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Fishery Bulletin 94(2), 1996 



ested in estimating the mean of all obser- 

 vations (including "true zeros") and in 

 modelling per-set discard, which requires 

 estimates of p and a. Therefore, we did 

 not follow their approach because we did 

 not have any a priori values forp and a. 

 Reducing counts to simple presence-ab- 

 sence would have decreased the informa- 

 tion in the sample such that estimation of 

 the full set of parameters would not have 

 been possible. 



Estimating variances for model 

 parameter estimates 



The analytic approximation formulae that 

 we used to estimate the variance of the 

 individual parameter estimates are based 

 on the asymptotic normality of MLE's. 

 Since this method uses the estimates for 

 p, a, and ji (rather than their unknown 

 "true" values) in the information matrix, 

 it suffers from the tendency for ML estimates 

 of variance to be biased downwards (e.g. 

 Efron, 1992). We did not attempt to "bias 

 correct" these variance estimates. 



When a variance estimate is based on a 

 normal approximation to the sampling dis- 

 tribution of the parameter, the accuracy 

 of the approximation should always be in- 

 vestigated. One way to help validate the 

 normality assumption is to use results 

 from bootstrapping to approximate the 

 sampling distribution. Figure 5 illustrates 

 some examples for the current data. His- 

 tograms of the bootstrap replicate parameter esti- 

 mates for dolphin set data were very skewed. By 

 implication, the normal-approximation variance es- 

 timates for the dolphin data, while convenient, are 

 probably not satisfactory. For school set data, histo- 

 grams of the replicates were slightly skewed because 

 of a small number of unusually large observations. 

 Bootstrap standard errors were consistently higher 

 than the analytic approximations, indicating that the 

 latter may be optimistic. For log set data, histograms 

 were close to normality, and bootstrap standard er- 

 rors were very similar to those from the analytic 

 approximations. The analytic estimates in this case 

 are probably appropriate. 



Estimating variances for mean discard per 

 set estimates 



In an attempt to derive analytic formulae for the vari- 

 ance of our estimates of E\Y], we manipulated the 



Dolphin sets 

 All areas 



ill. 



Illlllllllllll.ll,.!.., 



M 



10 



School sets 

 All areas 



10 



Log sets 

 Area 3 



M 



Figure 5 



Sample histograms of 1,000 bootstrap replicates of estimates of the 

 negative binomial mean parameters, for dolphin, school, and log sets. 

 See text for a complete description of the parameter 



likelihood equations for the NBAZ and found simpli- 

 fied forms for the MLE of E\Y\. In some cases, 

 the simplified form reduces to the sample mean, 

 Equation 6, and the variance of that estimator is 

 simply (suppressing area and set type subscripts for 

 simplicity) 



var 



(KIVI 



■(l/«)var|Y| 



:i//;i(l-/j>)(// + (a + /j)/r). 



(9) 



which can be estimated by substituting MLE's for a, 

 ft, and p. More simply, by using the fact that the es- 

 timator is just the sample mean, the minimum vari- 

 ance unbiased estimate of Equation 9 is the sample 

 variance, 



var 



(E[Yl): 



l (V < 



/</?-!), 



(10) 



