4.13 Estimating the Poisson Regression parameters

The calculation of $\lambda$ can then be used in the formula of the Poisson Distribution, allowing the Maximum Likelihood approach to be used in estimating the parameters, $\beta_0$, $\beta_1$,…, $\beta_p$:

Poisson Distribution Formula: $Pr(Y = k) = \frac{e^{-\lambda}\lambda^k}{k!}$ for $k$ = 0, 1, 2, …

Maximum likelihood: $l(\beta_0, \beta_1, ..., \beta_p) = \Pi_{i=1}^n\frac{e^{-\lambda(x_i)}\lambda(x_i)^{y_i}}{y_i!}$

where $\lambda(x_i) = e^{\beta_0 + \beta_1x_{i1} + ... + \beta_px_{ip}}$

Coefficients that maximize the likelihood $l(\beta_0, \beta_1, ..., \beta_p)$ (make the observed data as likely as possible) are chosen.