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Fisheries Research 159 (2014) 88–94

Contents lists available at ScienceDirect

Fisheries Research

j ourna l ho me pa ge: www.elsev ier .com/ locate / f i shres

hort Communication

omparing growth curves with asymmetric heavy-tailed errors:pplication to the southern blue whiting (Micromesistius australis)

avier E. Contreras-Reyesa,∗, Reinaldo B. Arellano-Valleb, T. Mariella Canalesa

División de Investigación Pesquera, Instituto de Fomento Pesquero, Blanco 839, Valparaíso, ChileDepartmento de Estadística, Pontificia Universidad Católica de Chile, Santiago, Chile

r t i c l e i n f o

rticle history:eceived 21 February 2014eceived in revised form 15 May 2014ccepted 16 May 2014andling Editor A.E. Punt

a b s t r a c t

Von Bertalanffy growth models (VBGMs) have been used in several studies of age, growth and naturalmortality. Assuming that the residuals about this growth model are normal is, however, question-able. Here, we assume that these residuals are heteroskedastic and follow a log-skew-t distribution,a flexible distribution that is asymmetric and heavy-tailed. We apply the proposed methodology tolength-at-age data for the southern blue whiting (Micromesistius australis) collected from Chilean australcontinental waters between 1997 and 2010. The estimates of the VBGM parameters were L∞ = 57.042 cm,

eywords:on Bertalanffyog-skew-t distributionouthern blue whitingeteroskedasticityeavy-tails

K = 0.173 yr−1, t0 = −2.423 yr for males, and L∞ = 61.318 cm, K = 0.163 yr−1, t0 = −2.253 yr for females. TheBIC criteria suggest that females grow significantly faster than males and that length-at-age for bothsexes exhibits significant heteroskedasticity and asymmetry.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Somatic growth is a fundamental property of fish populationshat is to be understood to comprehend their life history, popula-ion demography and vulnerability to exploitation as well as fisheryustainability (Pauly, 1980; Jennings et al., 1998; Beddington andirkwood, 2005). The von Bertalanffy growth model (VBGM) (vonertalanffy, 1938) is the most widely used model in the litera-ure on fisheries to describe the individual somatic growth of fishKatsanevakis and Maravelias, 2008). A common goal in the litera-ure is to describe somatic growth by fitting VBGMs to age-lengthata by maximum likelihood methods (Kimura, 1980).

Several authors have studied the VBGM assuming independencend normality for the distribution of the response variable (see,.g., Kimura, 1980, 1990). However, because the growth sampleistribution usually departs from normality due to the presencef asymmetries or extreme observations producing heavy tails,on-Gaussian models have also been considered in a number oforks. For example, Wang and Ellis (1998) used the log-normal,

amma and truncated normal models; Millar (2002) considered log-normal growth model within a Bayesian framework; andope and Punt (2007) utilized exponential and gamma models.

∗ Corresponding author. Tel.: +56 032 2151 513.E-mail addresses: jecontrr@uc.cl (J.E. Contreras-Reyes), reivalle@mat.puc.cl

R.B. Arellano-Valle), mariella.canales@ifop.cl (T.M. Canales).

ttp://dx.doi.org/10.1016/j.fishres.2014.05.006165-7836/© 2014 Elsevier B.V. All rights reserved.

Recently, Contreras-Reyes and Arellano-Valle (2013) implementeda non-linear regression analysis for VBGM and considered addi-tive random errors and assumed that they were independent,heteroskedastic and skew-t distributed, thereby introducing a flex-ible class of VBGMs to model these asymmetries and heavy-tails(Branco and Dey, 2001; Azzalini and Capitanio, 2003; Contreras-Reyes, 2014).

In this paper, we study VBGMs using a heteroskedastic skew-t and log-skew-t non-linear regression analysis, and we comparethe additive and multiplicative stochastic structures of the errors.Based on these models, we investigate inferential and infor-mational methods to compare VBGM and their parametric andstochastic structures. Specifically, we use the standard criteriafor model selections recommended in the literature, such as theAIC and BIC (Burnham and Anderson, 2002; Katsanevakis, 2006).We apply our findings to length-at-age data of the southern bluewhiting (Micromesistius australis Norman 1937), for which thedetermined longevity is 24 years and for which differential somaticgrowth by sex has been identified (Aguayo et al., 2010).

