Poisson Regression Analysis Under Alternate Sampling Strategies

Abstract

Poisson processes where the mean is a function of known exogenous variables and a vector of unknown parameters have been widely studied and used. However, the existing theory of estimation of the parameters has assumed a random sampling process. This paper extends the estimation theory to two other sampling processes: grouped dependent variables and sampling where the likelihood of an observation is proportional to the endogenous variable. The likelihood functions for the former case are proven to be globally concave for three functions of interest. In the latter case (termed proportionate endogenous sampling or PES), the likelihood function appears intractable but an alternate, computationally simple estimator is found. This estimator is computed by subtracting one from each dependent variable in the PES and treating it as though it were randomly drawn. It is consistent and asymptotically normal, though less efficient than the maximum likelihood estimator. Some numerical examples on simulated data are conducted to explore the effects of grouping and proportionate endogenous sampling on the estimated coefficients.