FISHERY BULLETIN: VOL. 85, NO. 2 



versus numbers of breeding males for those years 

 in which data were collected on all rookies indicated 

 a strong relationship that could be used for predic- 

 tion of total pup production if only subsamples of 

 rookeries were censused. The relationship held for 

 those years when censuses of pups were conducted 

 by counting (Fig. 1), and for later years when the 

 shearing-sampling method was used (Fig. 2). Al- 

 though the slopes varied substantially from year to 

 year (they ranged from 71 in 1913 to 29 in 1963), 

 the variance about the regression line within any 

 particular year was very small. 



We compared the various estimators and sampling 

 plans by analyzing the bias and variance of the esti- 

 mates and the half-width and coverage properties 

 of nominal 95% confidence intervals for 3 years 

 (1965, 1970, 1975) of data when all rookies were 

 sampled. Detailed statistics on the performance of 

 the estimators under all sampling plans appear in 

 a manuscript report available from the authors^. 



Under simple random sampling, the "blow-up" 

 estimate is unbiased. The various ratio estimates are 

 all slightly biased (in most cases less than 1%) with 

 the regression estimate exhibiting the largest 

 degree of bias. In Figure 3 the percentage of bias 

 of the three ratio estimates for the 1975 data is 

 shown as a function of sample size (under simple ran- 

 dom sampling). Estimates based on 1975 data were 

 the most biased among the 3 years analyzed, and 

 these biases are exhibited as a worst case. The 

 regression estimate was the most biased, and for 

 these data the bias increased as the sample size in- 

 creased; however, the bias was only about 1% and 

 is not serious. 



Confidence intervals were constructed for each 

 subsample and a count was made of the number of 

 nominal 95% confidence intervals containing the 

 "true" population. The observed coverage was near 

 95% for most procedures. Confidence intervals for 

 the regression estimate tended to be conservative, 

 i.e., a higher than 95% coverage rate, while the 

 coverage rate for the ordinary ratio estimate tended 

 to be less than 95%. Coverage rates for the jack- 

 knife and blow-up estimates were near 95% or a bit 

 higher. This indicates that the estimate of the vari- 

 ance of the regression estimate tended to be too 

 large, that of the ordinary ratio estimate was too 

 small, and that the estimates of the variance of the 



^York, A. E., and P. Kozloff. 1985. Estimation of numbers of 

 fur seal pups bom on St. Paul Island, 1980-84. Unpubl. manuscr. 

 Available National Marine Mammal Laboratory; 7600 Sand Point 

 Way N.E., Seattle, WA 98115. (Background paper submitted to 

 the 28th Annual Meeting of the Standing Scientific Subcommit- 

 tee of the North Pacific Fur Seal Commission, March-April 1985, 

 Tokyo, Japan.) 



blow-up and jackknife estimates tended to be un- 

 biased. The half-widths of confidence intervals for 

 the ratio estimates were nearly equal. All were less 

 than one-half the length of the half -width of the con- 

 fidence interval of the blow-up estimate. 



The rookeries were stratified by population size. 

 Two methods for stratifying the rookeries were in- 

 vestigated: one using two strata (small and large 

 rookeries) and the other using three strata (small, 

 medium, and large rookeries). As in the case of sim- 

 ple random sampling, the ratio estimators were 

 superior to the blow-up estimates. The estimators 

 under the three-strata sampling plans were less 

 variable than under the two-strata sampling plans. 

 In addition, the computed levels of the nominal 95% 

 confidence intervals were higher and the size of the 

 confidence intervals smaller. Under the three-strata 

 sampling plans, the standard deviations of the esti- 

 mates were about 10% smaller than under simple 

 random sampling with the same size sample. This 

 resulted in a similar reduction in the size of the con- 

 fidence intervals. These results indicated that 

 reasonable estimates of the size of the pup popula- 

 tion can be made using any of the ratio estimators 

 under various sampling plans. The superior plans 

 use three strata: two small, one medium, and one 

 large rookery; one small, two medium, and one large 



l.5r 



, Jacknife 



0.9 



c 



Q. 



m 



■0.3 



-0.9 



-1.5 



12 



15 



Sample size 



Figure 3.— Percent bias of the jackknife estimates ( ), ordinary 



ratio estimates ( • • • ), and regression estimates ( ) based on sim- 

 ple random sampling of 1975 northern fur seal data. 



372 



