FISHERY BULLETIN: VOL. 73, NO. 1 



An arithmetic average rather than a geometric 

 average is suggested because most appHcations 

 are on catch in weight, i.e. while year classes de- 

 cline exponentially in terms of numbers they con- 

 comitantly increase in terms of mean weight per 

 individual. 



The weighting procedure can be more precise 

 if it is knowTi when during the year of record that 

 recruitment occurs. For example, if recruitment 

 occurs at midseason during the year of record 

 for a fishable population of three year classes, 

 f, changes from ( 3 /; + 2 /": _ ^ + /; _ 2 ) /6 to 

 {2.5/; + 1.5 /; - 1 + 0.5/; 2! /4.5. Further pre- 

 cision is gained if k is variea from year to year 

 with the level of fishing effort, since at high fish- 

 ing rates fewer year classes will contribute sig- 

 nificantly to the catch than at low fishing rates. 

 Further adjustments can be made for unequal 

 catchability among the year classes. 



The unweighted method of averaging the 

 fishing effort. Equation (7), and the new weighted 

 method, Equation (9), will be compared in a sub- 

 sequent section of this paper. 



Estimation Procedure 



critical points in terms of the parameters of Equa- 

 tion (11) are 



/"opt = (a - a/n)/(m|3) 



C^opt = (aim) 



Vim - 1) 



(12) 



(13) 



and 



(a — am) ia/m) 



l/(m - 1) 



(14) 



Given the data set {u, , f,] , where i = 1. . .n 

 observations, the least-squares criterion for es- 

 timating the parameters a, /3, and m is to 

 minimize the function 



•5 = 2 w^.(u, - U,)2 



(15) 



i = 1 



where the W, are statistical weights, and f7, are 

 the predicted equilibrium catches per unit effort 

 from Equation (11). The statistical weights. 



w^ = iuy\ 



(16) 



Gulland (pers. commun.)^ prefers an eye-fitted 

 curve for estimating the equilibrium relationship 

 between C/, and /", because of the over- 

 simplification of the method and the errors as- 

 sociated with usual catch and effort data. How- 

 ever, these reasons should not defer the seeking of 

 a more precise method of fitting a curve nor the 

 taking advantage of error estimation schemes, if 

 the simplifications and assumptions are kept in 

 mind. On the contrary, it will be demonstrated 

 that, at least for some controlled conditions, the 

 equilibrium approximation approach provides 

 reasonably good results. 



Equation (5) may be written in terms of catch 

 per unit effort and averaged fishing effort as 



m - 1 



U, = [iKq IH) + {q IH)f^ ] 



- .1/ (m - 1) 



or simply 





1) 



(10) 



(11) 



Equation (11) is a nonlinear function with three 

 parameters which does hot require simultaneous 

 estimation of the catchability coefficient, q. The 



*John A. Gulland, Food and Agricultural Organization, Rome, 

 Italy. 



are derived from the assumption of the multiplica- 

 tive error structure as suggested by Fox (1971). 

 Weighting as in Equation (16) will usually give 

 the greatest weight to observations at the highest 

 level of averaged fishing effort; in many cases 

 these also will be the most recent observations. 

 Giving greater weight to observations at high ef- 

 fort levels will tend to give the greatest weight to 

 observations vidth the greatest temporal and spa- 

 tial coverage of the population. In addition, giving 

 the greatest weight to the most recent data is 

 especially advantageous when approaching the 

 ^max level during a period increasing fishing effort 

 because the observations nearest the Fmax level 

 receive the greatest weight. 



Up to now no mention has been made on the 

 estimation of the catchability coefficient, q. This is 

 because experience with GENPROD and stochas- 

 tic simulation studies have indicated that poor 

 results are frequently obtained from the simul- 

 taneous estimation of q (Pella and Tomlinson 

 1969; Fox 1971). Once that a, |8,andm have been 

 estimated, q may be treated as a conditional prob- 

 abilistic variable and estimated as a mean value. 

 Two tacks were selected, the difference method 

 and the integral method. 



26 