The VBGM explains the length of an individual in terms of itsage using a non-linear function that utilizes three parameters (L∞,K, t0): L∞ (cm) is the asymptotic length of the species, K (yr−1) isthe growth rate coefficient, and t0 (yr) is the age at zero length. Let

L(x) be the expected value of the length of a subject at age x; theVBGM is given by

L(x) = L∞(1 − e−K(x−t0)). (1)

sheries

nr

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itif

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J.E. Contreras-Reyes et al. / Fi

To fit the model (1) to an empirical dataset, (yi, xi), i = 1, . . .,, this model can be described in terms of an additive non-linearegression:

i = Li + εi, (2)

here yi is the ith observed length at age xi, Li = L∞(1 − e−K(xi−t0)),nd the εi are independent and identically distributed (iid) N(0, �2)andom errors (Kimura, 1980). In contrast, Millar (2002) considers

multiplicative structure for the errors:

i = Liεi, (3)

here εi are non-negative random errors and their transformationsre given by ε′

i= log εi are iid N(0, �2). Under this last assump-

ion, the model (3) corresponds to the multiplicative non-linearegression with log-normal random errors. Applying the naturalogarithm to both sides of Eq. (3), we recovered an additive struc-ure for model (2) as y′

i= L′

i+ ε′

i, with y′

i= log yi and L′

i= log Li, Li > 0,

> 0 and, consequently, t0 < min {x1, . . ., xn}.

. Methods

The skew-t approach used by Contreras-Reyes and Arellano-alle (2013) is considered for the additive case. Specifically, wessume that the random errors εi in the model (2) are independent,eteroskedastic and distributed according to a skew-t distribu-ion (Azzalini and Capitanio, 2003). We denote this assumption asi∼ST(�i, �2

i, �, �), where −∞ < �i < ∞ is an appropriate location

arameter, �2i

> 0 is a scale parameter controlling heteroskedas-icity, −∞ < � < ∞ is a shape parameter controlling skewness and

represents the degrees of freedom controlling kurtosis. In thisork, we assume �i = −

√2/��1�iı, where

1 =√

�

2 ((� − 1)/2)

(�/2), � > 1, and ı = �√

1 + �2,

n order to have errors with a zero mean. Under these assumptions,he response variable yi in model (2) has a mean of Li for � > 1 ands skew-t distributed, and yi∼ST(Li + �i, �2

i, �, �), with a density

unction given by

(yi; Li, �2i , �, �) = 2

�it(zi; �)T

(�zi

√� + 1

� + z2i

; � + 1

), (4)

ith zi = (yi − Li − �i)/�i, where

(zi; �) =

((� + 1)/2

) (�/2)

√��

(1 + z2

i

�

)−(�+1)/2

, −∞ < zi < ∞,

s the Student-t density function with � degrees of freedom, and(zi ; �) is the corresponding Student-t cumulative distributionunction. Additionally, for the variance of yi, we have

ar[yi] =(

�

� − 2− 2

��2

1ı2)

�2i , � > 2.

For � = 0, the skew-t density (4) clearly reduces to the symmetrictudent-t density t(zi ; �), which in turn converges to the normal

ensity (zi; Li, �i) = (2��2i

)−1/2

e−z2i

/2�2i as �→ ∞. For � /= 0, the

odel (4) converges to the skew-normal density 2(zi ; Li, �i)˚(�zi)s �→ ∞, which reduces again to the symmetric normal density(zi ; Li, �i) when � = 0 (Branco and Dey, 2001; Contreras-Reyes,014).

The maximum likelihood estimate (MLE) values of � =

ˇ�, �2, �, �)

� = (L∞, K, t0, �2, �, �)�

are obtained under thessumption that � is known. To compute the MLE of � for annknown �, Lange and Sinsheimer (1993) proposed varying � on

grid of discrete values larger than 1 and choosing the value that

Research 159 (2014) 88–94 89

maximized the likelihood function. Additionally, if � is the MLE of� for a given �, its covariance matrix � can be estimated using

� = J (�)−1

, where J(�) = − ∂2 � (�)/∂�∂�� and �(�, �) is the skew-tlog-likelihood function obtained from Eq. (4).

