Lopez Quintero et al.: Incorporating uncertainty into a length-based estimator of natural-mortality 
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with y\ = log y i; L'ix^ = log Lixj), and = log £, which 
are independent random errors. The estimated param- 
eters can be used to compute a M-at-length estimator. 
Gislason et al. (2010) proposed an indirect method 
for estimating M-at-length, using previous estimations 
of M, habitat temperature and the VBGF parameters. 
Their proposed M-at-length estimator is only a func- 
tion of the VBGF parameters and habitat temperature. 
Charnov et al. (2013) re-analyzed the data set assem- 
bled by Gislason et al. (2010) and concluded that coeffi- 
cients of growth parameters of the previous M-at-length 
estimator did not differ from the general life history 
theory described in Charnov (1993). The analysis in 
Charnov et al. (2013) also provides a theoretical basis 
for estimating indirectly M-at-length from life history 
theory. In this reductionist approach, the M-at-length 
in fish populations is based entirely on the VBGF pa- 
rameters and can be estimated by the expression 
M(L) = ET(%) 3/2 , (3) 
where M = the natural mortality rate at length L; and 
the other parameters are estimated from 
the VGBF. 
Natural mortality model 
Method error was incorporated in this model by follow- 
ing a log-normal regression model of the form 
log M(L) = P 0 + Px log% + P 2 log K + log 77, (4) 
where log 77 = an independent additive Gaussian error, 
log ri~N(0,tf). 
Parameter cr% corresponds to the uncertainty given by 
the mortality expression (Eq. 3), which came from the 
Gislason et al. (2010) database and comprises 168 more 
or less independent estimates for marine and brack- 
ish water fish. In order to have an approximation of 
tf, as well as p 0 , Pi, and p 2 , we estimate the log-normal 
regression model with normal prior distributions with 
mean 0 and large variance for p h Ni0,a%), i = 0, 1, 2, 
and an inverse gamma distribution, Inv-r(a,/?), for with 
small values for the scale and rate hyperparameters, 
(a, P) on a Bayesian framework. 
In the Results section, the estimated parameters 
are presented with this approach for the same data set 
used by Charnov et al. (2013). 
Life history parameters 
Bayesian approach Assuming the multiplicative struc- 
ture of Equation 3, Contreras-Reyes et al. (2014) con- 
sidered a skew-£ distribution for the errors £\, denoted 
by £j' =5 log £; ~ ST(n i ,af,X,v),i = l,...,n. More specifi- 
cally, we assumed that the original multiplicative er- 
rors £;, i = 1 ,..., n, are independently distributed ran- 
dom variables following a log-skew-^ distribution (Az- 
zalini et al., 2002; Marchenko and Genton, 2010) with 
parameters of location, dispersion, shape and degrees 
of freedom given by a *, A, and v>2, respective- 
ly. Then, the log-transformed lengths are y{ = log y, 
~ STip { -l-LijofjAjV), i = 1,..., n, with L{ = log L(x ;). 
Thus, the density of y{ = log y ; is 
/(^il > Ai , , A, v) = 
^-^(^ i ;v)T|A2: i ^■^-;v + l|,y i , eR, (5) 
where z x = (y;' — - L{) / Gj is a standardized ver- 
sion of y{, and t(z; v) and Hz; v ) represent the usual 
symmetric Student’s t density and cumulative distri- 
bution function, respectively. In this case, we assume 
that the original lengths follow a log-skew-^ distribu- 
tion, denoted by yj ~L<ST(/q +L i ,cr i 2 ,A,vj, i = 1 ,..., n. 
The power function mip; *;) = xf p introduces heterosce- 
dasticity into the dispersion parameter erf = a 2 m(p; jc;) 
(Contreras-Reyes and Arellano-Valle, 2013). 
A Bayesian analysis for the log-skew-t VBGF is pro- 
posed. If independence is assumed, the likelihood func- 
tion f (y' | x, 9) of the unknown parameter vector 9 = 
(L„, K, t 0 , a 2 , p, A, v) T = (j8 T , cr 2 , p, A, v) T is considered 
with Equation 5. The Bayesian model specification in- 
volves a prior distribution for each parameter of 9 to 
be inserted into Equation 3 in the case of L„ and K, 
respectively and inference on 6 rests on posterior dis- 
tribution n(6\x, y') af(y'\x, 6) u{6). 
In the specific case of southern blue whiting (Lopez 
Quintero et al., 2017), the prior distributions were 
the following: L X ~TN ( 0>oo) (0,100); tf~r(15,100); -t Q ~ 
r(14,4); n{p) a 1 (a non-informative prior density); 
cr 2 ~r(0.1,0.1); A~IV(0,100); and v~TE [2 >oo) (0.5), where 
TN( 0<oo) (0,<r 2 ) denotes the truncated normal density at 
interval (0,oo) with zero mean and variance a 2 , TE[ Cao ) 
(A) denotes the truncated exponential density at inter- 
val [c,°°) with parameter A, and Ha, /?) represents the 
gamma distribution whose shape and rate parameter 
are a and P, respectively. 
Copula approach The copula method is used when con- 
sidering 2 random variables represented by their joint 
(cumulative) distribution function, H(u,v) = H(F(x), 
G(y)), where u and v denote the values of Fix), and 
Giy), respectively. Our main objective is to simulate 
random variables from H(u,v). 
Most of the sampling methods are based on the as- 
sumption that Fix) and Giy) are known but the joint 
distribution HiFix), Giy)) is not. A naive framework 
would be that in which samples are drawn indepen- 
dently from each marginal distribution, yet this ap- 
proach is would be misleading because underlying de- 
pendent structures in H will be lost. Alternatively, a 
copula C between F and G can be used as a function 
such that CiFix), Gix)) = Hiu,v). This function can be 
recognized as a distribution function on [0,1 ] d (for a 
dimension d) whose arguments are the marginal distri- 
bution of the joint distribution. Thus, as in any statisti- 
cal distribution, C depends on some parameters, say 6. 
A copula can be simulated by using different ap- 
proaches depending on the family they belong to, e.g., 
Gaussian, Archimedean, Extreme-value, and others. 
