FOX: RANDOM VARIABILITY AND PARAMETER ESTIMATION 



The referee of this paper has raised an im- 

 portant point regarding- application of the var- 

 ious statistical models to actual fishery data. 

 In a non-overexploited fishery, generally, the 

 quality and level of catch and effort values in- 

 crease with time. Eelatively speaking, Model 

 in this case places greater weight on more recent 

 data than do Models 1, 2, or 3, and in the ab- 

 sence of any other criteria it might represent 

 the intuitive choice. However, if the quality 

 of the data were a more significant contributor 

 to unequal residence variance than the statistical 

 model, one would expect, in this case, a decrease 

 in the residuals plotted for Model against time, 

 catch, and fishing effort in contrast to the appar- 

 ent increase for the yellowfin tuna fishery (Fig- 

 ure 3) . If one has reason to suspect a significant 

 difference in quality of the data, as would be 

 suggested by a decrease in the residual plots 

 of Model 0, perhaps a solution is to partition 

 the data at the point in time where a significant 

 quality increase occurs. Then fit each set of 

 data individually placing greater weight on the 

 parameter estimates for the more recent set. 

 The specter of the suitability of employing pro- 

 duction models over long time periods is also 

 raised by this point. But it is outside the scope 

 of this paper and the reader is referred to the 

 papers cited previously. 



SUMMARY 



In using a least-squares procedure for esti- 

 mating parameters of a mathematical model, 

 such as the Pella-Tomlinson technique, there are 

 three assumptions about the residuals for ob- 

 taining the best least-squares estimates: 1) the 

 residuals are independent, 2) the residuals have 

 an expected value of zero, and 3) the variance 

 of the residuals is constant (Anscombe and 

 Tukey, 1963; Draper and Smith, 1966; Snedecor 

 and Cochran, 1967). We have observed from 

 the simulation study that two (of four alterna- 

 tive) simple statistical models which are bio- 

 logically sound — Model 2 (using a logarithmic 

 transformation) and Model 3 (weighting by the 

 inverse of the squared deterministic catch) — 

 fulfill the statistical assumptions for obtaining 



good least-squares estimates of the generalized 

 production model parameters over a wide range 

 of fishing effort. 



On applying these four statistical models in 

 estimating the parameters of the generalized 

 production model for the eastern tropical Pacific 

 yellowfin tuna fishery, residuals e.xamination re- 

 vealed that the same two statistical models. 

 Models 2 and 3, fulfilled the least-squares esti- 

 mation assumptions. Models (assumed by 

 Pella and Tomlinson, 1969) and 1 did not. Model 

 3 was selected as the best model since it involves 

 the direct minimization of the actual residual 

 variance, and is therefore considered to be theor- 

 etically superior to Model 2. 



Finally, anyone using the generalized pro- 

 duction model and the Pella-Tomlinson estimat- 

 ing technique should be aware of, in addition 

 to the proper statistical model, the effect of the 

 value of A'^ in equation (5) on the parameter 

 estimates. 



ACKNOWLEDGMENTS 



I wish to express my appreciation to Dr. 

 Douglas G. Chapman, Director of the Center for 

 Quantitative Science in Forestry, Fisheries and 

 Wildlife, University of Washington, for the con- 

 siderable consultation he gave during the course 

 of this study and for his review of the final 

 manuscript. Many others, especially the referee, 

 who offered suggestions on clarification and im- 

 provement of the value of this study, are also 

 extended my gratitude. 



LITERATURE CITED 



Anscombe, F. J. 



1961. Examination of residuals. Proceedings of 

 the Fourth Berkeley Symposium on Mathematical 

 Statistics and Probability. 1 : 1-36. 

 Anscombe, F. J., and J. W. Tukey. 



1963. The examination and analysis of residuals. 

 Technometrics 5: 141-160. 

 Beverton, R. J. H., AND S. J. Holt. 



1957. On the dynamics of exploited fish populations. 

 Fish. Invest. Min. Agr. Fish. Food (G.B.), Ser. II, 

 19, 533 p. 



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