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Fishery Bulletin 94(3). 1996 



exp(2p + cx 2 ) 

 var(c) = — - — - x 



(6) 



-E m , [m 2 {exp(CT 2 /m)£ ra (CTV2/?!)-l}]+p(l-p; 



p m (l-p)"""x 



where 



A ,( n 



{exp(cr / m)g m (o* 1 2m) - l} . 



It can be shown using results from Bradu and 

 Mundlak (1970) that the var(c) is always less than 

 or equal to vari x ), both decrease as n increases, but 

 var(c) decreases more quickly than does vari x ). For 

 values of a 2 typical for marine data, varic) is consid- 

 erably smaller than varix) (Pennington, 1986; 

 Smith, 1988). This can be seen in Figure 1 which 

 contains plots of varic) divided by varix) versus 

 sample size for a range of <r"s appropriate for ma- 

 rine survey data. 



Tracking trends in abundance 



For a series of marine surveys, it is usually assumed 

 that the mean catch per tow is proportional to popu- 

 lation size. If this is the case, then the estimator, c, 



Figure 1 



The relative efficiency of the estimators c and x for esti- 

 mating the mean of the A-distribution. The plots show the 

 varic) divided by the vari x ) as a function of sample size 

 when the variance of the nonzero logged values, cf , equals 

 2, 3, 4, and 5, and when the proportion of nonzero values, 

 p, is 0.8. 



is an index of abundance. The mean of the lognor- 

 mal distribution is given by exp(p + cr/2). McCon- 

 naughey and Conquest (1992) have suggested that 

 for lognormally distributed survey data, exp( y ), a 

 slightly biased estimate of exp(p), may be a more 

 stable index for following trends in abundance than 

 estimates of the mean. That is, if cr is constant over 

 time (which is equivalent to the coefficient of varia- 

 tion of the untransformed variable being constant) 

 then exp(p), the median of the lognormal distribu- 

 tion, will also be proportional to abundance. The vari- 

 ance of exp( y ) can be considerably smaller than the 

 variance of c. 



The mean of the A-distribution isp[exp(p + cr/2)]. 

 If the mean is proportional to population size and cr 

 is constant for a survey series, then plexp(p)] will 

 also be an index of abundance. It can be shown with 

 techniques similar to those in Pennington ( 1983 ) that 

 the minimum variance unbiased estimator, k, of 

 p[exp(p)] is 



n 



II 



o, 



exp( y)g n 



{2im-D 



m > 1 



m = l 

 w=0 



(7) 



and the minimum variance unbiased estimator of the 

 variance of & is given by 



var„,l*) = 



"' m-\\ m 



— exp(2vK — t 



n n 



2im - 1) 



-2s 2 



m > 1 



m = 1 

 m =0 



(8) 



As before, if m = n, then Equations 7 and 8 reduce to 

 the lognormal case (Bradu and Mundlak, 1970). 



Confidence intervals 



If /? is large, then c ± 2[i;ar Js , ic)] m and k ± 2[var est 

 (£)] 1/2 are approximately 95% confidence intervals. 

 For smaller n, a conservative approach for construct- 

 ing confidence intervals is to calculate separate in- 



