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# 统计代写|广义线性模型代写Generalized linear model代考|STAT458 Small values of the total sample size,N

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## 统计代写|广义线性模型代写Generalized linear model代考|Small values of the total sample size,N

Sections $3.6$ and $4.8$ considered the situation of small sample sizes for data from a Bernoulli distribution. Here, suppose that, at the $i$ th support point $\boldsymbol{x}{i}(i=1, \ldots, s), n{i}$ observations are taken from a Poisson distribution with mean $\lambda_{i}=\exp \left(\eta_{i}\right)=\exp \left(\boldsymbol{x}{i}^{\top} \boldsymbol{\beta}\right)$. If $y{i j}$ denotes the $j$ th observation at $\boldsymbol{x}{i}$, then the likelihood of the overall sample is \begin{aligned} L\left(\boldsymbol{\beta} ; y{11}, \ldots, y_{s n_{s}}\right) &=\mathrm{e}^{-\lambda_{1}} \frac{\lambda_{1}^{y_{11}}}{y_{11} !} \times \ldots \times \mathrm{e}^{-\lambda_{1}} \frac{\lambda_{1}^{y_{1 n_{1}}}}{y_{1 n_{1}} !} \times \ldots \times \mathrm{e}^{-\lambda_{s}} \frac{\lambda_{s}^{y_{s n_{s}}}}{y_{s n_{s}} !} \ &=\exp \left(-\sum_{i=1}^{s} n_{i} \lambda_{i}\right) \prod_{i=1}^{s} \lambda_{i}^{y_{i}} /\left(\prod_{i=1}^{s} \prod_{j=1}^{n_{i}} y_{i j} !\right) \end{aligned}
which implies that the $\log$ likelihood, $\ell\left(\boldsymbol{\beta} ; y_{11}, \ldots, y_{s n_{s}}\right)$, is given by
$$\ell\left(\boldsymbol{\beta} ; y_{11}, \ldots, y_{s n_{s}}\right)=-\sum_{i=1}^{s} n_{i} \lambda_{i}+\sum_{i=1}^{s} y_{i} \cdot \ln \left(\lambda_{i}\right)-\sum_{i=1}^{s} \sum_{j=1}^{n_{i}} \ln \left(y_{i j} !\right)$$
In addition, by $(1.21)$, the $(j, k)$ element of the matrix $\mathcal{I}$ equals
$$\mathcal{I}{j k}=\sum{i=1}^{s} n_{i} \frac{f_{i j} f_{i k}}{\operatorname{var}\left(Y_{i}\right)}\left(\frac{\partial \mu_{i}}{\partial \eta_{i}}\right)^{2}=\sum_{i=1}^{s} n_{i} \frac{f_{i j} f_{i k}}{\lambda_{i}} \lambda_{i}^{2} \quad j, k \in{0, \ldots, p-1}$$

## 统计代写|广义线性模型代写Generalized linear model代考|Modelling data from a multinomial distribution

In a multinomial experiment, the response variable $Y$ takes a value from one of a fixed number of categories. These categories may be nominal in nature (the categories are labels that cannot be put in a meaningful order), or ordinal (where there is a meaningful ordering). Examples of nominal categories are the political parties for which a person may vote. These parties may be listed in alphabetical order, but this is not a meaningful order with regard to (say) the parties’ policies. If it were possible to order these parties from “most extreme left-wing views” to “most extreme right-wing views,” this could be a meaningful ordering. Examples of ordinal categories are the severities of a disease suffered by patients attending an out-patient clinic: mild, moderate, and severe.
If there are $k$ categories, they are frequently numbered from 1 to $k$, whether or not this ordering is meaningful.

Suppose that an experiment is carried out independently $n$ times and that, on each occasion, the result will be exactly one of these $k$ categories.

Let $Y_{i}(i=1, \ldots, k)$ be the number of times that an observation in category $i$ is observed. The random variables $Y_{1}, \ldots, Y_{k}$ are said to have a multinomial distribution
$$\left(Y_{1}, \ldots, Y_{k}\right) \sim \operatorname{Multinomial}\left(n ; \pi_{1}, \ldots, \pi_{k}\right)$$
where $\pi_{i}(i=1, \ldots, k)$ is the probability that an outcome is in category i. The observed values $y_{1}, \ldots, y_{k}$ sum to $n$. The $\pi_{i}$ satisfy
$$\pi_{i}>0(i=1, \ldots, k) \quad \text { and } \quad \sum_{i=1}^{k} \pi_{i}=1$$

## 统计代写广义线性模型代写Generalized linear model代考|Small values of the total sample size,N

$$L\left(\boldsymbol{\beta} ; y 11, \ldots, y_{s n_{s}}\right)=\mathrm{e}^{-\lambda_{1}} \frac{\lambda_{1}^{y_{11}}}{y_{11} !} \times \ldots \times \mathrm{e}^{-\lambda_{1}} \frac{\lambda_{1}^{y_{1 r_{1}}}}{y_{1 n_{1}} !} \times \ldots \times \mathrm{e}^{-\lambda_{s}} \frac{\lambda_{s}^{y_{s n_{s}}}}{y_{s n_{s}} !} \quad=\exp \left(-\sum_{i=1}^{s} n_{i} \lambda_{i}\right) \prod_{i=1}^{s} \lambda_{i}^{y_{i}} /\left(\prod_{i=1}^{s} \prod_{j=1}^{n_{i}} y_{i j} !\right)$$

$$\ell\left(\beta ; y_{11}, \ldots, y_{s n_{s}}\right)=-\sum_{i=1}^{s} n_{i} \lambda_{i}+\sum_{i=1}^{s} y_{i} \cdot \ln \left(\lambda_{i}\right)-\sum_{i=1}^{s} \sum_{j=1}^{n_{i}} \ln \left(y_{i j} !\right)$$

$$\mathcal{I} j k=\sum i=1^{s} n_{i} \frac{f_{i j} f_{i k}}{\operatorname{var}\left(Y_{i}\right)}\left(\frac{\partial \mu_{i}}{\partial \eta_{i}}\right)^{2}=\sum_{i=1}^{s} n_{i} \frac{f_{i j} f_{i k}}{\lambda_{i}} \lambda_{i}^{2} \quad j, k \in 0, \ldots, p-1$$

## 统计代写广义线性模型代写Generalized linear model代考|Modelling data from a multinomial distribution

$$\left(Y_{1}, \ldots, Y_{k}\right) \sim \operatorname{Multinomial}\left(n ; \pi_{1}, \ldots, \pi_{k}\right)$$

$$\pi_{i}>0(i=1, \ldots, k) \quad \text { and } \quad \sum_{i=1}^{k} \pi_{i}=1$$

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