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.