FISHERY BULLETIN: VOL. 84, NO. 3 



a proportion of zeros and the nonzero values are 

 distributed lognormally. The minimum variance un- 

 biased estimates of the mean, c, and its variance, 

 var(c), for the A-distribution are given by (Penning- 

 ton 1983), 



c = 



and 



var(c) = 



ra 

 n 



n' 



0, 



exp(^) G m (s 2 /2), ra > 1, 



ra = 1, 

 ra = 0, 



(1) 



f ™ exp(2^) 



= <*<**> •■feff 



x G r 



I m - 2 

 m - 1 



, m > 1, 



ra = 1, 



(2) 



EFFICIENCY OF x 



Figure 1.— The efficiency of x and s 2 (the sample mean and 

 variance, respectively) for the A-distribution with 50% zeros. 



o, 



ra = 0, 



where n is the number of tows, m is the number of 

 nonzero values, y and s 2 are the sample mean and 

 variance respectively of the nonzero log f values, x 1 

 is the single (untransformed) nonzero value when 

 ra = 1, and 



GJx) = 1 + 



+ I 



ra 



1 



ra 



x 



(m - l) 2 -? -1 x> 



j=2 jyO (ra + 1) (ra + 3). . .(ra + 2j> - 3) j\ ' 



The series defining G m {x) is a function of x [e.g., # 

 = s 2 /2 in Equation (1)] and ra which is easily 

 evaluated for particular values of x and ra using a 

 computer. 



Figure 1, which is an extension of a graph in Ait- 

 chison and Brown (1957, p. 98), shows the large sam- 

 ple efficiency of the ordinary sample statistics as 

 compared with their most efficient estimates for the 

 A-distribution with 50% zeros. Estimates of a 2 , the 

 variance of the nonzero log e values, are often be- 

 tween 1 and 2 for trawl surveys. Thus (Fig. 1) the 



sample mean is a fairly efficient estimator of the 

 mean for trawl surveys, but the sample variance is 

 highly inefficient. Though for larger values of o 2 , 

 which, for example, are common for egg surveys 

 (Pennington and Berrien 1984), the sample mean is 

 also very inefficient. It does not follow that the 

 variance of c is necessarily small, but it is smaller, 

 and as o 2 increases, much smaller than the variance 

 of the sample mean. However, it should be noted 

 that if the sample variance is used to estimate the 

 variance of the sample mean for moderate sample 

 sizes because of the inefficiency of the sample 

 variance, the estimated variance of c will often be 

 greater than the estimated variance of the sample 

 mean. 



Estimating the Index of Abundance 



As an index of abundance, the series of yearly 

 catch per tow estimates, y h (based, e.g., on the A- 

 distribution theory if appropriate) has two draw- 

 backs. First, its estimated variance when derived 

 from the within survey variance can be an under- 

 estimate since catchability may vary from year to 

 year. The second and more serious deficiency is that 

 the index for a particular year is based only on that 

 year's survey which disregards relevant information 

 contained in the surveys for other years. 



520 



