MENDELSSOHN: ESTIMATING POPULATION SIZES 



to be far too smooth. The results of this present ex- 

 ample may explain their observation. If these values 

 were used in a subsequent analysis, say to determine 

 the role of the environment on recruitment, a totally 

 false picture of this relationship could emerge. I em- 

 phasize that the predicted estimates are calculated 

 from the previous season's filtered values, whereas 

 it is often true in the fisheries literature that the 

 predicted values are estimated from the previous 

 predicted values, rather than from the filtered 

 values. Direct comparison with the estimate in 

 Parrish and MacCall (1978) are difficult because 

 restricting recruitment to be of the form (Equation 

 (29)) with k fixed, rather than the more general form 

 (Equation (30)) with k variable, may not be appro- 

 priate for the Pacific mackerel data. But overall, 

 their estimates tend to resemble the smoother trend 

 of my predicted estimates. 



RELATIONSHIP TO OTHER LITERATURE 

 AND A NEW PARAMETERIZATION 



If the models are restricted to additive errors, 

 then most of the simpler difference equation models 

 proposed in Collie and Sissenwine (1983), Deriso 

 et al. (1985), Fournier and Archibald (1982), Four- 

 nier and Doonan (1987), and among others can be 

 formulated as I did. Some of these models assume 

 recruitment is a nonlinear function of the underly- 

 ing population, which cannot be handled in this 

 model without some modifications (suggested 

 below). However, all of these authors treat the 

 underlying population sizes as parameters of the 

 data rather than as missing data. As discussed 

 earlier, it is questionable whether this will produce 

 proper estimates of the underlying populations. 

 Biases from treating missing data as parameters in 

 a regression setting are explicitly discussed in Little 

 and Rubin (1983, 1987). 



A very broad class of possible models that can be 

 selected to model catch-at-age data are given by 

 Schnute (1985). He correctly identified the values 

 that are parameters of the difference equations he 

 discusses, and these are sufficient for estimating the 

 likelihood (if evaluated properly). However, if we 

 assume observation error, then it can be shown (see, 

 for example, Shumway 1988) that the innovations 

 are determined by predictors calculated from the 

 previous period's filtered, rather than predicted 

 values. Moreover, the minimum mean squared error 

 estimates of the underlying populations, as dis- 

 cussed earlier, are the smoothed estimates. It ap- 

 pears that Schnute (1985) used the predicted or. 



at best, the filtered estimates of the underlying 

 populations. 



A popular parameterization that appears to have 

 been first suggested by Doubleday (1976) is to 

 assume that the observation matrix H{t) is of the 

 form H{t) = {s{a)f{t)} where s{a) is an age-depen- 

 dent selectivity factor and/(0 is a time-dependent 

 exploitation rate. These values can be found by using 

 a minimization routine during the M-step of the 

 algorithm. However, it should be noted that the 

 estimate of/(0 for each t will depend on R and that 

 the estimate ofR will depend on both s(-) and/(-), 

 so that either R, s, and / should be solved for 

 together, or else they should be successively solved 

 for using Equation (7) while holding the other 

 parameter values fixed. 



Alternatively, Equation (7) can be differentiated. 

 Then for given values of/(0, the optimum value of 

 the vector s(a) at each iteration are the diagonals 

 of the matrix S given by 



S = AB' 



(33) 



where A = 1 y{t)x{t\Tyfit), and 5 = 1 f^{t) 



t = i 



( = 1 



{P{t\T) + x{t\T)xit\Ty. However, this is an un- 

 constrained estimate and does not guarantee that 

 s(a) is between (0,1). It can be shown that the op- 

 timal solution is to set s(a) at zero if s (a) is negative 

 or to 1 if s(a) is greater than 1. 



For fixed values of sia) and R, Equation (7) is 

 maximized when/(0 takes the value 



fit) 



tr{R-^y{t)x{t\TYS') 



tr{R-^S{P{t\T) + xit\T)xit\Ty)S') 



(34) 



where the matrix iS is as above. This is the uncon- 

 strained solution. The constrained solution again is 

 to force the estimate to He within the closed inter- 

 val (0, 1) as with the estimate of the s{a). Since the 

 estimates oiR, s{a), and/(0 are interrelated, I have 

 found it to be a workable procedure to first estimate 

 Q as given above and then for a given number of 

 iterations, iteratively solve for /(f) then s(a). When 

 these values stabilize, estimate R using the formula 

 given above. While this procedure does not neces- 

 sarily maximize Equation (7), it is sufficient for the 

 generalized EM algorithm the new values increase 

 the function given in Equation (7). 



As with the original estimates, the smoothed and 

 filtered estimates of recruitment (Fig. 3) are close 



625 