In turn, if we assume model (3) with ε′i= log εi∼ST(�i, �2

i, �, �),

then the transformed response variables y′i= log yi, 0< yi < ∞, which

also have a skew-t density given by Eq. (4), where zi is replacedby z′

i= (y′

i− L′

i− �i)/�i. Strictly speaking, we assume in this case

that the original response yi follows a log-skew-t distribution(Azzalini et al., 2003; Marchenko and Genton, 2010), denoted byyi∼LST(Li, �2

i, �i, �). This case will be referred to in the following

as the log-skew-t model. Analogously, the log-likelihood functionfor the transformed responses y′

iis represented using the log-

likelihood function �(�) with z′i

used instead of zi. In both cases,the parameter estimates are obtained using the expectation condi-tional maximization estimation (ECME) algorithm (see Labra et al.,2012, for more details).

Without any loss of generality, in the next sections we investi-gate additional statistical tools for the skew-t model.

2.1. Heteroskedastic functions

With the purpose of modeling heteroskedasticity, we assumethat �2

i= �2m(�; xi), where m(� ; xi) is a nonnegative function that

depends on the age xi of the ith subject, such that m(0 ; xi) = 1(homoskedasticity condition), and it is called the heteroskedas-ticity function. Specifically, we consider the exponential functionm(� ; xi) = e�xi (Cook and Weisberg, 1983) and that the power func-tion m(�; xi) = x2�

i(see, e.g., Labra et al., 2012). Both functions

allow us to model the heteroskedasticity through a single param-eter �. For � = 0, we recover the homoskedastic case; for � < 0,we observe a decreasing variance trend; and for � > 0, we findan increasing variance trend. Thus, the estimation of this param-eter can be easily incorporated into the ECME algorithm for theskew-t.

2.2. Model selection criteria

In the context of fish growth studies, the chosen heteroskedas-ticity function allows us to interpret the variability in length-at-ageamong ages and to test the appropriateness of the chosen m(� ; x)function. For this purpose, it is recommended that models befit with different m(� ; x) functions before using suitable modelselection criteria to identify the “best” model (Katsanevakis andMaravelias, 2008; Katsanevakis, 2006). In this sense, we selectedthe “best” model by using the AIC and BIC values, defined asfollows:

AIC(�) = −2�(�, �) + 2q,

BIC(�) = −2�(�, �) + q log(n),

where � is the MLE of �, �(�, �) is the log-likelihood function, n is thesample size, and q is the number of estimated parameters. The AICand BIC criteria have proven to be convenient and robust tools formodel comparisons (Burnham and Anderson, 2002), and they canalso be used to compare disjoint models. For large samples, the BICis more appropriate because it penalizes the log-likelihood func-tion using the sample size. Thus, for a better interpretation of the

BIC value, we consider the difference �i (�) = BICi (�) − min{BICi (�)}associated with the ith competing model, where min{BICi (�)} is thesmallest BIC value of all of the competing models (Katsanevakis,2006).

90 J.E. Contreras-Reyes et al. / Fisheries Research 159 (2014) 88–94

(a)

Length(cm)

frequency

0100200

1

20 30 40 50 60 70

2

30100200

4

0100200

5 6

7

0100200

8

0100200

9 10

11

0100200

12

0100200

13

14

15

0100200

16

0100200

17

18

19

0100200

20

0100200

21

22

20 30 40 50 60 70

23

0100200

24

(b)

Length(cm)

frequency

050100150

1

20 30 40 50 60 70

2

3

050100150

4

050100150

5 6

7

050100150

8

050100150

9 10

11

050100150

12

050100150

13

14

15

050100150

16

050100150

17

18

19

050100150

20

050100150

21

22

20 30 40 50 60 70

23

050100150

24

and (

3

tas5tFb(rUed

iveC(av

4

ew3boa

taedt(

Fig. 1. Histograms of length-at-age for (a) male

. Data and software

The data set used for evaluating the performance of our sta-istical findings corresponds to a sample of 24,942 (12,938 malesnd 12,004 females) southern blue whiting age-length pairs. Theamples were collected from a region spanning latitudes 46 ◦S to6 ◦S over the period 1997–2010 and were obtained using the rou-ine sampling program of the fishery conducted by the Instituto deomento Pesquero (http://www.ifop.cl/). The age was determinedy reading the annual growth increments in the sagitta otolithsLeguá et al., 2013). The annual formations of a hyaline and opaqueings were determined from the analysis of the otolith margins.sing the corrected back-calculated length, the lengths at differ-nt ages were estimated. A more detailed description of the ageetermination can be found in Aguayo et al. (2010).

