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## 统计代写|广义线性模型代写Generalized linear model代考|Maximum penalised likelihood estimation

Recall from Section $1.5$ that the model parameters $\beta_{0}, \ldots, \beta_{p-1}$ are generally estimated by the ML estimators, which maximise the likelihood given in (1.16). For very large values of $N$, the vector of estimators, $\hat{\boldsymbol{\beta}}$, satisfies $\mathrm{E}(\hat{\boldsymbol{\beta}})=\boldsymbol{\beta}$ and $\operatorname{cov}(\hat{\boldsymbol{\beta}})=\boldsymbol{I}^{-1}$.
If $N$ is “small,” these results may be inaccurate. In particular, the result for $\operatorname{cov}(\hat{\boldsymbol{\beta}})$, on which the calculation of a locally D-optimal design is based, may lead to a design which is not as desirable as hoped. To reduce the bias of each $\hat{\beta}{i}$ (discrepancy between $\mathrm{E}\left(\hat{\beta}{i}\right)$ and the true value $\left.\beta_{i}\right), i=0, \ldots, p-1$, Firth (1993) introduced the maximum penalised likelihood (MPL), $L^{}\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)$, whose logarithm (denoted by $\ell^{}$ ) is given by
\begin{aligned} \ell^{*}\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right) &=\ln L\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)+\frac{1}{2} \operatorname{det}\left[\mathcal{I}\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)\right] \ &=\ell\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)+\frac{1}{2} \operatorname{det}\left[\mathcal{I}\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)\right] \end{aligned}
Example 4.8.1. Consider a Bernoulli random variable, $Y$, with $P\left(Y_{i}=1\right)=\pi_{i}$ $(i=1,2)$. Model $\pi_{i}$ using $\ln \left[\pi_{i} /\left(1-\pi_{i}\right)\right]=\eta_{i}=\beta_{0}+\beta_{1} x_{i}$. Take $n_{i}$ observations at $x=x_{i}$, and write $y_{i}$ for the total of those observations $(i=1,2)$. This scenario was investigated in detail by Russell et al. (2009a).

The $M P L$ estimators of $\beta_{0}$ and $\beta_{1}$ are the values, $\beta_{0}^{}$ and $\beta_{1}^{}$, that maximise $\ell^{}$. $I$ will occasionally write them as $\beta_{0}^{}\left(y_{1}, y_{2}\right)$ and $\beta_{1}^{}\left(y_{1}, y_{2}\right)$ to show their dependence on $y_{1}$ and $y_{2}$. They can be shown to be \begin{aligned} &\beta_{1}^{}\left(y_{1}, y_{2}\right)=\frac{\ln \left[\left(y_{1}+0.5\right) /\left(n_{1}-y_{1}+0.5\right)\right]-\ln \left[\left(y_{2}+0.5\right) /\left(n_{2}-y_{2}+0.5\right)\right]}{x_{1}-x_{2}} \ &\beta_{0}^{*}\left(y_{1}, y_{2}\right)=\frac{x_{1} \ln \left[\left(y_{2}+0.5\right) /\left(n_{2}-y_{2}+0.5\right)\right]-x_{2} \ln \left[\left(y_{1}+0.5\right) /\left(n_{1}-y_{1}+0.5\right)\right]}{x_{1}-x_{2}} \end{aligned}

## 统计代写|广义线性模型代写Generalized linear model代考|IMSE-optimality

To compare designs on the basis of plots of $\operatorname{MSE}(x)$ vs $x$, it is usual to compare the areas that are beneath the curves and above the horizontal axis. That is, for each plot we consider
$$\int_{-\infty}^{\infty} \operatorname{MSE}(x) d x \approx \int_{x_{0.0001}}^{x_{0.9999}} \operatorname{MSE}(x) d x,$$
which is known as the integrated mean square error (IMSE). From amongst a set of candidate designs, the design for which the IMSE is least is said to be IMSE-optimal.
As it is not possible to write an expression for $\operatorname{MSE}(x)$ in terms of $x$, it becomes necessary to evaluate the integral in (4.26) using numerical integration. As Figures $4.9$ and $4.10$ suggest that the curve $\operatorname{MSE}(x)$ is reasonably smooth, Simpson’s rule (see Section 2.5) will be used to evaluate (4.26).

## 统计代写|广义线性模型代写Generalized linear model代考|Maximum penalised likelihood estimation

$\ell^{}\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)=\ln L\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)+\frac{1}{2} \operatorname{det}\left[\mathcal{I}\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)\right] \quad=\ell\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)+\frac{1}{2} \operatorname{det}\left[\mathcal{I}\left(\boldsymbol{\beta} ; y_{1}, \ldots, y_{N}\right)\right]$ 例 4.8.1。考虑一个伯努利随机変量, $Y$ ，和 $P\left(Y_{i}=1\right)=\pi_{i}(i=1,2)$. 模型 $\pi_{i}$ 使用 $\ln \left[\pi_{i} /\left(1-\pi_{i}\right)\right]=\eta_{i}=\beta_{0}+\beta_{1} x_{i}$. 拿 $n_{i}$ 观察在 $x=x_{i}$ ，和写 $y_{i}$ 对于这些对䕓的总数 $(i=1,2)$. Russell等人详细研究了这祌情况。 (2009a) 。 这 $M P L$ 估计者 $\beta_{0}$ 和 $\beta_{1}$ 是价值观， $\beta_{0}$ 和 $\beta_{1}$ ，最大化 $\ell$. I偶尔会把它们写成 $\beta_{0}\left(y_{1}, y_{2}\right)$ 和 $\beta_{1}\left(y_{1}, y_{2}\right)$ 显示他们的依赖 $y_{1}$ 和 $y_{2}$. 它们 可以显示为 $\beta_{1}\left(y_{1}, y_{2}\right)=\frac{\ln \left[\left(y_{1}+0.5\right) /\left(n_{1}-y_{1}+0.5\right)\right]-\ln \left[\left(y_{2}+0.5\right) /\left(n_{2}-y_{2}+0.5\right)\right]}{x_{1}-x_{2}} \quad \beta_{0}^{}\left(y_{1}, y_{2}\right)=\frac{x_{1} \ln \left[\left(y_{2}+0.5\right) /\left(n_{2}-y_{2}+0.5\right)\right]-x_{2} \ln \left[\left(y_{1}+0 . t\right.\right.}{x_{1}-x_{2}}$

## 统计代写|广义线性模型代写Generalized linear model代考|MSE-optimality

$$\int_{-\infty}^{\infty} \operatorname{MSE}(x) d x \approx \int_{x_{00001}}^{x_{0.9000}} \operatorname{MSE}(x) d x$$

## MATLAB代写

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