FISHERY BULLETIN: VOL. 73. NO. 1 



12-yr catch and fishing effort history by both 

 methods for a range of averaging times are given 

 in Table 2. Examination of the appropriate row for 

 each method in Table 2 clearly reveals that for this 

 example the superior estimates were produced by 

 the weighted average method. In fact, the max- 

 imum error among the weighted average esti- 

 mates of m, Fmax, and Umax (the parameters used 

 in searching for the least-squares solution) at 

 /e = 4 is only 1%. 



The effect of different averaging times on the 

 estimates of the parameters m, Ymax, and C/max is 

 the same for both effort averaging methods. By 

 increasing the averaging time, the estimates of m 

 and Ymax decrease and the estimate of Umax in- 

 creases. The residual sum of squares is minimum 

 at the appropriate averaging time for the weight- 

 ed method (i.e. at /s = 4). For the unweighted 

 method, however, the minimum residual sum of 

 squares is at 1 yr greater than the appropriate 

 criterion. 



Another way of comparing the weighted and 

 unweighted averaging methods is to examine how 

 well they estimated the equilibrium fishing effort. 

 The equilibrium fishing effort was computed for 

 each year's observed catch per unit effort (Figure 

 3), using the production model fitted to the 

 equilibrium data (Table 1) for interpolation. The 

 estimated equilibrium fishing effort by the 

 weighted average method was closest in 10 of the 

 12 yr and had a mean absolute error of less than 

 one-third of the unweighted method (Table 3). 



Estimates of the catchability coefficient, q, by 

 the integral method, Equation (22), — geometric 

 and arithmetic means — and the difference 



Table 2. — Empirical and estimated parameters for the simu- 

 lated pandalid shrimp catch history using the equilibrium 

 approximation approach and two methods of averaging fishing 

 effort. 



Method 



Averaging 



time 



{k or T) 



Mean 

 squared 

 'q error 



Empirical 



Weighted 

 average 

 fishing 

 effort^ 



Unweighted 

 average 

 fishing 

 efforts 



20.60 5.60 17.96 



1.00 — 



1 

 2 



3 



M 



5 



6 



1 

 *2 



3 

 4 



1.16 

 1.35 

 0.86 

 0.60 

 0.53 

 0.51 



1.16 

 1.09 

 0.35 

 0.28 



7.26 

 6.48 

 6.02 

 5.67 

 5.21 

 4.76 



7.26 

 6.17 

 6.10 

 5.36 



17.63 

 1761 

 17.82 

 17.97 

 18.01 

 18.02 



17.63 

 17.69 

 18.07 

 18.14 



0.42 

 0.62 

 0.88 

 0.87 

 0.75 

 0.62 



0.42 

 0.70 

 1.12 

 0.69 



1 .3830 

 0.3293 

 0.0686 

 0.0500 

 0.0913 

 0.1236 



1 .3830 

 0.1323 

 0.0529 

 0.2797 



'Integral method, geometric mean. 



^Estimated, Table 1. 



^Equation (9); Program PRODFIT, unweighted estimates option. 



^Appropriate averaging time. 



'Equation (7); Program PRODFIT, unweighted estimates option. 



Table 3. — -Comparison of two estimates of equilibrium fishing 

 effort for the simulated pandalid shrimp population catch his- 

 tory. 



i 



'Equation (9); /< = 4. 

 ^Equation (7); 7 = 2. 

 ^Calculated from Table 1 parameters. 



method. Equation (18), for the weighted {k = 4) 

 and unweighted (T = 2) fishing effort averaging 

 techniques are given in Table 4. The best es- 

 timator within either effort averaging method 

 was the integral method's geometric mean, with 

 the weighted average fishing effort method being 

 closest to the assumed value, 1.0. 



Stochastic Comparison 



In the deterministic comparison, the catch and 

 fishing effort data were known precisely, the catch 

 per unit effort was always exactly proportional to 

 the average population size, and the population 

 did not fluctuate. However, the stochastic nature 

 of population processes, temporal and spatial 

 changes in the availability and vulnerability of 

 the population to fishing, and the use of sample 

 data to represent an entire fishery all introduce 

 considerable variability in real catch and effort 

 data. Under the assumption that the component 

 sources of variability are independent and random 

 variables with constant expected values and vari- 

 ances, an approximation of the overall variability 



Table 4.— Estimates of the catchability coefficient, q, by three 

 methods for the weighted and unweighted fishing effort averag- 

 ing techniques. Actual value of q is 1.0. 



'Equation (22). 

 ^Equation (18). 

 ^Equation (9); k = 4. 

 "Equation (7); 7 = 2. 



30 



