Bayesian Inference of a Finite Population Mean Under Length-Biased - - PowerPoint PPT Presentation

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

Zhiqing Xu, Balgobin Nandram and Binod Manandhar

Estimating regrowth in a quarry

• 2 sets of three transects
• We count n from a finite

population N

• Sampling biased towards large x

Bayesian methodology

• Distribution of shrubs as a function of width (GG)

𝑔(𝑦|𝛽, 𝛾, 𝛿)

• Probability of counting, given width x

𝑄 𝑦 𝐽 = 1 = 𝑦 𝑥 𝑔 𝑦 ׬ 𝑦 𝑥 𝑔 𝑦 d𝑦

• Bayes→Generalised gamma with new parameters.

𝛽𝑐𝑗𝑏𝑡 = 𝛽 + 1 𝛿 , 𝛾𝑐𝑗𝑏𝑡 = 𝛾, 𝛿𝑐𝑗𝑏𝑡 = 𝛿

Estimating the finite size distribution

• Best estimate of N (Horvitz-Thompson)

෡ 𝑂 = 𝑥 ෍

𝑗=1 𝑜 1

𝑦𝑗

• Assuming

𝑄 𝑜𝑗 𝑂𝑗 ~𝐶𝑗𝑜𝑝𝑛𝑗𝑏𝑚(𝑂𝑗, 𝜈0)

• Using Bayes yields negative binomial distribution

Parameter estimation

• Bayes applied to find posterior for parameters

𝜌 𝛽, 𝛾, 𝛿 𝑦 = 𝐻𝐻 𝑦 𝛽, 𝛾, 𝛿 𝜌 𝛽, 𝛾, 𝛿 ׬ 𝐻𝐻 𝑦 𝛽, 𝛾, 𝛿 𝜌 𝛽, 𝛾, 𝛿 d𝛽d𝛾d𝛿

• Noninformative priors used
• Reduces effects from over fitting.
• Can be sampled using Markov-Chain MC or similar sampling schemes

𝛽 ҧ 𝑦 𝛿 𝛾1 𝛾2 𝛾3 Density Density

Summary

• Biased model fit yields better LH than unbiased
• Better sampling algorithms makes the sampling more effective
• Plans to include covariates in the analysis