FISHERY BULLETIN: VOL. 76, NO. 3 



In a few additional simulations (replicates 13 

 and 15), parameters obtained by the five-para- 

 meter procedure were ill determined. A parameter 

 is considered ill determined if its estimated 

 value responds unreasonably to seemingly insig- 

 nificant variations in the data (Bard 1974). The 

 basic difficulty is that the model is extremely gen- 

 eral and capable of several types of behavior over 

 the space of 0. In the Pella-Tomlinson system, 

 ill determination often occurs whenever an itera- 

 tion of the algorithm (11) gives an estimate of O 

 such that the point (m , /msy* o^ ^he yield-effort 

 plane lies outside the concentration of data. In 

 such a circumstance the exponent n takes on small- 

 er and smaller values in the successive iterations 

 and the solution of system (1) and (3) degenerates 

 to an exponential form for which only four 

 parameters are required for uniqueness. That is, 

 as/? ->0. in Equation (1), then (B/By.)" -►I and y 

 -► - 1 . The five-parameter procedure then over- 

 prescribes the system, which in turn predisposes 

 the coefficient estimates to extremely large var- 

 iances. The ultimate irony here is the fact that 

 wholly unrealistic parameter estimates still gen- 

 erate good fits to the catch-effort history (i.e. small 

 residuals). For example, in Figure 2 the fitted 

 five-parameter curve predicts /"jv^gy near infinity 

 while in the true model /'msy actually corre- 

 sponds to 174,000 units of effort. However dif- 

 ferent the equilibrium curves are, the five- 

 parameter procedure still generates a good fit to 

 the catch history (S(0) = 1.10). Incompleteness of 

 information over a wide range of effort values, as 



well as excessive noise in the catch-effort data, 

 will tend to bring about such pathological condi- 

 tions. 



To overcome these difficulties, reformulation of 

 the estimation problem is necessary. By the fol- 

 lowing considerations, the five-dimensional 

 parameter space can be reduced to three dimen- 

 sions. First, we will approximate £„ by Equation 

 ( 20). Furthermore, if the data contain information 

 on the yields under low exploitation, we may 

 define Sx as 



B 



= MAXiYJq f.) 



r. 



(21) 



By using Equations (20) and ( 21 ),fi, I andfixcanbe 

 deleted from G, leaving only m, q, and n as the 

 independent parameters requiring estimation. It 

 is important to understand at this point that B^ 

 and fix are not fixed; they are reevaluated by 

 Equations (20) and (21) at each iteration, along 

 with the parameters tu ,q, and n . In fact, the solu- 

 tion of Equations ( 1) and (3), as well as Equations 

 (20) and (21), specify a new model with unknowns 

 O =[m,q, nY. By this restructuring, much of the 

 degeneracy associated with the model can be 

 eliminated. As shown in Figure 2, this procedure 

 also provides a closer correspondence between the 

 "estimated" and the "true" equilibrium model. 

 Furthermore, the three-parameter procedure still 

 generates an adequate nonequilibrium catch his- 

 tory (S(G) = 1.40). In a Monte-Carlo simulation 

 study, parameter estimates obtained by using 

 these transformations fell within reasonable 



FlOURE 2. — Comparison of the "true" 

 model with the models obtained by 

 using the estimation procedure on three 

 and five parameters respectively- Solid 

 lines show equilibrium yield curves; 

 data points show nonequilibrium simu- 

 lated (dots) yields and predicted (circles) 

 yield values from the three-parameter 

 approach. Dashed vertical lines indi- 

 cate the magnitude of residuals. 



UJ 



528 



