Kimura et al.: Kalman filtering the delay-difference equation 



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ment model because it can lead to overfishing. How- 

 ever, simulation results also indicate that under the 

 presence of process error, nonlinear least-squares 

 methods provide biomass estimates with comparable 

 bias, and have much poorer RMSE properties. We 

 therefore conclude that the Kalman filter method 

 provides superior parameter estimates in the face of 

 process error. 



Simulation results indicate that Schnute's form of 

 the delay-difference equation can be used with the 

 Kalman filter despite violation of the independent 

 process error assumption. Although it appears diffi- 

 cult to estimate r directly with maximum-likelihood 

 estimation, simulation results indicate that the 

 Kalman filter can be used to estimate biomass when 

 only r = o 2 p I a 2 m is known, rather than the individual 

 variances. 



It is important to acknowledge that simulation 

 results will differ if natural mortality, catch levels, 

 growth parameters, biomass, and recruitment, or er- 

 ror variances, are changed. We believe that custom- 



ized simulation studies should be a routine part of 

 production modeling. The simulations performed in 

 this paper favored the Kalman filter in that the true 

 measurement and process error variances were as- 

 sumed to be known to the Kalman filter fitting algo- 

 rithm. Under these circumstances, the performance 

 of the Kalman filter method might be described as 

 less than fully satisfactory. 



There appears to be few published simulation stud- 

 ies describing the performance of production models 

 in the presence of process error. Process error is par- 

 ticularly troublesome because errors are propagated 

 through the years rather than being quickly forgot- 

 ten as is measurement error. Our results, and those 

 of Polacheck et al. (1993), indicate that parameter 

 estimation in the presence of process error is inher- 

 ently difficult for production models. However, the 

 process error method of Polacheck et al. (1993) is 

 regression-based and may not provide fully efficient 

 parameter estimates. Because real data contain pro- 

 cess error, we feel process error must be dealt with, 



