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Fishery Bulletin 97(3), 1999 



Both u\,Js) and «' ., depend upon the data trunca- 

 tion distance (Burnham et al., 1980), denoted by w 

 and equal to 5.5 km in this analysis. ;r*(s) was esti- 

 mated from the observed school sizes. However, to 

 estimate ;rts), we also needed to estimate «' .is), as 

 described in "Estimation" section. 



Anecdotal reports consistently imply that tuna 

 vessel captains do not search for or set on dolphin 

 schools at random when fishing on dolphin in the 

 ETP. Because larger dolphin schools are observed to 

 carry more tuna, they are presumably sought out and 

 set upon preferentially, and the set data would have 

 a selection bias towards large schools. We modeled 

 the schools associated with purse-seine sets (both 

 observed and unobserved) as a biased sample, with 

 replacement, of unknown size from the true popula- 

 tion of schools. To characterize these schools, both the 

 total number of sets, A'^^.^,,, and the effective probability 

 density from which their sizes were drawn, p(s), needed 

 to be estimated, pis) represents the superposition of 

 the tuna fishermen's school-size selection preference 

 upon nis). Sizes were recorded for all sets on trips car- 

 rying observers, and therefore there was no additional 

 observer selection bias (in relation top(s)). Assuming a 

 random selection of trips, we treated the observed sets 

 as an unbiased subsample of size n^^^^ and estimated 

 pis) directly from the observed sizes. There was some 

 concern with serial correlation between sets (see "In- 

 dependence of observations" section). 



Because the number of dolphin schools is not con- 

 stant over time, we interpreted A^jj.;,,,,,/,, as the time- 

 averaged expected number of schools, and in particu- 

 lar did not use a finite population estimator. Simi- 

 larly, we interpreted A,,.,,, as the expected number of 

 sets rather than making finite population estimates 

 of the actual realized number of sets.'* 



Estimation 



U.S. vessels— study period The observed school-size 

 distributions from both the research vessel sighting 

 data and from the tuna vessel set data were roughly 

 lognormal in shape (Fig. 2). We estimated ;r*(s) and 

 p(s) using an adaptive kernel density estimator on 

 the logs of the observed school sizes and then trans- 

 formed the estimated density back to the original 

 scale (Silverman, 1986). This variable bandwidth 

 algorithm was chosen in order to make reasonable 

 density estimates in the right tails where there were 

 few data, while not oversmoothing near the modes. 

 We chose the bandwidth scaling parameters as a 

 trade-off between smoothness and fit to the data. We 

 treated the observer estimates (mean estimates in 

 the case of research vessel data) as exact measure- 



■* Technical details and a discussion of our estimators for Af,^^^,, 

 and A^ ,, can be found in Perkins. P. C. and E. F. Edwards, 

 1997, SWFSC Admin. Rep. LJ-97-03, Southwe.st Fisheries Sci- 

 ence Center, La Jolla CA, 36 p. 