All of the statistical methods considered in this study, includ-ng the parameter estimations using the ECME algorithm, theariance-covariance matrix computation, and AIC and BIC crit-ra, were computed using the skewtools package developed byontreras-Reyes (2012) and implemented using the software RR Development Core Team, 2013). For the ECME algorithm, wessumed independence of the sample points, given that each obser-ation was collected from a single otolith.

. Results

Fig. 1 shows the length frequency at different ages of the south-rn blue whiting at ages 1–24 years, for males and females. In males,e observed that the empirical distribution of the length at ages

–4, 6–8, and 11–15 years showed asymmetry and that the distri-ution at ages 7–20 years had a heavy tail. The latter feature is morebvious in females at ages 3–7 (asymmetry) and 8–19 (asymmetrynd heavy tails) years.

Table 1 summarizes the fit of the models using the Student- and log-t distributions with homoskedastic variance functionsnd those using the skew-t and log-skew-t distributions with

xponential and power variance functions. Concerning the skew-tistribution, we observed that the power heteroskedascity func-ion had the lowest �i (�) values with respect to homoskedasticStudent-t and log-t models) and exponential functions. The VB

b) female southern blue whiting (1997–2010).

parameter set estimates were also very similar. Note that the het-eroskedastic parameter � is almost zero, producing a constantheteroskedasticity function in the case of the males. Under bothheteroskedastic variance functions, � = 13 degrees of freedom arereported for the group of males. In the opposite case, amongfemales, we see more differences among the fits related with expo-nential and power functions. Such estimates show the presence ofa moderate asymmetry (� = −0.71) and heavy-tails (� = 11 degreesof freedom), but they also indicate the presence of slight het-eroskedasticity (� = −0.02).

We obtained similar estimates for the VB parameters fromall of the fitted models. However, the log-skew-t had the small-est standard deviations; therefore, the log-skew-t estimators aremore efficient. BIC selected models related to the power vari-ance function. For these, we obtained heteroskedastic (� = −0.447and −0.688), asymmetric (� = −0.921 and −1.23) and heavy-tailed(� = 13 and 10) error distributions (for males and females, respec-tively). Note that the variance estimates are almost zero, mainlybecause of the logarithmic transformation. Therefore, in the case ofthe male group, the log-skew-t fit has the highest heteroskedastic-ity but the lowest �2 values. By the selection criteria, the log-skew-tmodel is “better” than Student-t, log-t, and skew-t models (Table 1).Therefore, from this perspective, we concentrate on the analysis ofthe log-skew-t fit.

For the full sample, the power variance function produces thebest fit and yields the following VB estimates: L∞ = 59.565 cm,K = 0.162 yr−1, t0 = −2.449 yr. For the male samples, these esti-mates are L∞ = 57.042 cm, K = 0.173 yr−1, t0 = −2.423 yr, and forthe females, they are L∞ = 61.318 cm, K = 0.163 yr−1, t0 = −2.253 yr.Fig. 2 (panels (a) and (b)) shows the age–length relationship for eachsex with the corresponding log-skew-t fits, heteroskedastic func-tions �2

iand confidence intervals, computed as L′

i± z(1−˛/2)S.D[L′

i],

where z(1−˛/2) denotes the 1 − ˛/2 standardized normal percentile;here, the significance level is = 0.05. Primarily, the confidenceintervals are affected by the heteroskedastic parameter � (negativefor both groups). The blue points corresponding to the homoskedas-

tic log-t fit show similarities with the log-skew-t fit. We observesignificant differences in the predicted lengths for older species(17–24 ages) in a zoom plot of the female observations. Fig. 3considers the absolute residuals from the log-skew-t fit versus xi

J.E. Contreras-Reyes et al. / Fisheries Research 159 (2014) 88–94 91

Table 1Summary of Student-t (homoskedastic), log-t (homoskedastic), skew-t (heteroskedastic), and log-skew-t (heteroskedastic) fitted models for each specification of the variance

function and by sexes and full samples. All fits include the number of parameters q, the log-likelihood function �(�) and the respective AIC(�), BIC(�) and �i (�) values. Thestandard deviations of the VBGM estimates are given in brackets.

