FISHERY BULLETIN: VOL. 80, NO. 4 



Now the joint conditional likelihood of the re- 

 capture sample is 



£ = n 



i\»a.-! r a, ] - r di ] 



IPa,^ Pb/ Bi 



X (1 - P Al - P Bi ) 



r di 



In either case, once the underlying model and 

 the corresponding elements of 8 are identified, 

 the ML estimates of 8 may be computed by maxi- 

 mizing X directly using a variety of iterative 

 search procedures. In some situations the deriva- 

 tives of X with respect to 8 are easily derived, but 

 even then only numerical solutions are possible. 



For example, when A and B tags are identical 

 and J{t,) = 1 — p(exp(— Lt,-)), the ML estimates 

 of p and L are found by solving the system of 

 equations 



= X n 



C, and 



= X a 



where Q = 



(1 + P di ) (r di - P dl n) 



1 - p 



di 



= J(r,r{r ti 



(I ~ J(r,-) \ 

 ' \1 +J(r i )) 



= J(T,y l {rd, - E(r di )}. 



The asymptotic covariance matrix of L and p 

 may then be derived in the usual manner by in- 

 verting the corresponding negative information 

 matrix 



/ = 



2 rl A 

 i=l 



1 " 



n n 



- X r, D, - X Di 



__ ^ 1=1 p 1=1 



where A = 



r. f P d> (1 + P d ,f 

 1 - Pdi 



Explicit analytical solutions are possible when 

 there is only a single recapture period centered 

 at r and the model is reduced to a one-parameter 

 function of either the Type I or Type II shedding 

 rate, i.e., either J(t) = 1 — exp(— Lr) or J(t) 

 = 1 — p. In this event the ML estimate of L (with 

 p = 1) is 



2r d 



L = 



\n +2r d 



with asymptotic variance estimated by 



r s (r s + r d ) 



(15) 



'2 _ 



o~ 



r 2 r d (r s + 2r d Y 



or when shedding is a function of p only (with 

 L = 0) 



and 



(16) 



a^ — 



P (r, +2r rf ) 



3 • 



In the case of identical A and B tags a con- 

 venient alternative to direct maximization of 

 the likelihood function is to fit the recapture 

 data to their expectations using an iteratively re- 

 weighted Gauss-Newton algorithm. To accom- 

 plish this one may use routines available in cer- 

 tain standard statistical software packages, e.g., 

 BMDP 2 . Specifically, we find an admissible 

 value of 8 which minimizes the sum of squares 



S'= X Wi[r di - E{r di )f. 



i=i 



Since the r di are assumed to be binomial (i.e., of 

 "regular exponential" form), minimizing S' with 

 a Gauss-Newton routine is equivalent to maxi- 

 mizing the likelihood of Equation (14) provided 

 the weights used are the reciprocals of the vari- 

 ances of the r di and are recomputed at each itera- 

 tion based on the current parameter values 

 ( Wedderburn 1974; Jennrich and Moore 1975; 

 Jennrich and Ralston 1978). In this case the 

 weights must be fv, = [r, P di (1 - P dl )]'\ where 

 P di is the function P dl evaluated at the current 

 parameter estimates. Asymptotic standard er- 

 rors for the parameter estimates are also com- 

 puted by the BMDP routine. 



A similar device may be used when a distinc- 

 tion is made between A and B tags. Given r A , + r di 

 we assume r d , is binomial with expectation 

 E B (r dl ) = (r Al - + r lh ) (1 ■- J b (r,)). Analogously, 



2 Reference to trade names does not imply endorsement by 

 the National Marine Fisheries Service, NOAA. 



696 



