FOX: RANDOM VARIABILITY AND PARAMETER ESTIMATION 



Table 2. — Regression analysis for statistical Model 3 of standard deviation of catch residuals, S{ea), on determin- 

 istic catch, C, with levels of fishing effort below and above that which produces the maximum sustainable yield (MSY) . 



In conclusion, the assumption of statistical 

 Models and 1 were rejected by the simulation 

 study. Statistical Models 2 and 3 were found 

 to be valid over a wide and similar range of 

 fishing effort. Their range of validity includes 

 up to and well beyond the level of fishing effort 

 producing the MSY (/ = 40,000) , the most likely 

 range in which a fishery would operate. Em- 

 ploying Model 3 has a theoretical advantage over 

 Model 2 in a least-squares estimating procedure. 

 With Model 3, the actual residual variance is 

 minimized. Whereas with Model 2 the log-re- 

 sidual variance is minimized and the parameters 

 are best least-squares estimates only in the trans- 

 formed model. The theoretical advantage of 

 Model 3 may serve as a criterion for choosing it 

 when no other criteria exist. 



Several additional simulation trials were made 

 to demonstrate the relative degree of influence 

 that random variability in each parameter ex- 

 erts on the variance of the catch residuals. The 

 upper two standard deviations of each stochastic 

 variable was set equal to 25 '^r of their mean, 

 the level of fishing effort was set at 40,000 ( MSY- 

 producing level), and four trials of 500 time 

 periods each were made. Each parameter in turn 

 was allowed to vary with the remaining three 

 constant (Table 3). The variation in catch was 

 most sensitive to varying the exponent, m, and 

 least sensitive to varying the catchability coeffi- 

 cient, q. This, of course, implies the relative 

 precision of the parameters if they had been ac- 



Table 3. — Catch residual variance produced by variation 

 in each stochastic variable of the generalized production 

 model. 



tual estimates. One should not, however, gener- 

 alize on the order of precision since these results 

 obtain specifically for the assumed probability 

 distributions and expected values. This exercise 

 does demonstrate a frequently employed method 

 for implying which parameters, given their esti- 

 mates, are most critical and perhaps deserving of 

 additional independent estimation. 



RESIDUALS EXAMINATION: 

 AN EXAMPLE 



The data of catch, catch per unit effort, and 

 fishing effort from the eastern tropical Pacific 

 yellowfin tuna fishery (Pella and Tomlinson, 

 1969; Table 6) are plotted in Figure 2. Appar- 

 ently the population and fishery dynamics are 

 250r . A 



E 200 



150 



.= 100 



if 50 



t 



_L 



_L 



J_ 



_L 



10 20 30 40 50 



Fishing Effort (Thousands of Boat Days) 



60 



1— o 



— DO 



t'f 



B 





60 



"0 10 20 30 40 50 



Fishing Effort (Thousands of Boat Days) 

 Figure 2. — Data from the eastern tropical Pacific yellow- 

 fin tuna fishery, 1934-67, plotted as (A) catch vs. fishing 

 effort, and (B) catch per unit effort vs. fishing eflTort. 



575 



