232 



Fishery Bulletin 100(2) 



The covariance between two total estimates of season-per- 

 mit anglers is estimated by (see Appendix 1, from Cochran, 

 1977, p. 25) 



CovCC;' 



N(N-n 



^C-, = i^^^lZii!^,C„-C,)(C,,-C,). (18) 

 nin-1) ^ 



The fourth term of Equation 17 is equal to if a different 

 sample of day-permit anglers is drawn in each month. 

 The total catch in weight is estimated by 



K K K 



w = ^Wi = ^w;;' + '^Wi';" = w"-' + w"^' (i9) 



and the variance estimator is similar to Equation 17 but 

 has a slightly different covariance which is 



Cov(w;-^',w;'') - w^m^.Cov{c;f\ci- 



(20) 



This covariance estimator was derived by the delta method 

 (Seber, 1982, p. 7, see Appendix 2). which coincides with a 

 covariance when Wj and w,. are constant. 



The mean annual catch rate is estimated for season-per- 

 mit anglers or boats by 



The estimator of covariance between C"" and M"' is simi- 

 lar to Equation 18 (see Appendix 1): 



C^{C"",M"")= ^'^~"' y (C ,-C)(M ,-M). (25) 



where C, and M, are the number of fish caught and the 

 number of fishing days of (th season-permit angler or boat 

 owner throughout the season, respectively, and 



^" = l.lf-"^"- 



and 



M 



y" M,l n. 



The mean number of fishing days per sampling unit 

 (season-permit angler or boat) is estimated by 



— M" 

 M = 



N 



and the variance estimator is 



V(M) = -^ViM'"}, 



R 



C" 



M'" 



(21) 



where M'" = ^M^ = iV^M* 



Here M /, is given by Equation 4. The total effort is esti- 

 mated by M'^'+ D. The approximate variance of ¥(/?'"") is 

 given by the delta method (Seber, 1982, p. 7; Appendix 3), 

 that is 



ViR"") 



M' 



V(C"") + 



c 



\2 



M'- 



ViM'") 



M' 



(Cov(C'",M'-') 



(22) 



where V(C' ' ) is given by Equation 17, and the variance of 

 the total number of fishing days is estimated by 



V(M"') = iV-'K^V(M;,)-^2^Cov(M^,M;..) . (23 



Here ViM^.) is given by Equation 5 and the covariance of 

 two sample means is estimated by (Cochran, 1977, p. 25) 



Cov(M,. M.. ) = ^ " T^M,, - M.)(M,,. - My ). (24 



where M'"'and V(M'"') are already derived from Equation 

 21 and Equation 23, respectively. 



The catch rate of day-permit anglers over the season is 

 estimated by 



R 



r'' 



D 



and the variance is 



V{k•") = A^V^C■-"\, 

 D- 



where C'"'' and V(C"'') are given in Equation 8 and Equa- 

 tion 10, respectively. 



Method 2 (based on total annual catch of each angler or 

 boat owner) Another procedure for estimating annual 

 catch is to use annual data rather than monthly catch for 

 individual anglers or boats. The advantages of this proce- 

 dure are that covariances between months do not have to 

 be considered and estimators are much less complicated 

 than those obtained using method 1. Equations derived for 

 monthly estimation can be used without modification for 

 this procedure. 



Modified estimators for day-permit anglers 



We could not conduct a sui-vey of day-permit anglers, so 

 we substituted i?^ for R^" in Equation 8. In addition, the 

 total number of day permits sold in /;th month ^D^) was 



