574 



Fishery Bulletin 97(3), 1999 



where r~  = the proportion of variance due to the 



Po.P 



hnear relationship. 



Note in Equation 10 that when r=0, the mean of 

 the ^j for all^ would just be the /?j estimate, which 

 means that /^^ and /3, values are independently se- 

 lected in each trial; note also in Equation 1 1 that the 

 resampling variance of the estimate would be its to- 

 tal variance. On the other hand, when r= 1 1 1 , Equa- 

 ./tion 11 shows that the resampling variance of /3j 

 would be totally due to the mapping of p^ ^ onto p.^ ^, 

 which is expected when the linear relationship be- 

 tween two variables is deterministic. In this case, 

 the algorithm presented here would only perform one 

 Monte Carlo simulation, that on /J,,. Therefore, the 

 algorithm is flexible enough to cover the whole range 

 of correlation between both parameter estimates. 



A confidence interval for Ip,- may be obtained by 

 the percentile method (Casella and Berger, 1990; 

 Efron and Tibshirani, 1993), for which two computa- 

 tional alternatives are available. If the resampling 

 through the bivariate normal distribution is un- 

 bounded, then the 100( l-a)% confidence interval is 

 obtained by ordering the lp<,j from smallest to larg- 

 est, and taking as bounds the values at positions 

 Nmc(oc/2) and NMc(l-(oil2), where N}^c is number 

 of Monte Carlo trials. If the resampling through the 

 bivariate normal distribution is bounded, with 

 bounds al2 and l-aJ2, then the 100(l-a)% confi- 

 dence interval limits are obtained as the first and 

 last quantiles when ordering the //>.,; ^ from smallest 

 to largest. 



Monte Carlo simulation 



To test the performance of the three procedures in 

 estimating Ip,. for different sample sizes, we carried 

 out a simulation analysis of a model population with 

 known size-at-age structure, maturity-at-size, and 

 mortality parameters ( Table 2 ). We explored only the 

 behavior of the methods for median (50%) size at 

 maturity (/^^,, ). Performance was evaluated by us- 

 ing four criteria. First, as the proportion of times that 

 confidence intervals did contain the true (and known ) 

 parameter (Ir^Qc,^ ), which we call success: 



success = 1 - failure 



number\(true -luwer)iupper - true) < 0} 

 number of iterations 



(12) 



Our second criterion was bias, evaluated as the av- 

 erage, over trials, of the sufficient statistic: 



bias 



resampled median 



(13) 



true 



which is 1 for an unbiased estimator. The third crite- 

 rion was the length of confidence intervals: 



length = upper - lower. 



(14) 



and the fourth and final criterion was the shape of 

 the interval (Efron and Tibshirani, 1993): 



shape 



upper - median 

 median -lower 



(15) 



which measures asymmetry around the median. In 

 all four measures of performance, "upper" and "lower" 

 refer to the bounds of the confidence interval, "me- 

 dian" is the median Ipc^, and "true" refers to the true 

 value. The deterministic and stochastic features of 

 our simulation were chosen for a population with 

 features like those previously reported for the squat 

 lobster iPleuroncodes monodon) from the continen- 

 tal shelf off central Chile (Roa, 1993a, 1993b). 



To accomplish this task, we implemented the fol- 

 lowing three-step algorithm, which we called 

 MATSIMVL: step 1, generation of A^,^,,^=5000 random 

 samples of maturity-at-size data of sample size 

 N , = 500, 1000, 3000, 5000, and 10,000 individu- 



sample 



als (Eqs. 16-19); step 2, estimation of the parameter 

 vector and covariance matrix for each one of these 

 samples (Eqs. 1 and 2); and step 3, running each of 

 the three methods to obtain the 2.5%, 50%., and 97.5%' 