Model Estimates Males Females Full sample

Student-t L∞ (cm) 57.137 (0.096) 61.408 (0.089) 59.722 (0.081)

K (yr−1) 0.172 (0.002) 0.164 (0.001) 0.161 (0.001)t0 (yr) −2.464 (0.051) −2.279 (0.043) −2.513 (0.040)�2 6.298 (0.340) 5.663 (0.347) 8.083 (0.309)� 13 6 16

�(�) -31,277.68 −29,355.88 −63,030.20q 4 4 4

AIC(�) 62,563.35 58,719.76 126,068.4

BIC(�) 62,593.22 58,749.33 126,100.9

�i (�) 99,082.45 93,586.37 192,679.40

Log-t L∞ (cm) 56.880 (0.095) 61.152 (0.100) 59.418 (0.080)

K (yr−1) 0.177 (0.002) 0.167 (0.001) 0.166 (0.001)t0 (yr) −2.329 (0.040) −2.185 (0.038) −2.374 (0.031)�2 0.003 (2e−04) 0.003 (1e−04) 0.003 (1e−04)� 9 10 10

�(�) 18,081.94 16,988.52 32,998.36q 4 4 4

AIC(�) −36,155.89 −33,969.03 −65,988.72

BIC(�) −36,126.01 −33,939.46 −65,956.22

�i (�) 363.22 897.58 622.25

Model Estimates Males Females Full sample

Exponential Power Exponential Power Exponential Power

Skew-t L∞ (cm) 57.121 (0.092) 57.120 (0.092) 61.375 (0.077) 61.309 (0.078) 59.716 (0.073) 59.715 (0.073)

K (yr−1) 0.172 (0.002) 0.172 (0.002) 0.164 (0.001) 0.165 (0.001) 0.161 (0.001) 0.161 (0.001)t0 (yr) −2.453 (0.049) −2.452 (0.049) −2.269 (0.039) −2.248 (0.039) −2.496 (0.035) −2.496 (0.036)� −1e−04 (0.005) −1e−04 (0.013) −0.018 (0.005) −0.119 (0.011) −1e−04 (0.004) −1e−04 (0.009)�2 6.656 (0.359) 6.651 (0.358) 9.585 (0.544) 10.379 (0.490) 9.266 (0.353) 9.258 (0.353)

� −0.309 (0.151) −0.309 (0.150) −0.709 (0.106) −0.709 (0.093) −0.509 (0.079) −0.509 (0.079)� 13 13 11 11 16 16

�(�) −31,274.89 −31,274.81 −29,282.94 −29,289.46 −63,016.69 −63,016.24q 6 6 6 6 6 6

AIC(�) 62,561.78 62,561.62 58,577.88 58,590.92 126,045.40 126,044.50

BIC(�) 62,606.59 62,606.43 58,622.24 58,635.28 126,094.10 126,093.20

�i (�) 99,095.82 99,095.66 93,459.28 93,472.32 192,672.60 192,671.70

Log-skew-t L∞ (cm) 57.111 (0.069) 57.042 (0.071) 61.553 (0.060) 61.318 (0.059) 59.664 (0.055) 59.565 (0.055)

K (yr−1) 0.172 (0.001) 0.173 (0.001) 0.160 (0.001) 0.163 (0.001) 0.161 (0.001) 0.162 (0.001)t0 (yr) −2.442 (0.035) −2.423 (0.036) −2.356 (0.029) −2.253 (0.030) −2.484 (0.024) −2.449 (0.025)� −0.051 (0.005) −0.447 (0.010) −0.076 (0.005) −0.688 (0.009) −0.039 (0.004) −0.358 (0.007)�2 0.006 (3e−04) 0.010 (3e−04) 0.008 (4e−04) 0.019 (4e−04) 0.007 (2e−04) 0.011 (3e−04)

� −0.866 (0.102) −0.921 (0.080) −1.088 (0.112) −1.230 (0.096) −1.050 (0.076) −1.096 (0.068)� 11 13 9 10 13 15

�(�) 18,255.05 18,273.02 17,426.95 17,446.70 33,286.19 33,319.61q 6 6 6 6 6 6

AIC(�) −36,498.10 −36,534.04 −34,841.90 −34,881.40 −66,560.38 −66,627.22

−3

39.

(fts

5

niit

BIC(�) −36,453.29 −36,489.23

�i (�) 35.94 0

panels (a) and (c)) and L′i= log Li (panels (b) and (d)) for males and

emales, respectively. Generally, the residual values are concen-rated at approximately 0.5 and indicate a constant trend, therebyuggesting uncorrelated observations.

