I 12 



Fishery Bulletin 91(1). 1993 



mean density and, as such, is a measure of accuracy. 

 For any unbiased estimator (e.g., the AM or FM, where 

 the expected value of the estimator is the parameter 

 itself), the RMSE is the same as the variance of the 

 estimator, in terms of expected value. For a biased 

 estimate (recall that GM has positive bias in estimat- 

 ing e MLN , which decreases as n increases), RMSE incor- 

 porates both bias and variance. 



BIAS= 



y (estimated parameter 



^ from i' h data set — true value) 



1000 



= (average value of estimated parameter — 

 true parameter). 



Results 



Monte Carlo simulations 



Root mean squared error The RMSE was consis- 

 tently lower for the GM than for the other measures of 

 central tendency (Fig. 2). The FM provided point esti- 

 mates of p that were consistently more accurate (ex- 

 cept at very small sample sizes) than the AM method, 

 particularly as skewness of the density data (Fig. 1) 

 increased. The RMSE of GM estimates and of p ob- 

 tained with the FM declined steadily as sample size 

 increased, whereas that for the AM, although gener- 

 ally declining, was somewhat less regular and much 

 more erratic (see Fig. 2a, n=40). Closer inspection of 

 the LOGN (4,2) data set revealed a single extreme 



The bias is the average amount by which the estimate 

 tends to "miss" its respective parameter. 



1000 1000 



£ (UL-LL\ £ length, 

 AVL = 



1000 



1000 



where (UL-LL), = length of a single 90% confidence 

 interval for the i ,h data set. The average length is a 

 measure of precision. 



SDCI 



£ (length, -AVL) 2 



1000 - 1 



The standard deviation is a measure of the spread 

 of the confidence-interval lengths around the average 

 length. An estimator with the most reproducible esti- 

 mate of the precision of the estimated mean would 

 have confidence intervals of relatively low variability 

 in length. 



The frequency with which a confidence interval in- 

 cludes the true value of the parameter defines the con- 

 tainment rate, PERCON. If the assumptions of sam- 

 pling and the appropriateness of statistical model are 

 met, 90% confidence intervals should contain the den- 

 sity parameter being estimated approximately 90% of 

 the time. 



The three estimators of central tendency (AM, FM, 

 GM) and their confidence intervals were also calcu- 

 lated for actual density data obtained during the 

 monthly trawl surveys. Two large systems, termed the 

 Coast and the Estuary, were considered. 



20 30 



SAMPLE SIZE 



Figure 2 



Comparison of root mean-square error (RMSE) values associ- 

 ated with the three estimators of central tendency according 

 to sample size, (a) LOGN (4,2) data representative of coastal 

 population of Cancer magister used for Monte Carlo simula- 

 tions. See text for outlier explanation, (b) LOGN (6,1) data 

 representative of estuarine population of C. magister used for 

 Monte Carlo simulations. 



