  We present a robust Bayesian method to analyze forestry data when samples are
selected with probability proportional to length from a finite population of
unknown size. Specifically, we use Bayesian predictive inference to estimate
the finite population mean of shrub widths in a limestone quarry dominated by
re-growth of mountain mahogany. The data on shrub widths are collected using
transect sampling and it is assumed that the probability that a shrub is
selected is proportional to its width; this is length-biased sampling. In this
type of sampling, the population size is also unknown and this creates an
additional challenge. The quantity of interest is average finite population
shrub width and the total shrub area of the quarry can be estimated. Our method
is assisted by using the three-parameter generalized gamma distribution,
thereby robustifying our procedure against a possible model failure. Using
conditional predictive ordinates, we show that the model, which accommodates
length bias, performs better than the model that does not. In the Bayesian
computation, we overcome a technical problem associated with Gibbs sampling by
using a random sampler.
