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Fishery Bulletin 97(1), 1999 



Because this posterior depends on 0, we plugged in 9 

 in the same way as we did for the ad hoc "Bayes" method 

 in the section "A simple point estimation example." 



In a simulation example, we let = 4, y = 4, a = 0, 

 and 6 = 8, so that the true value for /was at the 

 midpoint of its prior. The MLEs were = 4.06 and y 

 = 3.88. The 95'7f MCA interval for 0was ( 1.99, 87.55). 

 The interval obtained with the ad hoc "Bayes" method 

 was (3.78, 4.03). The MCA interval was 342 times 

 wider than the second interval, and 2.7 times wider 

 than the range of the observed data. 



Note that the MLE 6 is not contained in the sec- 

 ond interval above. This is an undesirable result of 

 the ad hoc "Bayes" method here. One possible rem- 

 edy is as follows: instead of simply plugging 9 into 

 the posterior for yeach time we sample from it, we 

 could attempt to "integrate over theta" by plugging 

 in a different estimate of 9 on each occasion. Each of 

 these plug-in values for 9 can be obtained by calcu- 

 lating the MLE of 9 from a nonparametric bootstrap 

 sample of the real data X The confidence interval 

 resulting from this strategy was (3.59, 4.08), so the 

 MLE is now contained in the interval. We stress that 

 this is again an ad hoc solution and we strongly fa- 

 vor standard maximum likelihood or Bayesian meth- 

 ods over either MCA or the ad hoc "Bayes" method. 



Finally, the example can also be twisted so that the 

 MCA interval is too narrow. Supposed, ~ U{ y- 9 y'^, y+ 

 9 Y^ ). This leads to 9y = ( max X, - min X, )/( 2y'- ), a cusp- 

 shaped function of /over any intei"val that includes 0. 

 Thus, considering / priors of the form U(-a,a) for a > 0, 

 we observed that MCA leads to the surprising result 

 that the width of a quantile-based confidence interval 

 for 9 approaches as a increases, while holding the ob- 

 served data fixed. \n other words, the width of the confi- 

 dence interval is entirely dependent on the prior, and 

 wider priors lead to narrower MCA confidence intervals. 



Application of MCA to bowhead whale 

 assessment 



Punt and Butterworth ( 1997) examined the applica- 

 bility of MCA in the assessment of the Bering- 

 Chukchi-Beaufort stock of bowhead whales. As with 

 the swordfish assessment of Restrepo et al. (1992), 

 the approach involved generation of pseudodata with 

 a parametric bootstrap. In the bowhead case, the real 

 data consisted of abundance estimates and corre- 

 sponding CV estimates for several years, and ob- 

 served age-class proportions. Thus, each MCA simu- 

 lation consisted of the following: 



1 Bootstrapping of data. A series of pseudo-abun- 

 dance estimates is bootstrapped from the observed 



data (Table 1 in Punt and Butterworth, 1997). 

 Each estimate is assumed to be independent and 

 from a lognormal distribution with mean and CV 

 equal to the observed estimates from that year. 

 Pseudodata for fractions of calves and matures 

 are generated from Table 4 of IWC (1995). 



2 Sampling of biological nuisance parameters from 

 priors. Parameters such as age at maturity and 

 natural mortality rates are generated from prior 

 distributions from IWC (1995). 



3 Conditional estimation. Conditional on the val- 

 ues of the nuisance parameters, maximum likeli- 

 hood estimation is used with an age-structured 

 density dependent population dynamics model to 

 obtain estimates of the parameters of interest: 

 carrying capacity (K) and a productivity param- 

 eter (MSYR). The likelihood contains contribu- 

 tions from both the abundance and proportion 

 data. 



4 Uncertainty estimation. The variation in condi- 

 tional MLEs is used to represent uncertainty. 



Note that K and MSYR are the parameters of inter- 

 est (denoted by 9 in our previous notation), whereas 

 the other biological parameters take the role of y. The 

 distributions from which they were simulated are the 

 prior distributions. The results of 1000 replications of 

 this procedure are used to form confidence intervals. 



A simple population dynamics model 



For the purposes of illustration, we applied MCA to a 

 simple population dynamics model (PDM). This is a 

 non-age-structured density dependent PDM given by 



P,,i = P,-C, + l.5{MSYR)Pi(l-(P,/K)'), 



(3) 



where P, = the population in year t, with / = cor- 

 responding to the baseline year before 

 commercial hunting started (here 1848); 



K(orPo) - the initial population size or carrying 

 capacity; 

 MSYR = the maximum sustainable yield rate of 

 production as a proportion of the popu- 

 lation aged 1+: and 

 Cf = the number of whales killed by hunting 

 in year t (known exactly). 



This model is much simpler than the BALEEN II 

 PDM (de la Mare and Cooke^ ) used by the IWC for 



^ de la Mare. W. K., and J. G. Cooke. 1993. "BALEEN II: The 

 population model used in the Hitter-Fitter Programs". Unpub- 

 li.'ihed manu.script available from the IWC Secretariat, The Red 

 House, 135 Station Road, Histon, Cambridge, UK CB4 4NP. 



