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

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.

Appendix 1: Uniform Boundedness of \(\Delta (\theta )\)

We need to show that

$$ \Delta (\theta ) = \tilde{\Gamma }(n\theta ) - n\tilde{\Gamma }(\theta ) - n \theta \ln (n) $$

is uniformly bounded in $\theta$. We will show that $\Delta (\theta )$ is asymptotically flat.

First, differentiating $\Delta (\theta )$, we have

$$ \Delta ^\prime (\theta ) = n\{\psi (n\theta )-\psi (\theta )-\ln (n)\}, $$

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.

Fig. 5

Line plots of $\Delta (\theta )$ for selected sample sizes (n)

Fig. 6

Line plots of $\Delta ^\prime (\theta )$ for selected sample sizes (n)

Second, differentiating $\Delta ^\prime (\theta )$, we have

$$ \Delta ^{\prime \prime }(\theta ) = \frac{n}{\theta } \{n\theta \psi ^\prime (n\theta ) - \theta \psi ^\prime (\theta )\}. $$

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.