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Fishery Bulletin 92(2). 1994 



avoided by use of a median correction, which is quite 

 resistent to outliers. However, the use of a median 

 correction implies that the estimated bias correction 

 will be nonzero for an estimator that is unbiased (in 

 the usual sense) but arises from a distribution in 

 which the median does not equal the mean. That is, 

 the use of a median bias correction transforms the 

 estimator into a median estimator. 



Several methods have been developed for comput- 

 ing bias-corrected confidence intervals from the boot- 

 strap (Efron, 1982; Efron, 1987; Noreen, 1989). The 

 most widely used at present appears to be the BC 

 method of Efron (1982). To compute a BC interval, 

 let Mz) be the cumulative distribution function (CDF) 

 of the standard normal distribution and \etN~ l be the 

 inverse— normal CDF. Let C be the empirical bootstrap 

 CDF of the parameter 9 ; i.e. C(g) is the proportion of 

 realizations of 9 in the bootstrap distribution that falls 

 below any arbitrary value g. Define the constant 



z*=N~ 



C(9) 



14) 



where 9 is the conventional estimator. Then, the 

 (1 - 2a) BC central confidence interval on 9 is de- 

 fined as 



0e{c- 1 [7V(22 o +iV- 1 (a))],C- I [iV(22 o -iV- 1 (a))]}. (15 



This method assumes that a transformation exists 

 under which the distribution of 9 becomes normal 

 and homoscedastic. However, the form of the trans- 

 formation need not be known (Efron and Gong, 1983). 

 Kizner (1991) constructed bootstrap confidence in- 

 tervals on production-model results, but he did not 

 state whether bias corrections were used. 



This discussion of bootstrapping has referred to 

 estimated "parameters" for simplicity, but the method 

 can be used to estimate bias corrections and bias- 

 corrected confidence intervals for any estimated 

 quantity. Such quantities might include estimates 

 of MSY, /" MgY , the population biomass in the final (or 

 any other) year, f 01 , projections of biomass levels 

 (discussed next), and so forth. 



Projections 



Because a production model implicitly includes a 

 recruitment function, it can be used to make projec- 

 tions based on hypothetical catch or effort quotas. 

 As noted above, the historical population biomass 

 trajectory is estimated during parameter estimation. 

 The modeled population can then be projected for- 

 ward in time by using the same population equations 

 (4, 6, 8), and a proposed set of yields or effort rates. 

 If the bootstrap is used following parameter estima- 



tion, the results of each bootstrap trial can be pro- 

 jected forward. From the results, it is possible to com- 

 pute bias— corrected point estimates and confidence 

 intervals on the projection results. 



Example: North Atlantic swordfish 



Many aspects of the production model discussed 

 above are illustrated in this example, which is loosely 

 based on swordfish, Xiphias gladius, in the North 

 Atlantic Ocean. The example comprises two analy- 

 ses, the difference between them being the use of an 

 abundance index for tuning the second analysis. Both 

 the base analysis and the tuned analysis used the 

 same yield and fishing-effort data (Table 1; Fig. 1); 

 the tuned analysis also used a hypothetical index of 

 abundance constructed for this purpose (Table 1; Fig. 

 1). In both analyses, errors were assumed to occur in 

 fishing effort and to follow a lognormal distribution; 

 in other words, the "second estimation approach" 

 described previously was used. Each analysis in- 

 cluded a projection of five years beyond the actual 

 data; during those five years, it was assumed that a 

 yield of 12,000 metric tons would be taken annually. 

 Each analysis included a bootstrap with 1,000 trials. 



This example is not intended as, and should not 

 be considered to be, a formal assessment of the sword- 

 fish fishery. Such an assessment would normally in- 

 clude additional information and analysis, including 

 age-structured population models and numerous sen- 

 sitivity analyses. Also, the abundance index used 

 here was developed solely to serve an example, and 

 is not believed to be an accurate reflection of abun- 

 dance over time. 



The North Atlantic swordfish fishery enjoyed a 

 high catch rate in 1962 and 1963, but it declined in 

 the late 1960s (Fig. 1). The U.S. and Canadian por- 

 tions of the fishery were sharply reduced in the early 

 1970s because of FDA regulations prohibiting inter- 

 state transportation or importation offish with mer- 

 cury concentrations exceeding the allowable level of 

 0.5 ppm (Hoey et al., 1989). In 1978, the FDA in- 

 creased the allowable mercury content to 1 ppm, and 

 since then, the catch has increased, but the CPUE 

 has slowly declined (Fig. IB). For the years 1971- 

 73, early years of the FDA regulations, data are avail- 

 able on catch but not on fishing effort. 



Results from the two analyses were similar, but 

 they illustrate how tuning can influence the results 

 of a production model. In each analysis, the model 

 fits the effort data reasonably well (Fig. 2); however, 

 because the hypothetical abundance index does not 

 match the observed CPUE well (Fig. IB), the fit in 

 the last years of the tuned model was a compromise 



