Fitting second-order finite mixture models to data with many censored values using maximum likelihood estimation

Finite mixture models, that is, weighted averages of parametric distributions, provide a powerful way to extend parametric families of distributions to fit data sets not adequately fit by a single parametric distribution. First-order finite mixture models have been widely used in the physical, chemical, biological, and social sciences for over 100 years. Using maximum likelihood estimation, we demonstrate how a first-order finite mixture model can represent the large variability in data collected by the U.S. Environmental Protection Agency for the concentration of Radon 222 in drinking water supplied from ground water, even when 28% of the data fall at or below the minimum reporting level. Extending the use of maximum likelihood, we also illustrate how a second-order finite mixture model can separate and represent both the variability and the uncertainty in the data set.