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# 统计代写|广义线性模型代写Generalized linear model代考|Derivation of the Bernoulli model

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## 统计代写|广义线性模型代写Generalized linear model代考|Derivation of the Bernoulli model

The binary or Bernoulli response probability distribution is simplified from the binomial distribution. The binomial denominator $k$ is equal to 1 , and the outcomes $y$ are constrained to the values ${0,1}$. Again, $y$ may initially be represented in your dataset as 1 or 2 , or some other alternative set. If so, you must change your data to the above description.

However, some programs use the opposite default behavior: they use 0 to denote a success and 1 to denote a failure. Transferring data unchanged between Stata and a package that uses this other codification will result in fitted models with reversed signs on the estimated coefficients.
The Bernoulli response probability function is
$$f(y ; p)=p^y(1-p)^{1-y}$$
The binomial normalization (combination) term has disappeared, which makes the function comparatively simple.
In canonical (exponential) form, the Bernoulli distribution is written
$$f(y ; p)=\exp \left{y \ln \left(\frac{p}{1-p}\right)+\ln (1-p)\right}$$
Below you will find the various functions and relationships that are required to complete the Bernoulli algorithm in canonical form. Because the canonical form is commonly referred to as the logit model, these functions can be thought of as those of the binary logit or logistic regression algorithm.
\begin{aligned} \theta & =\ln \left(\frac{p}{1-p}\right)=\eta=g(\mu)=\ln \left(\frac{\mu}{1-\mu}\right) \ g^{-1}(\theta) & =\frac{1}{1+\exp (-\eta)}=\frac{\exp (\eta)}{1+\exp (\eta)} \ b(\theta) & =-\ln (1-p)=-\ln (1-\mu) \ b^{\prime}(\theta) & =p=\mu \ b^{\prime \prime}(\theta) & =p(1-p)=\mu(1-\mu) \ g^{\prime}(\mu) & =\frac{1}{\mu(1-\mu)} \end{aligned}
The Bernoulli log-likelihood and deviance functions are
\begin{aligned} \mathcal{L}(\mu ; y) & =\sum_{i=1}^n\left{y_i \ln \left(\frac{\mu_i}{1-\mu_i}\right)+\ln \left(1-\mu_i\right)\right} \ D & =2 \sum_{i=1}^n\left{y_i \ln \left(\frac{y_i}{\mu_i}\right)+\left(1-y_i\right) \ln \left(\frac{1-y_i}{1-\mu_i}\right)\right} \end{aligned}

## 统计代写|广义线性模型代写Generalized linear model代考|The binomial regression algorithm

The canonical binomial algorithm is commonly referred to as logistic or logit regression. Traditionally, binomial models have three commonly used links: logit, probit, and complementary log-log (clog-log). There are other links that we will discuss. However, statisticians typically refer to a GLM-based regression by its link function, hence the still-used reference to probit or clog-log regression. For the same reason, statisticians generally referred to the canonical form as logit regression. This terminology is still used.

Over time, some researchers began referring to logit regression as logistic regression. They made a distinction based on the type of predictors in the model. A logit model comprised factor variables. The logistic model, on the other hand, had at least one continuous variable as a predictor. Although this distinction has now been largely discarded, we still find reference to it in older sources. Logit and logistic refer to the same basic model.

In the previous section, we provided all the functions required to construct the binomial algorithm. Because this is the canonical form, it is also the algorithm for logistic regression. We first give the grouped-response form because it encompasses the simpler model.

## 统计代写|广义线性模型代写Generalized linear model代考|Derivation of the Bernoulli model

$$f(y ; p)=p^y(1-p)^{1-y}$$

$$f(y ; p)=\exp \left{y \ln \left(\frac{p}{1-p}\right)+\ln (1-p)\right}$$

\begin{aligned} \theta & =\ln \left(\frac{p}{1-p}\right)=\eta=g(\mu)=\ln \left(\frac{\mu}{1-\mu}\right) \ g^{-1}(\theta) & =\frac{1}{1+\exp (-\eta)}=\frac{\exp (\eta)}{1+\exp (\eta)} \ b(\theta) & =-\ln (1-p)=-\ln (1-\mu) \ b^{\prime}(\theta) & =p=\mu \ b^{\prime \prime}(\theta) & =p(1-p)=\mu(1-\mu) \ g^{\prime}(\mu) & =\frac{1}{\mu(1-\mu)} \end{aligned}

\begin{aligned} \mathcal{L}(\mu ; y) & =\sum_{i=1}^n\left{y_i \ln \left(\frac{\mu_i}{1-\mu_i}\right)+\ln \left(1-\mu_i\right)\right} \ D & =2 \sum_{i=1}^n\left{y_i \ln \left(\frac{y_i}{\mu_i}\right)+\left(1-y_i\right) \ln \left(\frac{1-y_i}{1-\mu_i}\right)\right} \end{aligned}

## MATLAB代写

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