LAI: ESTIMATING AGE COMPOSITION 



cept that a's and b's are weighted by Wj. In fact, 

 minimizing Vartot is a special case of minimizing 

 M{N,n) = Vartot for all w, = 1. 



Another argument may relate to the possibility 

 that the cost function may be more complicated so 

 that traveling and overhead costs can be taken into 

 account. In such cases, cost function may become 

 a nonlinear form, and the explicit expressions of n* 

 and A/'* cannot be obtained. However, the technique 

 of nonlinear programming can be applied to find 

 numerical solutions of w* and A'*. In general, it is to 



minimize M{N,n) = Z. w, Var(pj) 

 subject to C = c{N,n) 



where C is total cost and c{N,n) is cost function. 

 Many optimization programs can be employed from 

 popular computer software packages for main frame 

 computers. Bunday (1984) provided several BASIC 

 programs for constrained optimization, which may 

 be useful in personal computers. It should be noted, 

 however, that the sufficient and necessary condi- 

 tions of this constrained minimum must be proved. 

 Theoretically, there is a unique minimum if objec- 

 tive function is convex and constraint function is 

 concave (Bunday 1984). 



ACKNOWLEDGMENTS 



This study was funded by the Northwest and 

 Alaska Fisheries Center, NMFS, Seattle, WA. I am 

 deeply indebted to Donald R. Gunderson, Vincent 

 F. Gallucci, and Loh-Lee Low for their guidance and 

 Pat Sullivan for his comments. Many thanks to the 



referees for their helpful comments, and particularly 

 for rewording a more accurate description of Vartot. 



LITERATURE CITED 



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Cochran, W. G. 



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1977. Statistical assessment of age-length key. J. Fish. Res. 



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1963. Estimating absolute age composition of California 

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183 



