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# 统计代写|广义线性模型代写Generalized linear model代考|More Empirical Demonstration

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## 统计代写|广义线性模型代写Generalized linear model代考|More Empirical Demonstration

To demonstrate the potential improvement of the new CMMP procedure, another simulation study was carried out. In Jiang et al. (2018), the authors showed that CMMP significantly outperforms the standard regression prediction (RP) method. On the other hand, the authors have not compared CMMP with mixed model prediction, such as the EBLUP (see Sect. 2.3.2), which is known to outperform RP as well. In the current simulation, we consider a case where there is no exact match between the new observation and a group in the training data, a situation that is practical. More specifically, the training data satisfy
$$y_{i j}=\beta_0+\beta_1 w_i+\alpha_i+\epsilon_{i j}$$
$i=1, \ldots, m, j=1, \ldots, n_i$, where $w_i$ is an observed, cluster-level covariate, $\alpha_i$ is a cluster-specific random effect, and $\epsilon_{i j}$ is an error. The random effects and errors are independent with $\alpha_i \sim N(0, G)$ and $\epsilon_{i j} \sim N(0, R)$. The new observation, on the other hand, satisfies
$$y_{\text {new }}=\beta_0+\beta_1 w_1+\alpha_1+\delta+\epsilon_{\text {new }}$$
where $\delta, \epsilon_{\text {new }}$ are independent with $\delta \sim N(0, D)$ and $\epsilon_{\text {new }} \sim N(0, R)$ and $\left(\delta, \epsilon_{\text {new }}\right)$ are independent with the training data. It is seen that, because of $\delta$, there is no exact match between the new random effect (which is $\alpha_1+\delta$ ) and one of the random effects $\alpha_i$ associated with the training data; however, the value of $D$ is small, $D=10^{-4}$; hence there is an approximate match between the new random effect and $\alpha_1$, the random effect associated with the first group in the training data.

We consider $m=50$. The $n_i$ are chosen according to one of the following four patterns:

1. $n_i=5,1 \leq i \leq m / 2 ; n_i=25, m / 2+1 \leq i \leq m$;
2. $n_i=50,1 \leq i \leq m / 2 ; n_i=250, m / 2+1 \leq i \leq m$;
3. $n_i=25,1 \leq i \leq m / 2 ; n_i=5, m / 2+1 \leq i \leq m$;
4. $n_i=250,1 \leq i \leq m / 2 ; n_i=50, m / 2+1 \leq i \leq m$.

## 统计代写|广义线性模型代写Generalized linear model代考|Prediction Interval

Prediction intervals are of substantial practical interest. Here, we follow the NER model (2.67), but with the additional assumption that the new error, $\epsilon_{\mathrm{n}, j}$ in (2.58), is distributed as $N(0, R)$, where $R$ is the same variance as that of $\epsilon_{i j}$ in (2.67). Still, it is not necessary to assume that $\alpha_{\text {new }}=\alpha_I$ has the same distribution, or even the same variance, as the $\alpha_i$ in (2.57). This would include both the matched and unmatched cases. Consider the following prediction interval for $\theta=x_{\mathrm{n}}^{\prime} \beta+\alpha_{\text {new }}$ :
$$\left[\hat{\theta}-z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}, \hat{\theta}+z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}\right] \text {, }$$
where $\hat{\theta}$ is the CMEP of $\theta, \hat{R}$ is the REML estimator of $R$, and $z_a$ is the critical value so that $\mathrm{P}\left(Z>z_a\right)=a$ for $Z \sim N(0,1)$. For a future observation, $y_{\mathrm{f}}$, we assume that it shares the same mixed effects as the observed new observations $y_{\mathrm{n}, j}, 1 \leq j \leq$ $n_{\text {new }}$ in $(2.58)$, that is,
$$y_{\mathrm{f}}=\theta+\epsilon_{\mathrm{f}}$$
where $\epsilon_{\mathrm{f}}$ is a new error that is distributed as $N(0, R)$ and independent with all of the $\alpha$ ‘s and other $\epsilon$ ‘s. Consider the following prediction interval for $y_{\mathrm{f}}$ :
$$\left[\hat{\theta}-z_{a / 2} \sqrt{\left(1+n_{\text {new }}^{-1}\right) \hat{R}}, \hat{\theta}+z_{a / 2} \sqrt{\left(1+n_{\text {new }}^{-1}\right) \hat{R}}\right],$$
where $\hat{\theta}, \hat{R}$ are the same as in (2.72). Under suitable conditions, it can be shown that the prediction intervals (2.72) and (2.74) have asymptotically the correct coverage probability. Furthermore, empirical results show that the CMMP-based prediction intervals are more accurate than the RP-based prediction intervals. See Jiang et al. (2018) for details.

## 计代写|广义线性模型代写Generalized linear model代考|More Empirical Demonstration

$$y_{i j}=\beta_0+\beta_1 w_i+\alpha_i+\epsilon_{i j}$$
$i=1, \ldots, m, j=1, \ldots, n_i$，其中$w_i$是观察到的集群级协变量，$\alpha_i$是特定于集群的随机效应，$\epsilon_{i j}$是一个误差。随机效应和误差与$\alpha_i \sim N(0, G)$和$\epsilon_{i j} \sim N(0, R)$无关。另一方面，新的观察结果满足了
$$y_{\text {new }}=\beta_0+\beta_1 w_1+\alpha_1+\delta+\epsilon_{\text {new }}$$

$n_i=5,1 \leq i \leq m / 2 ; n_i=25, m / 2+1 \leq i \leq m$；

$n_i=50,1 \leq i \leq m / 2 ; n_i=250, m / 2+1 \leq i \leq m$；

$n_i=25,1 \leq i \leq m / 2 ; n_i=5, m / 2+1 \leq i \leq m$；

$n_i=250,1 \leq i \leq m / 2 ; n_i=50, m / 2+1 \leq i \leq m$．

## 统计代写|广义线性模型代写Generalized linear model代考|Prediction Interval

$$\left[\hat{\theta}-z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}, \hat{\theta}+z_{a / 2} \sqrt{\frac{\hat{R}}{n_{\text {new }}}}\right] \text {, }$$

## MATLAB代写

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