FISHERY BUI.I.KTIN VOI, 77. NO 



(1978> have misgivings of probable occurrence of 

 additional unaccounted stocks in the high-seas 

 mixture. 



BEHAVIOR OF ESTIMATORS AND 

 ASYMPTOTIC FORMULAS 



Of interest to investigators beginning studies of 

 stock composition of mixtures is the behavior of 

 our estimators as test and mixed sample sizes vary 

 for fixed rules and the influence of rules on the 

 estimators. Further, we remarked that our solu- 

 tion of the stock mixture problem assumes large 

 test and mixed samples. Of concern is how large 

 specifically the samples must be for the asymptotic 

 expressions to be reasonably accurate. This 

 examination will be restricted to the two-stock 

 case which is genera! for our purpose in that any 

 number of stocks can be partitioned into two 

 groups; that is. we can evaluate the estimators for 

 a particular stock when the remaining stocks are 

 lumped into a second group after assignment by 

 the rules to the individual stocks. Bias and var- 

 iance for the particular stock would be unchanged 

 then even if the stocks of the second group were 

 treated severally. 



We evaluate estimation behavior and asympto- 

 tic approximation for three choices of "t" represent- 

 ing rules of increasing accuracy: 



We let ()' = i0.6, 0.4) for all three cases. Based on 

 experience in identification of sockeye salmon in 

 Bristol Bay. Alaska, using discriminant functions 

 on scale features, the ranges of elements of 4> are 

 realistic. The choice of O is arbitrary, of course. 

 Given <t>, O, and sample sizes ?, , /., , and m . we 

 can enumerate all possible sample points — Z^. 

 /j._,, /^,, t.22, "i,, and in.^ — as well as compute their 

 probabilities of occurrence. In om- evaluations, we 

 always used equal test sample sizes. For each 

 sample point, we can compute B, 6, (). ii,, 1,-,, and 

 i,i. With these calculations for each pomt we can 

 compute the mean and variance of each estimator 



by weighting its value at a sample point by the 

 probability of that point. 



Estimation of 6 by 6, 0, or O requires the prob- 

 ability that I 4) ] = be zero; this condition is not 

 met. If we supplement the procedure by assigning 

 arbitrary values to the estimators when j 'l> j = 0, 

 means and variances of such modified estimators 

 will approach the values we obtained by omission 

 of such sample points. The probability that | <t> 1 = 

 rapidly decreases with increasing test sample 

 sizes. For case 1 with test samples of 20, it is <5 x 

 lO"*, and with test samples of 40, about 5 x lO''. 

 The probability also decreases with improved 

 identification of stocks. For case 3 with test sam- 

 ples of 20, the probability is <4 X 10''. Weighting 

 the arbitrary values of the estimators correspond- 

 ing to such points by their probabilities makes 

 their contributions to expectation computations 

 negligible. 



We found these numerical studies to be expen- 

 sive, especially with large sample sizes. Therefore, 

 we began omitting sample points whose probabil- 

 ity was small even if | <t) ] 9^^ 0. Criteria for omission 

 of points are indicated in our tables; the justifica- 

 tion is again their negligible contributions in ex- 

 pectation computations. Results will be discussed 

 in terms of the first stock only. 



We consider bias first. Bias of any estimator, », , I 

 t)j , or 6*1 , is unaffected by changes in mixed sam- 

 ple size; however, bias decreases with increasing 

 test sample size. For example, we computed biases 

 for case 1 with three mixed sample sizes— 20, 30, 

 and 40 — at each of two choices of equal sized test 

 .samples — 20 and 30 (Table 2, lines 1 to 6). The 

 occasional change in the last digit for biases at 

 varying mixed sample sizes within fixed test sam- | 

 pie sizes is probably caused by omissiim of improb- 

 able sample points in evaluation of expectations. 

 Bias of 0^ also is predicted by the asymptotic for- 

 mula [Equation ( 12)| to vary only with test sample 

 sizes, not mixed sample size I Table 2, last column). | 



Bias of (J, is of opposite sign from that of either f), 

 or ^, (Table 2, column bn^ as compared with col- 

 umns i>« and fc«,l. Absolute value of bias of ^, is 

 less than that of either f), or (\. Generally, abso- 

 lute value of bias of 6^ is also less than that of H, ; 

 the sole exception is case 1 with test samples of 

 only 20. 



Bias of W, or H^ decreases with improved rules as 

 we go from case 1 to case 2 to ca.se 3, holding test 

 and mixed sample sizes fixed. Biases computed for 

 f> decreased between case 1 and case 3 for which 

 the 4>-matrices are both symmetric; however, for 



394 



