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Fishery Bulletin 115(3) 
are size dependent or not, provide only point estimates 
of mortality. Uncertainty in M estimates from indirect 
methods have not been investigated in detail and have 
usually been based on ad-hoc approaches (Cubillos et 
al., 1999; Quiroz et al., 2010; Wiff et al., 2011). These 
methods incorporate uncertainty in M by taking growth 
parameters from the literature and their associated 
uncertainty (e.g. standard deviation, covariance, confi- 
dence interval), and then by drawing empirical distri- 
butions of these parameters to propagate uncertainty 
in M estimates, usually assuming Gaussian error. 
A promising method for estimating M-at-length 
based on life history theory, has recently been proposed 
by Gislason et al. (2010) and Charnov et al. (2013), and 
depends entirely on von Bertalanffy growth function 
(VBGF) parameters. For these researchers, appropriate 
uncertainty in growth estimates become a key issue 
in assessing uncertainty in M estimates. Most ad-hoc 
approaches to incorporate statistical variability in M 
are based on a 2-step model that keeps estimations of 
growth parameters and M separated. Therefore, in this 
article, we consider the inclusion of 2 different sources 
of uncertainty in the modeling, one over the growth 
parameters and another from the proposed structural 
M-at-length mortality estimator. 
Uncertainty in M estimates derived from indirect 
methods come from 2 main sources. First, uncertainty 
depends on the variability among species or stocks for 
which the empirical relationship has been proposed. 
This source of uncertainty is usually referred to as 
“method error” because it represents how “accurate” 
the empirical model is (Quiroz et al., 2012). The second 
source of uncertainty is related to the error within the 
species-specific parameters that feeds into the indirect 
method (e.g., growth parameters). This source of uncer- 
tainty is called “trait error” because it represents the 
uncertainty in life history parameters of the stock or 
population for which M is being estimated. 
The trait-error of estimated growth parameters 
can be converted into M estimates as the iterative 
parameter updates of a Bayesian estimation with a 
non-Gaussian distribution. In particular, Gaussian er- 
ror distribution for estimating growth parameters can 
resolve some even nonsensical shortcomings, for in- 
stance negative length values. Distributions based on 
age-length fishing selectivity are usually skewed as a 
result of a size-selective sampling process (Contreras- 
Reyes et al., 2014; Montenegro and Branco, 2016). In 
addition, in harvested fish populations, an accumula- 
tive effect of fishing on size-at-age exists. Growth rates 
vary among individuals (Sainsbury, 1980) and fishing 
selectivity removes faster growing individuals from 
each particular age class. Moreover, in some studies, in 
which VBGF parameters are estimated, the assumption 
of Gaussian error distribution sometimes lacks adequa- 
cy, especially with the presence of outliers, which could 
lead to questionable estimates (see Contreras-Reyes 
and Arellano-Valle, 2013; Contreras-Reyes et al., 2014, 
and references therein). 
In our analysis we used 2 methods to incorporate 
uncertainties in the M-at-length estimates. Both de- 
pend on information that can be drawn from a Bayes- 
ian estimation derived from VBGF parameters. The 
first method takes advantage of the Markov chains 
and, after convergence, those chains are directly incor- 
porated into the M calculation. In particular, we used 
the Bayesian results generated by Lopez Quintero et al. 
(2017) as a baseline. For the second method, we propose 
taking the dependent structure, which is concentrated 
in the posterior distribution of parameters, and using 
this structure for simulating a distribution with the 
same dependent features based on the copula method 
(Nelsen, 2006). This last approach has the advantage 
that, even without the precision in the joint distribu- 
tion of parameters, we can obtain samples that pre- 
serve the dependence between the observed variables 
by only approximating its marginal distributions, at a 
much lower computational cost. We apply the length- 
based M estimators, using a data set corresponding to 
24,942 individuals of southern blue whiting ( Microme - 
sistius australis ) collected from Chilean continental 
waters over the period 1997-2010 (J. Contreras-Reyes, 
unpubl. data). Their total lengths and ages were re- 
corded and assigned by studying their otoliths. Further 
information about this data set can be found in Contre- 
ras-Reyes et al. (2014) and references therein. 
Let L(x) be the theoretical expected value of the length 
related to an individual at age x. The (specialized) ' 
VBGF function is defined as 
L(x) = L 00 ( 1 — e- K(x - to) ). (1) ! 
This equation represents the simplest formulation of 
the VBGF (Essington et al., 2001), which is described 
by 3 parameters: L m represents asymptotic length (in f 
length units e.g. centimeters): K represents the growth 
rate coefficient usually expressed in inverse time units: | 
and t 0 is the theoretical age (usually in years) at length 
zero. Parameters of the VBGF are estimated from ob- 
served length-at-age pairs such as (x, y ), where y is the 
length at age x. 
Equation 1 can also be modeled in terms of a multi- 
plicative structure for random errors 
yi = L(x0£i, (2) 
where y ; = the length at age of the ith sampled sub- 
ject, i = 1 Lj> 0, K> 0, £<min{:JCi,...,x n }; and £, are 
independent (not necessarily identical) non-negative 
random errors (Contreras-Reyes et al., 2014). 
With this assumption, the VBGF in Equation 2 corre- | 
sponds to the multiplicative nonlinear regression with 
logarithmic random errors. The additive structure of 
the original model in Equation 2 is easily recovered by 
applying logarithmic properties, such as y\ = L'(x\) + £\, 
