Poisson Regression Proofs

Property 1: The maximum of the log-likelihood statistic for Poisson Regression occurs when the following k equations hold

Proof: As usual, our goal is to look at the x_ij and y_i values as fixed and determine the unknown β_j which maximize LL. We do this by setting the partial derivatives of LL with respect to the μ_i  to zero.

First, note that

Thus

Property 2: Let B = [b_j] be the k × 1 column vector of logistic regression coefficients, let Y = [y_i] be the n × 1 column vector of observed outcomes of the dependent variable, let X be the n × k design matrix (see Definition 3 of Least Squares Method for Multiple Regression), let P = [p_i] be the n × 1 column vector of predicted values of success and V = [v_ij] be the n × n diagonal matrix where v_ij = z_ij on the main diagonal and zeros elsewhere. Then if B₀ is an initial guess of B and for all m we define the following iteration

then for m sufficiently large  B ≈ B_m, and so B_m is a reasonable estimate of the coefficient vector.

Proof: The proof follows from the following observation

References

Hintze, J. L. (2007) Poisson regression. NCSS
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Poisson_Regression.pdf

Nussbaum, E. M., Elsadat, S., Khago, A. H. (2007) Best practices in evaluating count data, Chapter 21: Poisson regression.
http://www.academia.edu/438746

Penn State (2017) Poisson regression. STAT 504: Analysis of discrete data.
https://online.stat.psu.edu/stat504/lesson/9