FOX: FITTING THE GENERALIZED STOCK PRODUCTION MODEL 



Table 5. — Empirical and estimated parameters for the five rep- 

 licated stochastic catch histories using the equilibrium approx- 

 imation approach and two methods of averaging fishing effort. 



'Integral method, geometric mean. 



^Estimate, Table 1. 



^Assumed value. 



■•Equation (9); k = 4; Program PRODFIT, weighted estimates option. 



^Standard error of the mean. 



^Equation (7); 7 = 2; Program PRODFIT, weighted estimates option. 



4- 



g 6 



UJ 



> 



3 



m 



13 

 O 

 UJ 



J o J 9 ® 



J o o o 



« « 8 J • 



J L 



J_ 



X 



o o o 



8 ° 



_L 



J I L 



5 



_l_ 



0.0 0.4 0.8 12 0.0 04 0.8 12 



FISHING EFFORT 



Figure 4. — Results (dots) of five stochastic simulated catch 

 trials for the equilibrium approximation approach to fitting 

 the generalized stock production model with the weighted 

 averaging method. Circles are the true values. 



structure is the multiplicative error model (Fox 

 1971) 



C, = C- 



(28) 



where C, is the observed catch in year i, Ci* is the 

 expected catch, and e, is a random variable with 

 an expected value of 1 and standard deviation o . In 

 practice, however, the e, are usually correlated 

 because some (or all) of the component sources of 

 variability do not meet the assumptions. 



An ideal (i.e., in the sense that the e, are inde- 

 pendent and random) error structure was chosen 

 to illustrate the estimation ability of the two 

 equilibrium approximation methods, because the 

 "true" error structure of any given population and 

 fishery is unique and largely unknown. Five inde- 

 pendent sets of 12 pseudorandom, normally dis- 

 tributed variables, 6, as with an expectation of 

 zero and a standard deviation of 0.1 were pro- 

 duced with the Library Subroutine RAND (Uni- 

 versity of Washington Computer Center, Seattle). 

 The sets of 5's were used to produce five stochastic 

 catch data sets from the deterministic catch his- 

 tory (Figure 3) and Equation (28), with e, de- 

 fined as 1 + 6 , . 



The results of fitting the five replicate sets of 

 catch and effort data by the weighted (Equation 9) 

 and unweighted (Equation 7) averaging methods 

 are given in Table 5. The effects of even moderate 

 variability on the parameter estimates for both 

 averaging methods are apparent. On the average, 

 two (m and i^max) of the three determining 

 parameters (m, Ymax, and Umax) are closer to the 

 empirical values for the weighted effort averaging 

 method. The important observation, however, is 

 that all the unweighted estimates of i^max fall 

 above the empirical value and that the average 

 over the five replicates is significantly different 

 from the empirical value with probability greater 

 than 0.999. 



Plots of the empirical equilibrium yields and 

 those determined from the generalized stock pro- 

 duction model parameters estimated by the 

 weighted average method are compared in Figure 

 4. Equilibrium yield, for the most part, is esti- 

 mated reasonably well in each replicate for the 

 range of estimated "equilibrium" fishing effort, 

 0.0 to 1.0 (Table 3). The exception is replicate 4 

 where the empirical equilibrium yield is substan- 

 tially underestimated above f = 0.8. Beyond the 

 range of data, f = 1.0 to 1.3, the equilibrium yield 

 is estimated reasonably well on the average, but 

 not individually. None of the fitted models, of 



31 



