PELLA and ROBERTSON ASSESSMENT OF STOCK MIXTURES 



P(M) = 



AK 



Al A2 



(4) 



We can estimate the probabilities \, from M by 

 the maximum likelihood estimator 



A' = (mi/m , m2lm , .... m^^lm} 



(5) 



corresponding to the parameter vector A ' = ( \j , Xj- 

 . . . , A^-). A is unbiased and has the variance- 

 covariance matrix 



-A = 



Ai(l-Ai) 



m m 



A2A1 A2(l-As 



m 



(6) 



X 1 A 2 A 1 A K- 



m 

 I A2AK 



m    m 



Ak(I-Ak) 

 m 



A is a natural estimator of stock composition of 

 the mixture. Unfortunately as we see next, its 

 expected value, A, depends not only on stock 

 composition, but also on the behavior of the rules. 



Basic Relation Between Parameters of 



Test Samples and Those of the Sample 



from the Mixture 



We know the mixture consists of individuals 

 from at most K stocks. Let H,, be the proportion of 

 the individuals composing the entire mixture 

 which are of the ktVi stock, where 

 /,' and 



H. s 1 for all 



K 



1 . 



The parameter vector 9' = {H^.ft., W^i is un- 

 known: its estimation is our objective. If the indi- 

 viduals of each of the stocks occurring in the mix- 

 ture are a random sample from the character 

 distribution of that stock, then the probability 

 that a randomly sampled individual from the mix- 

 ture is assigned to thejth stock, A ,, is related to 



previously defined probabilities by the equation 



system 



l! = l 



J = 1,2, 



K. 



(7) 



The term, Wj.<f>^,, represents the probability a ran- 

 domly sampled individual from the mixture is of 

 stock k and assigned to stocky; summing over the 

 K stocks gives the total probability the individual 

 is assigned to stocky. This basic set of relation- 

 ships can be expressed in matrix notation, 



A = *e 



(8) 



where '1> 



011 012 .  .011 



<>K 1 0K 2  • • <>K K 



ESTIMATION OF STOCK 

 COMPOSITION OF MIXTURE 



If l<t>j 7^ 0, we can solve Equation (8) for 0, 



e = (<l>') ' A. 



(9) 



When the rules assign individuals from the stocks 

 without error, <J> = 7, and H = A. Then the natural 

 estimator .\ is appropriate. But the rules will usu- 

 ally be imperfect, yet Equation (9) shows we can 

 still solve for H without error provided <1> and A are 

 known. Unfortunately neither (t' nor A is known in 

 usual circumstances; however, we saw how to es- 

 timate them from the test and mixed samples 

 using Equations (2) and (5). When \ and <t> in 

 Equation ( 8 ) are replaced by estimates from Equa- 

 tions (2) and (5), the problem of estimating B is a 

 special case of estimation of the solution of a sys- 

 tem of linear equations with random coefficients. 

 Fuller'' has provided several solutions for the gen- 

 eral problem; these are applicable in the present 

 case for large test and mixed samples. Later we 

 indicate how large these samples must be. 



■"Fuller, W. A. 1970. Mimeographed class notes. Statistics 

 638, winter 1969-70. Iowa State Univ, Stat. Lab.. 56 p.. on file 

 at the library of Northwest and Alaska Fisheries Center Auke 

 Bav Laboratory. National Marine Fisheries Service, NOAA, 

 P.O. Box 1,5.5. Auke Bav. AK 99821. 



389 



