# Bayesian Inference of a Finite Population Mean Under Length-Biased Sampling

## TL;DR

This paper introduces a Bayesian approach for estimating the average shrub width in a finite population with unknown size, using length-biased sampling data, and demonstrates its robustness and improved performance over non-length-biased models.

## Contribution

The paper develops a Bayesian method that accounts for length-biased sampling and unknown population size, employing a generalized gamma distribution for robustness.

## Key findings

- Model with length bias performs better than without.
- The method effectively estimates total shrub area.
- Robust Bayesian inference via a random sampler enhances computation.

## Abstract

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.