304 



Fishery Bulletin 104(2) 



However, when exploitation rates are all known, as in 

 some ecological modeling exercises (e.g., u=C^IN^, i.e., 

 catch per population), then all instantaneous mortality 

 rates can be solved in closed form. 



Example: As an example, the logit and H-F parameter- 

 ization can be used to estimate the distribution mixture 

 problem for empirical length-at-age data. To describe the 

 mixture problem for empirical length-at-age data, let 



/ = 1 ^ be the length category index; 



7 = 1 a be the age category index; 



/, = the observed length frequency, for which 

 we wish to estimate the corresponding 

 age distribution; 

 /", = fjf. where f=^f,. the observed length 

 distribution; 

 = the unknown age distribution we wish 

 to estimate from the observed length 

 distribution; 

 = the observed distribution of length at 

 age, estimated from age-length data; 

 Ij = '^pfl,, = length distribution estimated from an 

 age distribution and observed length- 

 at-age distributions. 



Solutions to the empirical distribution mixture problem 

 can be stated as solving for that age distribution Ip^l, which 

 when combined with length / at agej, say l<j,^), provides the 

 "best" weighting to reproduce some length frequency If J. 

 "Best" may be defined as 



1 Maximum likelihood (Kimura and Chikuni, 1987): 

 L = const. + ^ /", log (/, ). 



Pj 



q,j = PT0hll\j] 



2 Minimum chi-square 



r = I</',-A''/<A> 



For both of these estimation problems, the estimated 

 Ip I distribution must be constrained to sum to one. The 

 logit or H-F parameterization simplifies and unifies the 

 estimation of Ip^l for these or any other objective func- 

 tion. Any multivariate function optimization program, 

 using these parameterizations, can generate estimated 

 probability distributions whose components are positive 

 and will be guaranteed to sum to one. Therefore any 

 multivariate optimization program with these param- 

 eterizations can be used to estimate Ip I. 



Results 



Greenland turbot {Reinhardtius hippoglossoides) length- 

 at-age data were originally used to illustrate the iterated 

 age-length key (lALK) (Kimura and Chikuni, 1987). 



We used the 1983 length-frequency data from that data 

 set, along with the length-at-age data, to illustrate the 

 methods of estimating mixture probabilities with the 

 logit and H-F parameterizations (Table 1). Except for 

 results from the lALK algorithm, all parameters were 

 estimated by using an optimization program with the 

 logit and H-F parameterizations (i.e., the latter two 

 parameterizations gave identical results). Results also 

 showed that maximum-likelihood estimates from either 

 the logit or H-F parameterizations provided maximum- 

 likelihood estimates identical to those estimated by 

 using the TALK algorithm. 



Discussion 



It is probable that the logit and H-F parameterizations 

 would provide nearly identical solutions for a given data 

 set and objective function. If maximum likelihood solu- 

 tions are unique, either parameterization should provide 

 the maximum-likelihood estimates because the repa- 

 rameterizations of probabilities are one-one mappings 

 that lead to invariance properties when optimization is 

 performed. The maximum-likelihood solutions will gen- 

 erally be unique when all p,>0 (Kimura and Chikuni, 

 1987). However, difficulties in searching in multivariate 

 spaces, and limited computational precision may cause 

 differences in the estimates. 



