FISHERY BULLETIN: VOL. 76. NO. 3 



CASE STUDIES 



We applied the three-parameter method to the 

 catch-efibrt data of the yellowfin tuna fishery of 

 the eastern tropical Pacific, 1934 through 1967 

 [the same data that were analyzed by Pella and 

 Tomlinson ( 1969) and by Fox ( 1971)]. Table 4 gives 

 a comparison of results, and our final equilibrium 

 model is shown by Figure 3. As indicated by Table 

 4, the parameter estimates of the Levenberg- 

 Marquardt method are comparable with the esti- 

 mates that Fox obtained with his search al- 

 gorithm. Pella and Tomlinson also employed a 

 searching algorithm but their minimization 

 criterion was an unweighted least-squares func- 

 tion. Our standard deviation estimate is very 

 small for w( MSY) but relatively large f or B^,n,q, 

 and fi„, which is a consequence of insufficient in- 

 formation in the yellowfin tuna data on yield at 

 high fishing rates. With such limited information, 

 one can anticipate that neither the shape nor the 

 location of the descending portion of the equilib- 

 rium curve (dashed in Figure 3) could be deter- 

 mined with much accuracy, and the large variance 

 estimates on the system coefficients reflect this 

 situation. Of course, the variance estimates for 

 /^j^Y ^^^ ^MSY ^^^ always be calculated by the 

 delta method, and to avoid the complex deriva- 



tions that accompany the presence of covariance 

 terms, an alternative would be to define a new 

 parameter space so as to estimate /"msy oi' ^^msy 

 directly. The variance-covariance matrix for the 

 coefficients would then provide the desired infor- 

 mation on the variability of those parameters. 



Our final example is based on the data from the 

 Pacific halibut fishery in International Pacific 

 Halibut Commission Area 2, as given in Ricker 

 (1975, table 13.1). To analyze these data, Ricker 

 derived an estimate of c/ from the age composition 

 of the catch. Then he obtained parameter esti- 

 mates for a Graham-Schaefer model by regressing 

 Ye/B against B and Yg/f against / (Ricker 1975, 

 examples 13.5 and 13.6). In both cases, Ricker 

 employed GM and Nair-Bartlett regression. The 

 results Ricker obtained by fitting the Graham- 

 Schaefer model were compared with the results we 

 obtained from fitting the generalized stock pro- 

 duction model by our three-parameter version of 

 the Levenberg-Marquardt method (Table 5). The 

 latter provided estimates of m, q, and n with rela- 

 tively small variance estimates. Furthermore, the 

 estimate of n appears to be significantly different 

 from 2.00, which validates the use of the Pella- 

 Tomlinson model. Nevertheless, estimates of m 

 are not significantly affected by the choice of the 

 wrong model, while estimates of /msy are slightly 



Table 4. — Comparison of parameter estimates obtained by Pella and Tomlinson (1969), by Fox ( 1971) and by the 

 Levenberg-Marquardt algorithm for the yellowfin tuna in eastern Pacific Ocean. Values that follow the ± signs 

 are the standard-deviation estimates for each parameter. 



Table 5.— Comparison between the estimates of Ricker ( 1975) for the Pacific halibut ( Interna- 

 tional Pacific Halibut Commission Area 2) and those obtained by the Levenberg-Marquardt 

 algorithm. Values that follow the ± signs are the standard-deviation estimates for each 

 parameter. 



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