LANDER: PROBLEM OF BIAS IN MODELS 



offshore.] Ai)plying Assumi)tion 2 in Equation 5: 

 (^2 +E^S2u)/Nq = s^^s^a = is^S2)fj. (21) 



The geometric mean of limits in Equations 19 

 and 21 is then taken arbitrarily to be the 

 estimator: 



= [{0.25EJN^ + [0.25iEJN^f + EJN^Y'^} 



A 



= ff-ZT(T - /„) 



= Sj^S2 = e ^T 



0' . 



(22) 



From Equations 1-3, 5, and 22 we finally have: 



S2 = is';s2-EJN^)/{EJN^) 



^ o-Z^(T - t-. ) 



Sj S2/S2 



e-^i^'i - '0), 



iVj = Ejs 



2 • 



(23) 



(24) 



(25) 



m = [I-EJEJ^]-^ = EJ{E^ +iV,). (26) 



Nr= Arje-^2('r- 'i>. 



(27) 



The form of the middle member of Equation 

 26 (from the ratio of Equation 5 to Equation 1) 

 is included for its practical import: if an offshore 

 catch is identified by origin from any technique 

 whatsoever, then subsampling it for maturity 

 gives an independent estimate (/») from which 

 .s-2 then can be estimated without data on the 

 number of smolts. As noted in connection with 

 Cleaver's model, an independent maturity 

 estimate also gives nominally unbiased esti- 

 mators for the system when No is known 

 (excluding, again, the subdivision of Zo into 

 F and M2 as evident from Equation 7 when 

 t f ^i). Even though all estimates contain 

 unknown bias, with no offshore fishery the 

 values of Z2 {^M^^) from Equation 23 and 

 of Ny. from Equation 27 might prove useful in 

 biomass computations. 



Offshore Catch Known 



It is not necessary to assume as in the Cleaver 

 model that Mz = during all of T - tx. 

 Assumption 3 is: .1/2 = during T — t when 

 offshore fishing occurs. This gives: 



NrL =C + E2. 



(28) 



Solving Equation 20 (as in the case of C known) 

 and inserting the result in Equation 2: 



^lu =NoS,u +^1- 



(29) 



Assumption 4 is: F = during T - 1^.. By 

 Assumptions 3 and 4 we would observe C + E-z 

 instead of just E-z at time f — T (similarly, 

 Ricker's model utilizes E\ + E-z at time t = T). 

 Because C -\- E-z and A^^^ are both too high, the 

 coefficient, X, relating these artificial population 

 sizes may be defined and used as below: 



X = -ln[(C + £2)/^iu]/(^-^i)- (30) 





(31) 



The geometric mean of limits in Equations 

 28 and 31 is taken arbitrarilv as the estimator: 



iNrjN,„t'=N, 



(32) 



We then have from Equations 4 and 5. 7, and 32: 



CjNj. = u . 



A 



-ln(i:2/iV^)/(^- tj.)= F + M^. 



(u)(F+M2)/[l-e"^^'^'"2)(^-'r)] 



(fC:i/2)-f = m2, 



(33) 



(34) 



F (35) 



(36) 



[M2(T-t,) + F{T-tr)]/{T-t,)= Z^. (37) 



Given Z-z from Equation 37, remaining estimates 

 for the case of C known mav be calculated in 



519 



