NOTE Kimura and Dorn: Parameterizing probabilities for estimating age-composition distributions for mixture models. 



305 



Simple parameterizations, like direct estimates of 

 Ip I, will allow solutions that have negative probabil- 

 ities. Constrained solutions, which allow maximum 

 likelihood estimates and other types of estimates, will 

 generally make these negative components of the prob- 

 ability distribution have zero values. Such solutions 

 are boundary value solutions and may not be unique 

 (Kimura and Chikuni, 1987). This is another reason 

 it is difficult to claim unique solutions for the mixture 

 problem. 



From the modeling perspective, we illustrate the use- 

 fulness of reparameterization to impose mathematical 

 constraints. In the context of the mixture problem the 

 suggested parameterizations are reasonably transpar- 

 ent and allow the modeler to use familiar software. The 

 reason we propose the methods described in this note 

 is not that these methods provide superior estimates to 

 those described in the literature, but that the procedure 

 for estimation may actually be more straightforward 

 and transparent for modelers more interested in solu- 

 tions than in theory. 



Because of its simplicity, effectiveness, and ready 

 applicability to different objective functions, modelers 

 may prefer optimization using the logit or H-F param- 

 eterizations to estimate probability distributions for the 

 mixture problem. Another advantage of these reparam- 

 eterizations is that they can be more generally applied, 

 for example, to estimate geographic distribution in mi- 

 gration models (Heifetz and Fujioka, 1991; Shimada 

 and Kimura, 1994). 



Acknowledgments 



We thank two anonymous referees for comments that 

 helped us clarify our presentation. 



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