MENDELSSOHN: ESTIMATING POPULATION SIZES 



While the parameterization H{t) = {f{t)s{a)} 

 greatly reduces the number of parameters, I still 

 face the same problem as when I treated the un- 

 observed population as parameters: each new period 

 adds another parameter to the model. The model 

 still appears to be overparameterized, and the 

 asymptotic theory for maximum likelihood estima- 

 tion may be invalid. 



As with the recruitment estimates, the number 

 of effective parameters can be reduced by adding 

 a smoothness prior on/(0- However, /(^ is a pro- 

 portion and not likely to be a normal variable. The 

 f{t) are also constrained to lie in the interval (0, 1), 

 so a transformation to an unconstrained variable 

 would also be desirabl e. If /(0 is a binomial random 

 variable then arcsin {\Jf{t )) is approximately normal 

 with nearly equal variance. This suggests the trans- 

 formation of variables 



y{a,t) = s{a)sin'^{eit))x{a,t). 



(37) 



fit) = sinHeit)) 



(35) 



where eit) is an unconstrained normal variable. 

 Then the smoothness prior becomes 



Vh{t) = w(t) 



(36) 



(a smoothness prior that includes a seasonal com- 

 ponent or irregular stationary part, as in Equation 

 (30), can also be used), and for any age class, the 

 observation equation becomes 



The underlying population dynamics must also be 

 expanded to include the smoothness prior constraint 

 (Equation (36)). 



Unfortunately, the observation equation is no 

 longer linear in the state vector. The smoothness 

 prior is a prior distribution on the f{t), and a full 

 Bayesian analysis can be done to obtain the overall 

 distribution. The variance o{w(t) is then treated as 

 a hyperparameter in the analysis. 



A simpler approach is to evaluate the filter equa- 

 tions approximately by using any one of a number 

 of nonlinear filters (see Anderson and Moore 1979). 

 One that is easy to implement, given the nonlinear- 

 ities in this problem, is the extended Kalman filter 

 (EKF), which at each time period just linearizes all 

 the nonlinear terms around the value of the pre- 

 dicted state vector. The EKF, however, can have 

 divergence problems and is not guaranteed to find 

 the true penalized likelihood estimates. 



When using the EKF, it works to make a forward 

 and backward pass of the filter given the current 

 estimates of f{t) and x{t\T), and then to estimate 

 s{a) and R as before. I tested the algorithm on the 

 mackerel data with k = 1 in the constraint (Equa- 

 tion (36)). The resulting estimates (Fig. 5) are similar 

 to the previous estimates, but the estimated values 

 of /(O (Fig. 6) are less variable with a stronger trend 

 than before. It is clear from Figure 5 that the 



400000 



00 



§ 300000 



u 



w 



O 



a: 200000 



w 



CO 



H 100000 



W 







SMOOTHED 



FILTERED 



PREDICTED 



1930 



1940 



1950 



1960 



1970 



TIME 



Figure 5.— Estimated number of recruits using a first order spline for estimating /{<). 



627 