. Discussion

In this paper, we considered the log-skew-t model as an alter-

ate to the skew-t model. Because the model error provides

nformation about the parameters, we consider ourselves justifiedn the specification of a log-distributed error. Given that the skew-

model confers skew-t errors to the length data, we believe that

4,797.54 −34,837.04 −66,511.63 −66,578.47

50 0 66.84 0

it is theoretically permitted for lengths to be negative; thus, weadmit that the asymptotic length L∞ can be negative. Even so, itis more reasonable to use log-skew-t distributed errors to avoidthis complication (Millar, 2002). In addition, the log-transformationof the lengths produces the smallest standard deviations of theparameter estimates; based on the selection criteria, the log-skew-t is a “better” model than the skew-t. In this sense, the presentapproach considers explicit parameters for the error distribution.

For instance, in the Wang and Ellis (1998) and Cope and Punt (2007)approaches, the shape and heavy-tail parameters do not appearexplicitly in the distribution, as in the cases of the log-normal andgamma distributions.

92 J.E. Contreras-Reyes et al. / Fisheries Research 159 (2014) 88–94

Fig. 2. Southern blue whiting observations (gray circles): (a) males and (b) females. The solid black lines correspond to the log-skew-t fits, and the dotted red lines correspondto the confidence intervals at a 5% significance level. The blue points correspond to the log-t fits with homoskedastic variance functions. Each log-skew-t fit includes ther

eibhltfiieo2tt

SF

espective sub-plots of the heteroskedastic variance function �i .

The modeling of the variability of length-at-age using het-roskedastic functions produced a good fit, and allowed for a goodnterpretation of the variability of length-at-age. Empirical distri-utions with a greater presence of extreme values, asymmetry, andeteroskedasticity are also simultaneously modeled well by the

og-skew-t distribution. In this sense, one reason for the asymme-ry in length-at-age distributions is the impact of length-specificshing selectivity, such as the implementation of a minimum land-

ng size of capture that introduces bias into the VB growth. Thisxample would produce a negative sign for t0 due the absence ofbservations in the first ages (Contreras-Reyes and Arellano-Valle,013). Therefore, further work should be directed toward this issueo explicitly model the impact of selectivity on length-at-age dis-

ributions.

The southern blue whiting is an important fishery resource inouth America (in the waters off the southern Chilean coast andalkland Islands-Patagonia shelf) and in the sub-Antarctic waters

around New Zealand (Arkhipkina et al., 2009). Recent studies haveprovide VBGM estimates of this species to further explore its biol-ogy and population dynamics. In addition, Aguayo et al. (2010)studied the estimated von Bertalanffy curves divided by sexes usinga sample with a maximum age of 18 years collected between1990 and 1995. They reported L∞ = 52.1 cm, K = 0.262 yr−1 andt0 = −1.685 yr for males, and L∞ = 55.7 cm, K = 0.239 yr−1 andt0 = −1.679 yr for females.

The differences in the estimates of our study and those of Aguayoet al. (2010) can be interpreted in light of our novel incorpora-tion of a wide range of ages (1–24 yr) and lengths (20–70 cm) andour larger sample size (n = 26,942). The approach developed here ismore accurate in terms of the description of the errors in the dis-

tribution, as we found a greater presence of extreme values andvariability of length-at-age data (Contreras-Reyes and Arellano-Valle, 2013). The comparison of the growth curves by selectioncriteria generated different growth curves between the sexes of

J.E. Contreras-Reyes et al. / Fisheries Research 159 (2014) 88–94 93

F c) xi ar

sfi

A

(wCc

R

A

A

A

A

B

B

ig. 3. Absolute residuals of log-skew-t fits vs. (a) xi and (b) L′i

for males and vs. (esiduals.

outhern blue whiting, with the faster growth provided by theemales. This result is similar to that stated by Aguayo et al. (2010),n which the comparison was based on a Hotelling T2 test.

cknowledgements

The authors are grateful to the Instituto de Fomento PesqueroIFOP, Valparaíso, Chile) to provide access to the data used in thisork. Arellano-Valle’s research was supported by Grant FONDE-YT (Chile) 1120121. We also thanks to André Punt for his usefulomments and suggestions.

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