FISHERY BULLETIN: VOL. 77. NO 2 



20, variance of decreases 249i when mixed sam- 

 ples increase from 20 to 30, but by only IGVi when 

 mixed samples increase further to 40. Similarly at 

 a mixed sample of 40, variance of 6^ decreases by 

 1 Kr and 8'i as test samples increase from 20 to 30 

 and 30 to 40, respectively. The return to sampling 

 effort of precision of estimation by increase in 

 mixed sample size with test sample sizes fixed 

 diminishes and is limited by test sample sizes. 

 Return of precision to increase in test sample sizes 

 is similarly related to and limited by mixed sam- 

 ple size. 



Overridinji; both test and mixed samples in de- 

 termining ultimate precision of estimation are the 

 rules characterized by the <t>-matrix. As rules of 

 assignment improve and the <t>-matrix approaches 

 the identity matrix, precision of estimation at 

 fixed test and mixed sample sizes increases. 



In our evaluations, variance of ^j is always less 

 than that of f)^. In this respect tt^ also enjoys con- 

 siderable advantage over fi, when rules are poor, 

 case 1 , and sample sizes are small. As test or mixed 

 samples increase, the advantage diminishes until 

 fi^ has the smaller variance. However, differences 

 among variances of the three estimators (W, , ()^, 

 and W, ) become negligible either as rules improve 

 or samples sizes become large. 



Predicted variance of the estimators of 0^ from 

 the asymptotic formula (Equation (11)] describes 

 variance of ^, remarkably well, even when rules 

 are poor and sample sizes are small (Table 3, com- 

 pare lines 1 to 7 of column cr, ,^ with column <t„ ^). 

 With improved rules, variances of each of 0,, if,. 

 and f)j are well described by the asymptotic var- 

 iance (Table 3, compare lines 8 to 12 of columns 

 <r„'^, (T„^, and ir,,'-^ with column fu'-^l. 



Two evaluations concerning adequacy of inter- 

 nal variance estimation by S,,, 1,-,, and i,, con- 

 clude our numerical studies. Computations are 

 heavy so the range of these studies is restricted. 

 First, we computed the mean of the internal var- 

 iance estimator <>„ - ( i.e., of the element in the first 

 row and first column of 1,,) for the cases of <t> and 

 sample sizes used in the previous evaluations of 

 bias and variance (Table 4). The mean of this in- 

 ternal variance estimator, £'(<t„ '^K generally ex- 

 ceeds the actual variance of ^, , ct„'-^. As rules im- 

 prove with sample size fixed, percent bias changes 

 from large positive values to small negative val- 

 ues. 



Percent bias under case 1 decreases sharply 

 with increase of test sample size, but increases 

 slightly with increase of mixed sample size (Table 



T.MiLE 4. — Variance, <t« ^. of estimator. B, ; mean. £u>(j^l, of 

 internal variance estimator. & n^; and percent bias for indicated 

 ■{'-matrices; for indicated test and mixed sample sizes; and O' = 

 (0.6,0.4). 



'Evaluated at all sample points except when | <i> | = 

 'Evaluated only at sample points tor wtiicti probability ot observing ttie 

 outcomes ot ttie test samples  10'^ and | "t" I ^0 



4, lines 1 to 7); conceivably omission of sample 

 points in our evaluations underlies the slight in- 

 crease with mixed sample size. Under any case of 

 <t>, the internal estimator or variance of W, becomes 

 nearly unbiased at the largest sample sizes 

 examined. 



Our last computations are of the mean and var- 

 iance of the internal variance estimators (o-,,^^, 

 (T„ ^, and (T „^) (i.e., of the elements in the first row 

 and column of i^-|, i,-,,and Xy, respectively) for the 

 three cases of <t> with test and mixed samples all of 

 size 20. Also we determined the actual probability 

 that 9(y7i and 95*^^ simultaneous confidence inter- 

 vals from Equation (16) using either (), i), or (), 

 each with its internal variance estimator, S,-,, i,-,, 

 or i||, cover the actual composition vector O' -= 

 (0.6, 0.4) (Table 5i. 



Comparison of actual variances of the es- 

 timators (6* ,, «,,and (*,) (Table 5, line 1) with the 

 mean of the corresponding internal variance es- 

 timators (Table 5, line 2) shows the positive bias of 

 each internal estimator diminishes as rules im- 

 prove. Only the internal estimator of variance of ^, 

 becomes negatively biased. Percent bias (Table 5, 

 line 3) of each estimator decreases sharply with 

 improvement of rules. 



Variance of the internal variance estimators of 

 «, and H^ are manyfold greater than that of ^, 

 under case 1 and case 2. With improved rules of 

 case 3, all internal variance estimators have com- 

 parable variance. 



Probabilities that simultaneous confidence in- 

 tervals for each estimator ((), O, and ()) cover the 



396 



