PELLA and ROBERTSON: ASSESSMENT OF STOCK MIXTURES 



Table 2. — Biases, ibt, , Hh . and 6^ *• of estimators and asymptotic bias. 61, from 

 Equation ( 12t for indicated 4>-matrices, indicated test and mixed sample sizes, and B' = 

 (0.6,0.4). 



Tesl 

 sample 

 sizes 



Mixed 



sample 



size 



[0.75 

 25 



25 

 75 



'20 

 230 

 240 

 '20 

 •30 

 •'40 

 •40 



-001077 



• 01076 



• 01076 



• 00637 



• 00637 

 - 00636 

  00438 



-0.00363 



- 00362 



- 00364 



- 00123 



- 00124 



- 00124 

 00063 



-0 01873 



- 01873 

 ' 01873 



- 00381 



- 00381 

 • 00382 



- 00108 



• 00750 



- 00750 



- 00750 



- 00500 



- 00500 



- 00500 



• 00375 



[■ Case 2 -, 

 75 25 



10 



90 



20 

 30 

 40 



'20 



'30 

 '40 



01053 - 00090 

 00660 - 00037 

 00481 - 00022 



- 00186 - 00905 



- 00060 - 00604 



- 00033 - 00453 



'Evaluated at all sample points except when 'l" 1 - 



'Evaluated only at sample points for whicti probability ot observing this outcomes of the test samples 

 ■10-6 and I*! * 



two of three combinations of test and mixed sam- 

 ple sizes, repeated under case 1 and case 2, bias 

 increased between case 1 and case 2, the latter not 

 having a symmetric ft>-matrix. 



The predicted bias of 0^ from the asymptotic 

 formula [Equation (12)] agrees with actual bias of 

 H reasonably well. The approximation obviously 

 becomes more accurate as size of test samples in- 

 creases or as rules improve. 



Biases would appear negligible in comparison 

 with magnitude of variances of the estimators 

 next considered. Absolute value of bias in the situ- 

 ations evaluated represents at most 3.1'V of the 



parameter value. ^, = 0.6. Random errors in esti- 

 mation are the main concern. 



Variances of the estimators (H. H, and ih de- 

 crease as test samples become larger, agreeing in 

 behavior with biases; in contrast to biases, var- 

 iances also decrease as size of mixed samples in- 

 creases. We computed variances under case 1 for 

 the same test and mixed sample sizes described for 

 bias evaluation (Table 3, lines 1 to 6). Although 

 variance of any of the estimators (^1, W, , and H, ) 

 decreases with size of test or mixed samples, the 

 rate decreases with size of either type when that of 

 the other is fixed. For example, at test samples of 



T.ABLE 3. — Variances ( th,'. <rn^^,and "-h^' ) of estimators and asymptotic variance (ct,j^i 

 from Equation (lit for indicated <J'-matrices. indicated test and mixed sample sizes, and 

 B' = (0.6,0.4). 



Tesl 



sample 



sizes 



Mixed 



sample 



size 



Case 1 



[0 75 25 

 25 75 J 



Case 2 

 75 25 

 10 90 



Case 3 



[0 90 10~| 

 10 9oJ 



'Evaluated at all sample points except when i 'I* ! = 0. 



^Evalualed only at sample pomis lor which probability ot observrng the outcomes of the test samples 



 1 0-& and I tp ! ^ 



395 



