470 



Fishery Bulletin 102(3) 



without which this numerical estimator did not con- 

 verge. 



The factorial terms in the binomial coefficients of 

 Equations 12 and 14 are defined only for natural num- 

 bers. However, in numerical minimization, factorials 

 must be replaced with continuously varying approxima- 

 tions because the negative log-likelihood objective func- 

 tion is minimized by using numerical derivatives. The 

 factorial z! was extended from natural numbers to the 

 real line by using the gamma function, T(z+1) and by 

 using an asymptotic approximation formula for In T(z) 

 (Eq. 6.1.41 in Abramowitz and Stegun, 1965): 



lnr<2) = 



1 



InO 



1, „ 1 



z + - ln( 2/r ) + 



2 12z 



1 



1 



691 



360z ! 



1 



(15) 



12602 s I68O2 7 11882 ! ' 3603602 11 1562 1 



The negative log likelihood was minimized numeri- 

 cally by using the AD Model Builder parameter estima- 

 tion software (http://otter-rsch.com/admodel.htm). 



Results 



The closed-form estimator for the proportion of lobsters 

 that moved from the sanctuary (P|j) gave an estimate of 

 0.6206; i.e., about 62% of the lobsters tagged in Gleesons 

 Sanctuary moved out in one year. The estimate obtained 

 numerically, by maximizing the double-hypergeometric 

 likelihood, yielded a value of 0.6212. 



The small difference between the analytic and nu- 

 merical estimates (0.09%) is presumably due to the 

 use of the numerical approximation for the log-gamma 

 function by the expansion of Equation 15. The close 

 agreement suggests that the error introduced by that 

 approximation is small. 



The AD Model Builder parameter estimation soft- 

 ware allows one to estimate confidence intervals of the 

 movement-rate estimate in two ways: asymptotically. 

 as diagonal elements of the covariance matrix, and by 



0.4 5 0.6 0.7 



Estimate values for Pf . 



0.8 



Figure 3 



Profile likelihood (solid line) and asymptotic normal 

 approximation (dashed line) for the likelihood confidence 

 range about the estimate of P M . 



using a profile likelihood. Confidence intervals for the 

 emigration rate estimate were thus obtained numerical- 

 ly from the hypergeometric likelihood by using both the 

 asymptotic normal approximation and an exact profile 

 likelihood. These gave 95% errors of 21.2% and 21.5% 

 of the estimate, respectively. The approximate normal 

 probability density function and the profile likelihood 

 probability density function were also plotted (Fig. 3), 

 yielding close agreement. Asymptotic confidence inter- 

 vals therefore appear satisfactory for emigration propor- 

 tion estimates not lying near the bounds of and 1. 



Intermediate calculation results (Table 2) included 

 the recovery rate and movement rate (>3 km) within 

 the fished zone. 



When independent estimates of exploitation rate are 

 available, typically from stock assessment, the rate of 

 tag reporting can be calculated from the tag-estimated 

 recovery rate. The exploitation rate (yearly proportion of 

 legal-size lobsters harvested) for the recapture year and 



