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Fishery Bulletin 115(3) 
Each of these families have their own set of parameters 
(dos Santos Silva and Lopes, 2008). Owing to an impor- 
tant result from copula theory, we are able to simulate 
from H, even if unknown, only requiring the marginal 
distributions, F and G, and the copula C e (see Sklar’s 
Theorem in Nelsen, 2006). 
In order to imitate the joint distribution H, the cop- 
ula C d and the marginal distributions F and G need 
to be estimated. Doing so could create various situ- 
ations, namely: if F and G belong to known families, 
we could use a parametric framework to obtain C e , by 
either estimating all parameters associated with both 
marginal and copula inside a single routine in a proce- 
dure known as full maximum-likelihood , or by estimat- 
ing the parameters in 2 stages. The first stage includes 
estimating parameters of marginal distributions and 
afterwards the parameters in copula. This approach 
is commonly known as inference functions for margins 
(Cherubini et al., 2004). There are also intermediate 
procedures where the marginal distributions are es- 
timated nonparametrically. Results are then plugged 
into the copula parameters obtained by the maximum 
likelihood estimation. This latter approach is called 
“canonical maximum likelihood”or “maximum pseudo- 
likelihood” and is based on the empirical estimation of 
the distribution functions F and G. This strategy is ap- 
plied when the families of marginal distributions are 
unknown (McNeil et al., 2005). In all these cases, to 
estimate the copula C we need approximations for F 
and G, denoted by F and G, respectively. These quanti- 
ties are known as “pseudo-samples” or “pseudo-obser- 
vations” and provide valuable information for the type 
of dependency between the original variables (see Fig. 
2 and McNeil et al., 2005). 
We obtained the empirical posterior distributions for 
K and L M , where the exact marginal and joint family 
distributions are unknown. We consider the Gaussian 
copula with parameter p, |p| <1, which is the usual 
correlation coefficient that measures the linear asso- 
ciation between 2 variables. 
The Gaussian copula has a nonclosed form, 
i i _ i w^-2pwz+z^ 
\ 1 r 0 ^ 2(1 -o 2 * * ) ^ ^ fn\ 
CJu,v) = , < / e dwdz, (6) 
and to estimate it, the likelihood function, 
^P) = £Hi lo g[c„ {E(Xi),G(yi)}], (7) 
needs to be maximized, where c p (a,b)=g^C p (u, v) 
and <P(.) represents the cumulative normal distribution. 
We are following the maximum pseudolikelihood ap- 
proach where the marginal distributions F(x) and G(y) 
are estimated nonparametrically as follows: 
= ( 8 ) 
= <9) 
71 + 1 
Using the copula estimate parameters, we draw 
samples from C d = C p , as suggested in Nelsen (2006). 
Furthermore, to obtain samples on the same scale of K 
and Loo, instead of the (0,1) interval, we apply the ap- 
propriate quantile functions to the simulated marginal 
copula. These quantile functions depend on the original 
marginal distributions. 
M-at-!ength estimation 
To obtain an estimator of Equation 3, we take N esti- 
mated values of parameters I.-,,, and K, represented as 
Loo (j) and }&\ respectively, j = 1 All these values ; 
are used for each of the values of fish lengths, y i( i = • 
1 ,...,J, where the value of J is one of the distinct val- 
ues of the age. The estimated values JRti) and LJj> are 
obtained from either the Bayesian chains values after 
convergence or from the copula simulation. To complete 
the estimator, it is then perturbed with the uncertainty 
rji (method error) of Equation 4, recovered from the M 
modeling, 
M (J) = #0) r 
Ui J 
GO) 
with j = 1,...,IV, i = 1 
In order to emphasize that the mortality estimator de- 
pends on individual i through the length variable, y, 
the subscript y, is added to the notation. Using the 
Equation 10, we draw samples at each length i. Uncer- 
tainty in the VBGF parameters (trait error) is guaran- 
teed by using Markov chains or copula iterations, the 
incorporation of 7| is whereas M estimators have their 
own uncertainty abridged a n parameter. 
Furthermore, to include the dependence structure 
between age and length in the mortality estimation, 
we use the predicted value L(x s ) instead of y- v Using j 
Equation 1, we obtain the mortality estimator with the 
following equation: 
AfJ f : 
i? (j) 
' Iff | 
L(x#)J 
( 11 ) 
where j = 1, . ,N; 
i = 1 and 
p = (L oo ,K,i 0 ) J are point estimates, such as median 
values from posterior distribution for (L m K, 
if) which can be taken from previous studies. 
Results from both approaches, with Bayesian chains or 
copula simulation, are then compared. 
All statistical methods used in this study were de- 
veloped with the software R 2 (vers. 3.1.0 or higher; 
R Core Team, 2014). Bayesian estimations were car- 
ried out with the program JAGS, vers. 3.4 or higher 
(Plummer, 2003). Copula estimation was conducted 
with the R package copula, vers. 0.999-12 or higher, 
developed by Hofert et al. (2015). The length and the 
2 Mention of trade names or commercial companies is for iden- 
tification purposes only and does not imply endorsement by 
the National Marine Fisheries Service, NOAA. 
I 
