FISHERY BULLETIN: VOL. 80, NO. 4 



we assume the exact recapture time is known, 

 i.e., we know {t sl , t s2 , .... tsr s } and {tdi, td2, ..., t drd \. 

 Let *«/ = Pr {fish is returned with both tags 

 intact in (0, T)} and <i% = Pr {fish is returned 

 with one tag remaining in (0, T)}. Then, assum- 

 ing independence between fish and between the 

 two tags initially applied, the numbers of returns 

 in the various classes are trinomial random vari- 

 ables, i.e., 



N d (0)\ 



Pr{Td ' n} r rf ! r s ! (N d (0) - r d - r.)! 



X 4>/ s <*>/"( 1 - 4> s <D d )' Vrf|0, - rs - rJ . 



Following principles set out previously we 

 write the recapture probabilities as 



4> s = 2/ F(m) S(m) J(m) (1 - «/(m)) dw 



and 



4> rf = / f\u) S(u) (1 - J(u)f du. 



Further, the conditional probability densities 

 for recapture times are 



f,(t) = 



m = 



2F(t) S(t) J(t)(l - J(t))/$ a < t < T 



otherwise 



f F\t) S(t)(l - J(t)?/* d < t < T 



otherwise. 



The joint likelihood function for the observed 

 numbers of single- and double-tag recoveries 

 and the respective sets of recapture times may 

 now be written as 



r s r d 



Z = Pr{r d , r s \. n Mtsd n f d (t d! ). 

 i=i i=i 



For specified forms of F{u), S(u), and J{u) com- 

 putation of ML estimates may now be contem- 

 plated, although the form of £ is apt to be exceed- 

 ingly complex in most situations. For example, 

 taking the most elementary case, assume that 

 J(u) = 1 - exp(-Lu), S(u) = exp[-(M + F)u] 

 and F(u) = FtorO<u<T. Also let T- «>. Under 

 these conditions the log-likelihood becomes 



In Z = K + r. InF + (NAO) - r. ) 



X lnh 



-(' 



2LF 2 



W 



(F + M + L)(F + M +2L) 

 - (F H- M) T. - L(T a + 2T d ) 



rs 



+ 1 ln(l - exp(-Lf„-)) 



i=i 



where K is a function of the observations only, 



rs 



T = S t 



1=1 



rd 



T d = z t d i, 



T = T s + T d , and r = r s + r d . 



Using numerical methods this may now be maxi- 

 mized as a function of F, M, and L in the usual 

 manner to yield ML estimates of these parame- 

 ters, as well as asymptotic variance estimates. 

 A simpler approach which yields information 

 on Z — F + M and L is to condition the likelihood 

 of r d and r s on the total number of recaptures, 

 r, i.e., 



Pr {r d , r s } = 



3>.s 



$d 



Jd 



r s \ r d ll\<P s + <$> d \ 4>,- + $ d 



This gives the log-likelihood 



In X = K' + r. {ln(Z + L) + ln(Z + 2L) 

 - ln(Z + 3L)} - ZT - L(T + T.) 



r s 



+ S ln(l - exp(-L^)) 



(18) 



i=i 



where K' is independent of Z and L. 



Differentiating Equation (18) with respect to 

 Z and L and setting the derivatives to zero one 

 finds that the ML estimates of Z and L satisfy, 

 among other relations, the equation 



T. 

 r 



1 



+ 



1 



Z + L Z + 2L Z + SL 



Combining this with the result at Equation (17) 

 leads immediately to a solution for Z, i.e., 



698 



