PELLA and ROBERTSON: ASSESSMENT OF STOCK MIXTURES 



TABLE 5— Variances of the estimators (»,.#,, and »,). means of mtemal variance estimators, percent bias of interna] variance 

 estimators, variances of internal variance estimators, anci probabilities of coverage of B' = (0.6, 0.41 by simultaneous 90 and 95-^ 

 confidence intervals for three cases of 4> when test and mixed samples are all of size 20.' 



'Evaluations only include sample points for wfiich probability ot observing the outcome of the test samples 10-' and | * | * 



parameter vector G' = (0.6. 0.4) approach the in- 

 tended levels of confidence as rules improve (Table 

 5. lines 5 and 6). For rules of case 1 or case 2. the 

 level of confidence provided by any of the es- 

 timators exceeds that intended; such is preferable 

 to the converse because the intervals provide at 

 least the level of confidence the investigator in- 

 tends. Our normality assumption used to con- 

 struct confidence intervals will be better satisfied 

 as mixed and test sample sizes increase. Appar- 

 ently the internal variance estimators become less 

 biased as test sample size increases. Therefore, we 

 anticipate the level of confidence of intervals from 

 any of the estimators will more closely approach 

 the intended level as test sample size increases 

 even when rules are poor. 



Limited as these numerical studies are, they 

 demonstrate that when sample sizes are small and 

 rules are poor, 6 should be used to estimate com- 

 position of a mixture. We found then that 6 is least 

 biased, has smallest variance, and its internal var- 

 iance estimator itself has smallest variance. With 

 larger sample sizes or good rules of assignment, 

 the estimators 6, B, and ("> appear more nearly 

 equivalent. 



Decisions on sample sizes depend on desired 

 precision and the rules characterized by <i>. The 

 closer O is to an identity matrix or, equivalently, 

 the better the identification of stocks, the fewer 

 required individuals in test and mixed samples to 

 achieve desired precision of composition estima- 

 tion. With an accurate initial estimate of <t> from 

 the learning samples, the corresponding asympto- 

 tic variance-covariance matrix at Equation ill) 

 can be used to estimate sample sizes needed to 

 achieve required precision. We recall that var- 

 iance of B^ is well described by the asymptotic 

 variance-covariance matrix even when rules are 



poor and sample sizes are smal 1, providmg another 

 reason for preferring H to () or () in that cir- 

 cumstance. 



AFTERWORD 



Withholdmg individuals of samples from the 

 separate stocks to form test samples must result in 

 less effective rules than if the learning and test 

 samples were pooled for rule formation. Although 

 the practice is repaid in part by the ability to 

 evaluate precision of composition assessment, the 

 penalty at rule development can be further al- 

 leviated. Roles of the two samples from each of the 

 separate stocks can be interchanged; either can be 

 the learning or test .sample. If each of the samples 

 from the segregated stocks is partitioned into two 

 approximately equal sized subsamples, two sets of 

 rules can be formed; two estimates of "t obtained; 

 two estimates of B computed by any of B. B, or G; 

 and two internal estimates of the variance- 

 covariance matrices (!„, I,-,, or S„) calculated. 

 The pairs of estimates are statistically dependent. 

 Nonetheless, means of pairs of estimates of B and 

 i have the same expectation and presumably 

 greater precision than the individual members of 

 the pairs. Exact evaluation of that enhanced pre- 

 cision for estimates of the composition vector 

 does not appear easy; however, use of the mean of 

 internal estimates of the variance-covariance 

 matrix in calculation of the confidence set Equa- 

 tion! 15) provides an unknown but greater level of 

 confidence than the indicated 100(1 - an value. 



LITERATURE CITED 



AN.\S, R. E., AND S. MURAl. 



1969. Use of scale characters and a discriminant function 



397 



