LAI: ESTIMATING AGE COMPOSITION 



APPENDIX R Derivation of Equation (7) 



The Cauchy-Schwarz inequality (Cochran 1977, p. 97) is 

 (I A i){LbI) - (I At,B^f = 11 {A,Bj - A^B^f > 0. (B.l) 



h h h i i>j 



Therefore, 



iLAl)(ZBl)>ii:A,B,f. 



h h h 



For a fixed age subsample, let 



Ai = \fajn; A2 = \/a^; B^ = \fc^; and B., = \Jc^N. 

 Applying Equation (B.l), the product of D'- and C is 



D'-C = \— + ~\ {c^n + CjN) > (\/a^2 + \f<hCiY- (B.2) 



\ f 



The product Z)-C will be minimized, provided that the equality of 

 Equation (B.2) holds. Setting equality of Equation (B.2) and ex- 

 panding both sides, we find the solution is 



r* = (n/N)* = Vai Cj/agCg (B.3) 



which is Equation (7). 

 For a random age subsample, let 



Ai = \/bJn; A.^ = V^^; B^ = \/c7w; and B^ = \/c^. 



The reader can derive Equation (11) by the procedures identical 

 to Equations (B.2) and (B.3). 



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