variability of marine organisms when the data 

 contain zero observations. 



The typical lognormal distribution is asymmet- 

 ric, with a long tail on the right-hand side, and 

 with a population mean that lies to the right of 

 the middle point. Because the distance between 

 the middle point and the mean increases with the 

 variability of the data, the geometric mean or its 

 estimator the sample median are often used to 

 describe central tendency in lognormal data. 

 However, there is no known estimator for the 

 variance of those statistics when the lognormal 

 data contain zeros. The basis for applying the 

 delta distribution to describe the abundance of 

 marine organisms is that, for approximately 

 lognormal data with many zeros, the best estima- 

 tors of the population mean and its variance are 

 the mean of the delta distribution {h-meati) and 

 its variance {h-variance). Like the sample mean 

 (i.e., the average or arithmetic mean of the sam- 

 ple), the h-mean estimates the population mean 

 rather than the mid-point of the data distribution. 

 Recent applications of the delta distribution to 

 describe the variability of ichthyoplankton and 

 fish in the MARMAP program (National Marine 

 Fisheries Service) have been reported by 

 Pennington (1983, 1986). 



^^mCF) = 1 + m + 



m\2^?)(m + 1) 



{m - \)'y^ 



m\y.){m + \){m + 3) 

 The constant y in these series is computed as 



(2) 



(3) 



where .!■ is the sample variance of the log- 

 transformed m nonzero observations. The num- 

 ber of terms needed in the series (Eq. 2) to achieve 

 reasonable accuracy was found to be six to ten, 

 depending on the number of significant digits in 

 the logarithmic mean [x). 



The unbiased estimator of the 5-mean variance 

 was also derived by Aitchison (1955). However, 

 for large n and a proportion of zeros (5) appre- 

 ciably less than 1.0, Owens and DeRouen (1980) 

 found that the approximate (asymptotic) variance 

 of h-mean given by Aitchison and Brown (1969) 

 was accurate enough and much easier to compute. 

 1 his simplified estimator is computed as: 



Estimation of the 8-mean and its variance: The 



minimum variance unbiased estimator of the 

 h-mean was derived by Aitchison (1955). Its 

 computation in practice is rather involved and it 

 generally requires the use of a computer program 

 to evaluate a series iteratively until enough accu- 

 racy is reached. The estimate of the h-mean for 

 a given data collection is computed as: 



h-mean = 



exp(jc) G^{y) , 



(1) 



where m is the number of nonzero values in the 

 data, n is the total number of observations in the 

 data, X is the arithmetic mean of the log- 

 transformed nonzero observations, and Cj„,{y) is 

 a Bessel function that is evaluated as the scries: 



h-var - ^exp(2Jc + .r^)|s{l - 8) + ^(I - 5)(2.v^ + j'')}(4) 

 for a proportion of zeros 



5 = 1 



(5) 



and where m, n, x, and s are as previously de- 

 fined for equations 1 through 3. 



In applications with actual data it is often of 

 interest to construct confidence intervals for the 

 h-mean estimates. Not knowing the exact statis- 

 tical distribution of the h-mean as estimated by 

 r.q. 1, it may be reasonable to assume asymptotic 

 normality for this estimator in order to form ap- 

 proximate confidence intervals. Owen and 

 DeRouen (1980) investigated this possibility in a 



312 



