578 



Fishery Bulletin 97(3), 1999 



3 

 C/5 



1.000 



0,975 - 



0.950 - 



0925 - 



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00 



1 - 



CQ 



1,0050 T 



1 0025 



1.0000 - 



9975 - 



B 



0,900 ^ 1 1 1 1 1 1 1 1 9950 H 1 1 1 1 1 1 1 ; 



1 0,2 0,3 0.4 5 0.6 7 8 9 1 2 3 0.4 0.5 0.6 7 8 9 



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0.1 0.2 0.3 0.4 0.5 0,6 0.7 08 09 



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1 0.2 3 4 0.5 0.6 0.7 0.8 9 



Natural Mortality 



Figure 3 



Effect on natural mortality rate on the performance of the Monte Carlo method. Open circles =1000 indi- 

 viduals; closed circles = 5000 individuals. A^,„„,p,,= 5000, N^i^= 5000. 



on Fieller's (1944) theorem and the logit hnk func- 

 tion, and two computationally-intensive methods 

 based on nonparametric bootstrap of the observa- 

 tional pair (/,,^,l and Monte Carlo resampling of pa- 

 rameter estimates from the logistic model. Although 

 this is not an exhaustive set of methods (for example, 

 we did not explore likelihood profiles), they repre- 

 sent a set of conceptually different alternatives to be 

 tested against the model we used to generate simu- 

 lated data. In particular, both resampling methods 

 are especially useful to obtain the distribution of a 

 function of estimated parameters, such as in Equa- 

 tion 3, mainly because of their mathematical sim- 

 plicity, which comes at the expense of extensive com- 

 putation. Our results indicate that the three meth- 

 ods to estimate size at P9c maturity perform almost 

 equally well in terms of bias, length, and shape of 

 the confidence interval, but that Monte Carlo per- 

 formed better in containing the true parameter 

 within its confidence bounds with the nominal 95% 

 rate. This greater accuracy is accompanied by Nf^^^^-1 

 times less computation than bootstrap. 



Bootstrap single assumption was that all observa- 

 tions from any given sample have the same prob- 

 ability to appear in a new sample. In contrast, the 

 Monte Carlo method assumed a bivariate normal 

 distribution of parameter estimates of the maturity 

 model. Having simpler assumptions, it is unclear why 

 the bootstrap method failed more than the nominal 

 5% of the times at low sample sizes. One reasonable 

 explanation is that for every sample, there would be 

 A^^^^^^, bootstrap samples, and therefore A^^^^^j, numeri- 

 cal solutions to the normal equations under the bi- 

 nomial likelihood model. We used here 5000 boot- 

 strap samples and the SIMPLEX algorithm (Press 

 et al., 1992). Small errors in the numerical algorithm 

 coupled with a minimum bias, may add to the nomi- 

 nal 57( , accounting for the I7r to 2% increase in fail- 

 ure rate. In contrast, the Monte Carlo algorithm re- 

 quires a single numerical solution so that it does not 

 accumulate numerical errors. On the other hand, 

 Fieller's analytical method requires a more compli- 

 cated set of assumptions than the Monte Carlo ap- 

 proach. Fieller's method requires normality of a lin- 



