Beyond the single biggest fish: robust inference of species maximum length (Lmax) from sample maxima

Abstract 1. Maximum body length ( $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> ) is a key trait for describing an animal species’ biological characteristics, ecology, and vulnerability to exploitation. This is particularly true for fish species, and $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> is widely used in fisheries and is a key parameter in population assessments. Yet $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> estimation is strongly contingent on sampling intensity, and uncertainty is rarely quantified or propagated in downstream applications. 2. We apply and develop two complementary estimators of $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> and its uncertainty using observations of the largest individuals recorded across multiple samples of approximately similar size. First, using Extreme Value Theory (EVT), a method widely used in insurance and finance, we use the Generalised Extreme Value distribution to model the probability of an extreme event, i.e., the observation of a certain individual body length. Second, we propose a new method, an Exact Finite Sample (EFS) approach, which estimates the most likely parameters of the underlying body‐size distribution that gives rise to the observed sample maxima. We use Bayesian inference for both methods to estimate the expected maximum individual body length for a given sampling effort (e.g., the expected maximum from 20 comparable samples) with credible intervals. 3. Sensitivity analyses show that both EVT and EFS recover unbiased $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> when samples arise from approximately truncated-normal population length-frequency distributions. For heavier right-tailed length-frequency distributions, both methods tend to underestimate $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> (5–15%), with EVT yielding wider uncertainty but less sensitivity to distribution misspecification. For animal, and especially fish, ecology and management applications, we recommend reporting a “20-sample maximum $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> ”, defined as the 95th percentile of the probability density function of the maximum lengths, as a practical benchmark that is comparable across studies and explicitly conditions on sampling effort. As a case-study, we use 14 fishing competition records for Australasian snapper ( Chrysophrys auratus ) and estimate its 20-sample $${L}_{max}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> as 139 cm (127–151 cm, 80% credible interval) using EVT, and 126 cm (121–133 cm, 80% credible interval) using EFS.