686 



Fishery Bulletin 94(4), 1996 



With variance assumption S2, we have only pro- 

 cess error. With this assumption both the Kalman 

 filter and nonlinear least-squares biomass estimates 

 are biased high (Table 2; Fig. 4). However, root mean 

 square errors show that the Kalman filter perfor- 

 mance is clearly superior to nonlinear least squares 

 for this case. It is interesting that simulation S2 also 

 shows that nonlinear least-squares estimates are not 

 necessarily more biased under process error but that 

 parameter estimates have larger variances. 



Variance assumption S3 represents a combination 

 of process and measurement error. Results from S3 

 are similar to those from S2, with biomass estimates 

 appearing to be biased high for both the Kalman filter 

 and nonlinear least-squares methods (Table 2; Fig. 5), 

 but with RMSE clearly favoring the Kalman filter 

 method over the nonlinear least-squares method. It 

 should be noted that in simulation S2 and S3, 1 or 2 of 



the nonlinear least-squares fits failed to converge prob- 

 ably owing to the inclusion of process error. These simu- 

 lations were thrown out and the runs were repeated. 

 Results from simulation SP1 (Table 3) indicate that 

 Schnute's form of the delay-difference equation can 

 be used with the Kalman filter, despite the violation 

 of the assumption that process error vectors are in- 

 dependent. And finally, results from simulation SP2 

 (Table 4) indicate that the Kalman filter parameter 

 estimates have smaller RMSE compared with non- 

 linear least-squares parameter estimates when only 



the value of r 



Op 1 *** 



is assumed in the model fit. 



Conclusions 



Simulations show that with the fit of the delay-differ- 

 ence equation, the Kalman filter and nonlinear least- 



