Population Density Estimators for Right-Censored Distance Sampling

Abstract

Distance-based point-centered quarter method (PCQM) is widely used for population density estimation, yet its performance is challenged by right-censored observations arising from a truncated search radius. Existing methods for addressing such right-censored data are predominantly developed under the assumption of complete spatial randomness (CSR) using a Poisson model, while approaches for spatially aggregated populations--despite the negative binomial distribution (NBD) being well-established for uncensored distance sampling--remain lacking a systematic framework. This study presents a systematic set of censored distance-based estimators under these two core frameworks. We develop both moment-based estimators and maximum likelihood estimators (MLEs) under these two core frameworks, extending classical results to the censored setting for CSR populations and providing new inference tools for aggregated populations under the NBD model. Extensive simulations and applications to fully-censused forest plot data demonstrate that the NBD-based MLE achieves the highest accuracy and robustness across a wide range of ecological scenarios, with a median relative bias below 20\% in most empirical scenarios--a level of estimation accuracy that cannot be consistently guaranteed by other competing methods, providing a rigorously validated toolkit for analyzing censored point-to-tree distance data.

Figures (10)

• Figure 1: Relative bias (rBias, points and solid lines) of three NBD-based censored estimators, $\widehat{\lambda}_{n}^{(c)}$ (red), $\widehat{\lambda}_{m1}^{(c)}$ (blue, not applicable for $\ell=1$), and $\widehat{\lambda}_{\text{n,MLE}}^{(c)}$ (green), across a gradient of cluster scales ($\sigma$) for nearest-neighbor orders $\ell = 1, 2,$ and $3$. Error bars represent $\pm$ one relative standard deviation (rSD). Simulations are based on a Thomas cluster process with intensity $\lambda = 0.05$ and censoring radius $C = 10$ m.
• Figure 2: Performance of seven censored distance-based estimators applied to tree populations in BCI. Left column: Relative bias (rBias) across nearest-neighbor orders $\ell = 1, 2, 3$ (rows) for a fixed censoring radius $C = 20 \text{ m}$. Right column: Corresponding relative root mean squared error (rRMSE). Estimators are grouped by underlying spatial assumption: Complete Spatial Randomness (CSR: $\widehat{\lambda}_{\text{DK}}$, $\widehat{\lambda}_{C}^{(c)}$, $\widehat{\lambda}_{P}^{(c)}$, $\widehat{\lambda}_{\text{MLE}}^{(c)}$) and Negative Binomial Distribution (NBD: $\widehat{\lambda}_{m1}^{(c)}$, $\widehat{\lambda}_{n}^{(c)}$, $\widehat{\lambda}_{\text{n,MLE}}^{(c)}$). Note that $\widehat{\lambda}_{m1}^{(c)}$ is not applicable for $\ell = 1$. The boxplots summarize the distribution of each metric across 200 sampling replications for 112 analyzed species (abundance $\geq$ 500 individuals).
• Figure SM.1: Performance of the seven estimators for the Barro Colorado Island (BCI) plot with a maximum search radius $C = 10 \text{ m}$. Left column: Relative bias (rBias); Right column: Relative root mean squared error (rRMSE). Rows from top to bottom correspond to $\ell = 1, 2, 3$, respectively.
• Figure SM.2: Performance of the seven estimators for the Barro Colorado Island (BCI) plot with a maximum search radius $C = 30 \text{ m}$. Layout is identical to Figure \ref{['fig:bci_c10']}.
• Figure SM.3: Performance of the seven estimators for the Barro Colorado Island (BCI) plot with a maximum search radius $C = 40 \text{ m}$. Layout is identical to Figure \ref{['fig:bci_c10']}.