provides numerical computations for three stocks 

 so that users may verify their understanding of 

 the formulas. 



Special Case of Two Stocks 



We assume a set of rules based on learning sam- 

 ples from each stock has been developed which 



FISHERY BULLETIN: VOL 77. NO 2 



The form of B has been chosen to agree with the 

 solution of Worlund and Fredin ( 1962); in develop- 

 ing this form, we used the facts for this special case 

 that 



022 = 1 —021 



012 = 1 —011 (22) 



and X2 " 1 — "^1 



G = 



(,11. 



1,22,, 



ll(l-01l)-0 011012 0'^02l(l-02l)-0 021022 



021012(1-012) -0'0 



12V1 1 



I-. 



1 2 J 



0^^022(1-022) -0'^0220 



22^21 



t-, 



(23) 



assigns each individual to one of the two stocks. 

 Individuals in two test samples, size /, from stock 

 1 and size /., from stock 2, are assigned by the rules 

 to either ofthe stocks. Of the f, individuals,/,, are 

 assigned to stock 1 and t^.^, to stock 2. Of the t., 

 individuals, t.,^ are assigned to stock 1 and t.^.,. to 

 stock 2. A sample from the mixture of size m is 

 assigned by the rules to the stocks — m , to stock 1 

 and w„ to stock 2. Then 



011 012 

 _021 022 _ 



'll/'l- 'l2/'l- 

 '2l/'2- ^22l^2- 



(18) 



-<!>■( 1 



XjXs 



m 



1 -1 



-1 1 



'^"011012 ^2 021022 



(24) 



1 -1 

 -1 1 



(25) 



G, 1^, and !„■,.„ are obtained by substituting 

 estimates for the corresponding parameters. Then 



('I' )' 



>i 



A, 



0" o'' 



_0 " <;>" 



011022 -012021 



O = (<l> )~iA 



m2lm 



U^ 



('1' )-' {^\ 



■())• 



(26) 



(19) 



(20) 



(21) 



provides internal estimates of the variance of W, or 

 H., as well as their covariance. 



Expressions for 6 and B follow directly from 

 specialization of Equations (13) and (14) to two 

 stocks; substitution of their elements into Equa- 

 tion (26) in place of those of B provides i;-, and Sj-^, 

 respectively. 



Numerical Computations for Three Stocks 



To illustrate the computations for a three-stock 

 situation, we use the information reported by 

 Cook and Lord ( 1 978) regarding stock composition 

 of high-seas mixtures of sockeye salmon, On- 

 vorhynchus nerka. Their purpose was to estimate 

 proportions of the mixture arising from each of 

 three river systems — Egegik, Kvichak, and Nak- 

 nek — of the Bristol Bay region of Alaska. Actu- 



392 



