A plotless density estimator with a Norton-Rice distribution for ordered distances

A Norton-Rice distribution (NRD) is a versatile, flexible distribution for k ordered distances from a random location to the k nearest objects. In a context of plotless density estimation (PDE) with n randomly chosen sample locations, and distances measured to the k = 6 nearest objects, the NRD provided a good fit to distance data from seven populations with a census of forest tree stem locations. More importantly, the three parameters of a NRD followed a simple trend with the order (1, horizontal ellipsis , 6) of observed distances. The trend is quantified and exploited in a proposed new PDE through a joint maximum likelihood estimation of the NRD parameters expressed as a functions of distance order. In simulated probability sampling from the seven populations, the proposed PDE had the lowest overall bias with a good performance potential when compared to three alternative PDEs. However, absolute bias increased by 0.8 percentage points when sample size decreased from 20 to 10. In terms of root mean squared error (RMSE), the new proposed estimator was at par with an estimator published in Ecology when this study was wrapping up, but otherwise superior to the remaining two investigated PDEs. Coverage of nominal 95% confidence intervals averaged 0.94 for the new proposed estimators and 0.90, 0.96, and 0.90 for the comparison PDEs. Despite tangible improvements in PDEs over the last decades, a globally least biased PDE remains elusive.

Automated summary

The Norton-Rice distribution (NRD) is a versatile distribution for estimating ordered distances from a random location to the k nearest objects. The proposed new PDE showed promising results in simulated probability sampling, but exhibited increased bias with smaller sample sizes. Despite advances in PDEs, achieving a globally least biased PDE remains challenging.