14 
Fishery Bulletin 115(1) 
age class. Hence, the bias in sampling for length at 
age that favors fast-growing individuals of each age 
class (Taylor et ah, 2005). Therefore, the assumption 
of Gaussianity to estimate parameters of the VBGF 
is not adequate (Contreras-Reyes and Arellano-Valle, 
2013; Montenegro and Branco, 2016). Moreover, the as- 
sumption of Gaussianity implies that length may take 
negative values and, therefore, is nonsensical (Xiao, 
1994; Millar, 2002). 
Different approaches have been proposed to over- 
come this drawback and fitting the VBGF. They can 
be separated roughly into 2 categories. In the first one, 
models, such as the one in Taylor et al. (2005), pro- 
vide a mechanistic approach to dealing with skewed 
length-at-age data, with a combined process of growth, 
selectivity, and mortality when fitting the VBGF. The 
second category is a more empirical approach in which 
skewed and heavy-tailed length-at-age data are mod- 
eled by using the maximum-likelihood method and as- 
suming a non-Gaussian distribution (Contreras-Reyes 
and Areliano-Valle, 2013) and by using Bayesian anal- 
ysis (Millar, 2002; Siegfried and Sanso, 2006). Millar 
(2002) proposed a Bayesian framework to estimate 
parameters of the VBGF, using a multiplicative error 
model with log-normal distribution. Contreras-Reyes 
and Arellano-Valle (2013) calculated the maximum- 
likelihood estimates for the VBGF with the family of 
skew- distributions (Azzalini and Capitanio, 2003), a 
flexible class that extends the known Student distri- 
bution (e.g., Geweke, 1993). Such models can incorpo- 
rate asymmetric and heavy-tailed errors, with presence 
of heteroscedasticity (Montenegro and Branco, 2016). 
Contreras-Reyes et al. (2014) reanalyzed the skew- ap- 
proach to incorporate a log-skew-^ distribution under 
multiplicative error distribution. 
In this study, we examined our proposed Bayesian 
method for estimating the VBGF parameters on the 
basis of a log-skew-^ distribution. This new framework 
merges the benefits provided by Bayesian analysis (Sieg- 
fried and Sanso, 2006) and the log-skew-i distribution 
(Contreras-Reyes et al., 2014) for estimating parameters 
of the VBGF for harvested fish populations. Addition- 
ally, our approach allows for heteroscedasticity in errors, 
modeled as power and exponential functions (Contreras- 
Reyes and Arellano-Valle, 2013). This Bayesian frame- 
work is applied to data of length-at-age composition of 
southern blue whiting {Micromesistius australis), an 
important species fished in the southeast Pacific. 
Materials and methods 
Log-skew-f von Bertalanffy growth model 
We let L(xi) be the expected value of the length related 
to an th individual at age Xj, L^>0, K > 0, tQ < min{xi, 
... Xj,}, and n is the sample size. The VBGF defines 
growth in length as 
L(xi) = L<^(l-e-^<"‘-'»^). 
Equation 1 represents the simplest formulation of the 
VBGF, described by 3 parameters: 
where = the asymptotic length (in length units, e.g., 
centimeters); 
K - the growth rate coefficient expressed per 
unit of time; and 
^o = the theoretical age (usually in years) when 
the length is zero. 
Parameters of the VBGF usually are estimated from 
observed length-at-age pairs, such as (x;, jj), i = 1 , n, 
where is the ith observed length at age xj. Equation 
1 was described in terms of multiplicative structure 
(Millar, 2002; Siegfried and Sanso, 2006; Contreras- 
Reyes et al., 2014) for random errors: 
= L(Xi)£i, (2) 
where £; = non-negative random errors, usually as- 
sumed to be independent, identically distributed errors 
with a mean of 1. Given this assumption, the VBGF 
in Equation 2 corresponds to the nonlinear regression 
with multiplicative random errors. We easily recovered 
the additive structure of the original model in Equa- 
tion 2 by applying log scale in the following way: 
y[ = L'(Xi ) + £,', with log y[, L'(x -^ ) = log L; = L[, and 
e( = log£i, i = 1, re, (3) 
in which el were assumed to be independent, identi- 
cally distributed, random errors with zero mean. 
Contreras-Reyes et al. (2014) assumed a log-skew-^ 
distribution (Azzalini et al., 2003) for the multiplica- 
tive and heteroscedastic random errors. Specifically, 
they assumed that the multiplicative errors Ei, i = 1, 
re, were independent random variables following a 
log-skew-^ distribution with parameters /fj G R (loca- 
tion), af > 0 (scale and dispersion), Aj G R (skewness 
and shape), and v > 0 (degrees of freedom), a distribu- 
tion that is denoted by 
el ~ LSTifii,af,X,v), i = l,...,n. (4) 
This approach is equivalent to considering that the 
transformed errors Sj, i = 1, ..., n, are independent and 
have a skew distribution (Branco and Dey, 2001; Az- 
zalini and Capitanio, 2003) denoted by 
el ~ ST(iii,af,X,v), i = 1, re. (5) 
In turn, this notation indicates that the transformed 
response variables (lengths) are derived from 
yl ~ STifi^ + Ll,af,X,v), i = l,...,n. (6) 
namely, that the density of yl is given by 
where 
— (yl ~Mi~ ) / (Tj is a standardized version of y ■ , 
( 1 ) 
