A gamma-Poisson distribution of point to k nearest event distance

Distance sampling of events in natural or seminatural populations often indicates a larger variance in the distance to the kth nearest event than expected for events distributed completely at random. Overdispersion contributes to the well-known bias problem of distance sampling density estimators. Distance distribution models that accommodate overdispersion in the data should lead to more robust estimators of density. To this end we propose a gamma-Poisson distribution model for distances from a point to k nearest events. The model assumes a gamma distribution of local densities of randomly distributed events. Properties of the distribution and estimation of the parameters and event density are detailed for both constrained and unconstrained sampling. Four examples, one with simulated data from a known negative binomial distribution and three with simulated distance sampling in natural and seminatural stem-mapped tree stands, illustrate the promising performance of this new distribution, both as a model for distances and for density estimation. The modeling approach extends to other mixing distributions.