PELLA and ROBERTSON: ASSESSMENT OF STOCK MIXTURES 



and <b" is the element in the ith row and /th col- 

 umn of (t" '. 



Improved estimators with smaller bias than H 

 can also be developed from Fullers ( see footnote 4 ) 

 general results. These are the new estimators: 



e = (*' +G) A and 



(13) 



e = [T - (<i>'r'(J] ($')' A. (14) 



it. When only two stocks occur in the mixture, this 

 set of simultaneous intervals reduces to the famil- 

 iar univariate normal approximation for setting 

 confidence intervals: 



^2 -2 a 12(022^)'^' <02<02 +^0/2(022 



2vV! 



(16) 



Here G is obtained by substituting the estimates 

 for the unknown parameters of G. To the order 

 of approximation provided by Fuller, these 

 estimators have the same variance-covariance 

 matrix as O. An internal estimate of this 

 variance-covariance matrix ^(, can be obtained by 

 substitution of observed values for parameters in 

 Equation (11). We can substitute in Equation (11) 

 for elements of O the corresponding elements of 

 either 0, O, or (i To distinguish between these 

 possibilities, we label the internal estimators of 

 X,-, as 1|-|, i|i, or i|,, respectively. With the internal 

 estimate of i|,, we can estimate not only H but also 

 how precisely the estimation is accomplished. 



To establish confidence intervals on the ele- 

 ments of B, we assume test and mixed samples are 

 sufficiently large so that the estimators B, B, or O 

 are each approximately distributed as the mul- 

 tivariate normal with mean B and known 

 variance-covariance matrix i,„ i,). or S,,, respec- 

 tively. Then a 100(1 - aW set of confidence inter- 

 vals such that all the unknown elements of B are 

 simultaneously covered by their respective inter- 

 vals with a probability 1 - « is for the estimator B 

 (say) as follows (see Morrison 1967, section 4.4): 



where^ ,„ is the standardized normal deviate such 

 that 100(a/2)'7f of the distribution lies below -2,^,2 

 and 100(a/2)"7r lies above Zui-z. These expressions 

 are in terms of the estimator B; they apply as well 

 to the other estimators when elements of B or B 

 replace those of B within them. 



Worlund and Fredin (1962) developed the es- 

 timator B in Equation (10). To translate their no- 

 tation to ours, let 



0,7 = Pij 



(17) 



ft, 



and permit the subscripts to take on letter values 

 a,h,c, . . . .In the special case when the mixture is 

 comprised of only two stocks, they developed an 

 asymptotic expression for the variance of H, (the 

 variance of H2 necessarily equals that of H, since 

 H2 = 1 - f^i ). In deriving the variance expression, 

 they assumed >!> is known without error so that 

 i,i,i, is a null matrix: such is approximately true 

 as t and I., become large. 



h-(Ollh.:K-l^f'<01 <»! +(Oii\„;K-i')'' 



2 - (022^,, ;K-1 ^Y" <- 6 2 < d 2 + (a22^X „ ;K-1 ^) 



Sk -{0,k\.:K-ir' <0, <e, ^ {d,,\.,,^_,^) 



2\V2 



(15) 



^K-(aK/c'x.;K-l')"'<0K-<OK+(6;,K-^„;/C-l^)"' 



where (x^.^^ is the element in the kth row and col- 

 umn of i,,, and x„ K-\^ '^ '-he value associated with 

 a chi-square distribution with A'-l degrees of free- 

 dom such that lOOa'/r of the distribution lies above 



We consider two examples now to illustrate our 

 notation and method in concrete terms. The first 

 case restricts our general approach to the simplest 

 situation of two stocks in the mixture; the second 



391 



