Confidence Intervals for Extinction Risk: Validating Population Viability Analysis with Limited Data

Abstract

Quantitative assessment of extinction risk requires confidence intervals (CIs) that remain informative with limited data. Their usefulness has long been debated because short observation spans can make uncertainty so large that population viability analysis appears impractical. I derive new CIs for extinction probability under the drift-Wiener process, a canonical model of extinction dynamics, by introducing transformed parameters $w$ and $z$ whose maximum-likelihood estimators follow noncentral $t$ distributions. The resulting $w$-$z$ method yields CIs with coverage close to the nominal level and shows that precision depends not only on data length but also on effect size: extinction probabilities that are sufficiently low or high can often be estimated reliably even from limited time series. I also propose an observation-error-and-autocovariance-robust (OEAR) estimator for settings with additive observation error and short-run dependence. Applied to two 64-year national harvest indices for Japanese eel (Anguilla japonica), the method gives Criterion E extinction probabilities far below the IUCN threatened-category thresholds, with narrow CIs, despite the species being listed as Endangered under Criterion A. These results show that extinction-risk CIs can be both statistically rigorous and practically informative for conservation assessment under limited data.

Figures (32)

• Figure 1: The $w$--$z$ method for CIs of the extinction probability $G(w,z)$. Schematic overview of the $w$--$z$ transformation and the mapping to $G(w,z)$ (left), and estimation from an observed time series (right). Hats denote maximum-likelihood estimators, and underlines/overlines denote confidence limits. For OEAR, replace $\widehat{\sigma}^2$ by the long-run-variance-based estimator $\widetilde{\sigma}^2$ and use the corresponding $\widetilde{G}$.
• Figure 2: Contour plot of extinction probability $G(w, z)$ in the $(w, z)$ plane, restricted to $w + z > 0$. Shading indicates extinction risk (light: low, dark: high), with contours labeled by $G(w, z)$. The red dotted line ($w = z$) separates regions of positive drift ($\mu > 0$, below) and negative drift ($\mu < 0$, above).
• Figure 3: Correlation landscape of ML estimators $\widehat{w}$ and $\widehat{z}$ in the $(w, z)$ plane, evaluated for $q = t_q = 63$ and time horizon $t^\ast = 25$. Shading indicates $\operatorname{Corr}(\widehat{w}, \widehat{z})$, from negative (light gray) to positive (dark gray). The red curve shows the zero-correlation hyperbola $w z = k$, where $k \approx 47.82$ (Equation \ref{['eq:k-def']}). Contours follow Equation \ref{['eq:corr-k']}.
• Figure 4: Geometry underlying the $w$--$z$ method when $z < 0$. (a) The red ellipse conceptually illustrates the joint 95% confidence region for the parameters $(w, z)$ when $z \ll 0$, typically elongated along slope $-1$ due to negative correlation between the ML estimators $\widehat{w}$ and $\widehat{z}$. The extinction probability $G(w,z)$ is shown as contours of constant $G$. As shown analytically in Appendix \ref{['appendix:slope_G']}, these contours have slope strictly less than $-1$, implying they are steeper than the ellipse orientation. Open white circles mark the two points where the ellipse is tangent to a $G$ contour, determining the exact CI for $G$. Filled black circles mark the diagonally opposite corners of the dashed rectangle formed by the 95% CIs for $w$ and $z$, which approximate the CI for $G$ (the $w$--$z$ method). (b) Monte Carlo sample cloud of $(\widehat{w},\widehat{z})$ for $(w,z) = (10, -5)$, overlaid on $G$ contours. $q = t_q = 63$ and time horizon $t^\ast = 100$. Although not a true confidence region, this bootstrap-type sample cloud approximates the joint confidence region for $(w, z)$.
• Figure 5: Empirical rejection rates of CI methods, based on Monte Carlo simulations over 2,688 parameter combinations. Each point shows the proportion of 10,000 replicates in which the true extinction probability $G(w,z)$ fell outside the estimated interval. Parameters were varied over $\mu \in \{-0.5,-0.3,-0.1,0,0.1,0.3,0.5\}$, $\sigma^2 \in \{0.001,0.01,0.1,1\}$, $x_d \in \{3,5,7,9,11,13\}$, $t^{\ast} \in \{10,20,50,100\}$, and number of increments $q \in \{10,20,50,100\}$. The dashed red line marks the nominal significance level $\alpha=0.05$. Among the methods compared, the $w$--$z$ interval maintains rejection rates consistently close to the nominal level; the delta/logit method deviates in both directions depending on the parameter regime; and the percentile bootstrap and TMU (theoretical minimum uncertainty) methods are predominantly anti-conservative.