FISHERY BULLETIN; VOL. 76, NO. 1 



The maximum likelihood estimators of the 0^ are: 



(9) 



(10) 



(11) 

 (12) 



(13) 

 (14) 

 (15) 

 (16) 



A maximum likelihood estimator of a function of 

 the parameters 6, is obtained by replacing the 

 parameter values by the corresponding maximum 

 likelihood estimates, 6, (Graybill 1961). Beyond 

 that, however, there exists no unique transforma- 

 tion, or function, to obtain maximum likelihood 

 estimates of mi, ma, mg, F^Fg, Fg, Mj, Mg, M^, and 

 Af 4. Any given set of observed data can generate a 

 variety of combinations of parameter estimates. 



Since no unique solution exists, the only practi- 

 cal solution is to assume values for one of the 

 unknown parameters and solve the equations for 

 the remaining parameters. Thus Cleaver (1969) 

 and Henry (1971) assumed values for M, (natural 

 mortality) for hatchery chinook salmon and calcu- 

 lated values for the remaining parameters. How- 

 ever, they assumed M to be constant (Mj) 

 throughout the life of the salmon to simplify com- 

 putations. Lander and Henry (1973), on the other 

 hand, assumed values for m (proportion of fish 

 that mature annually) for coho salmon and then 

 calculated the remaining parameters. 



Assuming fixed values for the proportion offish 

 that mature annually (m,) permits a unique solu- 

 tion to Equations ( l)-(8), combined with Equations 

 (9)-(16), so that with: 



mi = nil (fixed) {di<mi<l). 

 ^^2 " '^2 (fixed) (^3<m2<l). 



A, 



m^ = m3(fixed) (05<m3<l). 



(17) 



(18) 

 (19) 



18 mi 

 ln/e2+12M2 ^^ .^^. 



(20) 

 (21) 



(22) 



(23) 



(24) 



(l-e^nkQ-lnk^+M^-^ ^ ^^ 



(25) 



where k^ 



de 



F 



(l-mi)(l-m2)(l-m3) 



{l-mi)il-m2){l-ms) 

 ln/e6-In/?4+12A/4 



18M1 



,18Mi 



6 



(26) 



The derivations of Equations (17)-(26) are ver- 

 ified in the Appendix. For a particular value of mj 

 (Equation (17)), one solves Equation (20) 

 explicitly for Mj. Then using these values of mi 

 and Ml plus a selected value for m2 in Equation 

 (18), M2 in Equation (21) is found by iteration. 

 ThenFj is computed from Equation (22). Next, for 

 a particular value of mg in Equation (19) plus the 

 other values already determined, Mg in Equation 



48 



