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The Poisson distribution $P(\mu)$ is often used to model count data. If $Y$ is the number of occurrences, its probability distribution can be written as
where $\mu$ is the average number of occurrences. And we can obtain
Let $Y_1,\ldots, Y_N$ be independent random variables with $Y_i$ denoting the number of events observed from exposure $n_i$ for the $i$-th covariate pattern. We have
where $\theta_i$ is usually modeled by
Therefore, the generalized linear model is
The natural link function is the logarithmic function
where $\log(n_i)$ called the offset, it is a known constant.
Measure the goodness of fit
Hypotheses about the parameter $\beta_j$ can be tested using the wald, score or likelihood ration statistics. Approximately, we have
The fitted values are given by
The pearson residuals are
where $o_i$ is the observed value of $Y_i$. The chi-squared goodness of fit statistics are related by
The deviance for a Poisson model can be written in the form
For most models, we have $\sum o_i=\sum e_i$, so
The deviance residuals are the components of $D$,
so that $D = \sum d_i^2$.
Using the Taylor series expansion,
The example which implement in C++ and R can be found on gsl_lm
You can see the code in my github repository gsl_lm/poisson.