模型代写Generalized linear model代考|Point Estimation

〈left 的分隔符缺失或无法识别

$$l(\beta, \theta)=c-\frac{1}{2} \log (|V|)-\frac{1}{2}(y-X \beta)^{\prime} V^{-1}(y-X \beta),$$

〈left 的分隔符缺失或无法识别

$$\hat{\beta}=\left(X^{\prime} \hat{V}^{-1} X\right)^{-1} X^{\prime} \hat{V}^{-1} y,$$

（1.9）计算。至于 MLE $\theta$ ，由（1.7）和（1.8），很容易证明它满足
$$y^{\prime} P \frac{\partial V}{\partial \theta_r} P y=\operatorname{tr}\left(V^{-1} \frac{\partial V}{\partial \theta_r}\right), \quad r=1, \ldots, q,$$

$$P=V^{-1}-V^{-1} X\left(X^{\prime} V^{-1} X\right)^{-1} X^{\prime} V^{-1} .$$

统计代写|广义线性模型代写Generalized tinear model代考|Asymptotic Covariance Matrix

$$\operatorname{Var}\left(\frac{\partial l}{\partial \psi}\right)=-\mathrm{E}\left(\frac{\partial^2 l}{\partial \psi \partial \psi^{\prime}}\right)$$

$$\mathrm{E}\left(\frac{\partial^2 l}{\partial \beta \partial \beta^{\prime}}\right)=-X^{\prime} V^{-1} X, \mathrm{E}\left(\frac{\partial^2 l}{\partial \beta \partial \theta_r}\right) \quad=0, \quad 1 \leq r \leq q \mathrm{E}\left(\frac{\partial^2 l}{\partial \theta_r \partial \theta_s}\right)=-\frac{1}{2} \operatorname{tr}\left(V^{-1} \frac{\partial V}{\partial \theta_r} V^{-1} \frac{\partial V}{\partial \theta_s}\right), \quad 1 \leq r, s \leq q$$

MATLAB代写

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