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, by using Bayesian predictive inference, we estimate the finite population mean of shrub widths in a limestone quarry area with plenty of regrown 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 the 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.
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Acknowledgements
The authors thank the two referees for their comments. Balgobin Nandram’s work was supported by a grant from the Simons Foundation (#353953, Balgobin Nandram).
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Appendix 1: Uniform Boundedness of \(\Delta (\theta )\)
Appendix 1: Uniform Boundedness of \(\Delta (\theta )\)
We need to show that
is uniformly bounded in \(\theta \). We will show that \(\Delta (\theta )\) is asymptotically flat.
First, differentiating \(\Delta (\theta )\), we have
where \(\psi (\cdot )\) is the digamma function. Now, using the duplication property (Abramowitz and Stegun 1965, Chap. 6) of the digamma function, one can show that \(\Delta ^\prime (\theta ) \ge 0\). That is, \(\Delta (\theta )\) is monotonically increasing in \(\theta \); see also Fig. 5.
Second, differentiating \(\Delta ^\prime (\theta )\), we have
Using a theorem (Ronning 1986) which states that \(x\psi ^\prime (x)\) decreases monotonically in x, we have \(\Delta ^{\prime \prime }(\theta ) \le 0\). That is, \(\Delta (\theta )\) is concave and the rate of increase of \(\Delta (\theta )\) decreases; see Fig. 6.
Therefore, \(\Delta (\theta )\) asymptotes out horizontally and \(\Delta (\theta )\) must be bounded, so is its exponent.
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Xu, Z., Nandram, B., Manandhar, B. (2020). Bayesian Inference of a Finite Population Mean Under Length-Biased Sampling. In: Chandra, G., Nautiyal, R., Chandra, H. (eds) Statistical Methods and Applications in Forestry and Environmental Sciences. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1476-0_6
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