McConnaughey and Conquest: Comparative trawl-survey estimation based on lognormal theory 



109 



sample variance) are estimators of u and o 2 , the popu- 

 lation mean and variance of the untransformed data. 

 Letting Y = In (X), Yand s 2 Y are estimators of u LN and 

 o 2 ln the population mean and variance of the log- 

 transformed data. Note that in Eq. (A3) and (A4) both 

 u and o 2 for the lognormal distribution are functions 

 of two parameters, u LN and a 2 LN , making the former 

 parameters difficult to estimate. In particular, any es- 

 timate of u involves both location and dispersion pa- 

 rameters. Therefore, variation in estimating u will come 

 from two sources: variation in estimating u LN and 

 variation in estimating a 2 LN . 



The arithmetic mean (AM) The ordinary sample mean 

 is an unbiased estimator of u regardless of the under- 

 lying frequency distribution. When the underlying dis- 

 tribution is normal, the sample mean is also the mini- 

 mum variance unbiased estimator (MVU, the one with 

 the smallest variance of all unbiased estimators) of u. 

 However, the sample mean does not have this MVU 

 property when the underlying distribution is lognor- 

 mal (Gilbert 1987). Moreover, the AM is sensitive to 

 the presence of one or more large data values, particu- 

 larly for small sample sizes. For lognormal data, these 

 extreme values are not outliers; they simply reflect 

 the right-skewed nature of the distribution. Finney 

 (1941) demonstrated the inefficiency of the sample 

 mean when the variance of the natural logarithms is 

 greater than 0.69, and Koch & Link (1970) suggested 

 using the sample mean only when the coefficient of 

 variation is believed to be less than 120%. For highly- 

 skewed distributions such as the lognormal, sample 

 sizes in excess of 200 may be necessary to invoke the 

 Central Limit Theorem, which justifies use of the 

 sample mean for inferences about means of popula- 

 tions that are not normally distributed (Sissenwine 

 1978, Jahn 1987). 



The Finney-Sichel estimator (FM) Among alternative 

 estimators that have been investigated is an MVU es- 

 timator of u (Finney 1941, SicheJ 1952), which also 

 has been described as equivalent to a maximum- 

 likelihood estimator for lognormal data (Aitchison & 

 Brown 1969). The Finney-Sichel method adjusts the 

 geometric mean upwards and is commonly used in gold 

 and trace-mineral assay work, where ore concentra- 

 tions are typically lognormally distributed ( Sichel 1952 ). 

 If Yand s 2 Y represent the ordinary sample mean and 

 variance of the log-transformed values, the Finney- 

 Sichel estimate for \i is 



1 (n-l)t 

 1 + + 



(n-l)H 2 



(n-l)V 



FM = exp(Y)y n (r) 



(1) 



where n is the sample size and \j/ n (t) is the infinite 

 series 



2\n-(n+l) 3ln 3 (n+l)(n+3) 



(n-Vt 4 

 4!n 4 (n+l)(rc+3)(n+5) 



(2) 



+ .. 



with t = —^— . The function \|/ n (t) is defined such that 

 E[\|/ n (s 2 )] = exp f-^p o 2 ) and ^^ [\j/ n (s 2 >] = exp(o 2 ); 



it is used extensively with the lognormal distribution 

 (Smith 1988). In their book, Aitchison & Brown (1969) 

 included tables of y„ for computing the Finney-Sichel 

 estimate. More extensive tables are provided in Link 

 et al. (1971), who claim that linear interpolation be- 

 tween tabled values gives close approximation for esti- 

 mates of u. They also include a FORTRAN program 

 for calculating the \\i n function, which we used in com- 

 puting FM, the Finney-Sichel estimate of the popula- 

 tion mean. (A version of this program may be obtained 

 from the authors. ) 



Confidence limits for the lognormal mean are not 

 symmetric because of the skewed nature of the under- 

 lying distribution. Hence, it becomes necessary to com- 

 pute separate upper and lower confidence limits. Land 

 (1971, 1975) obtained upper one-sided 100(1-00% and 

 lower one-sided 100a% confidence limits for the log- 

 normal mean, where a is the frequency of type I error: 



C/L, 



exp 



Y + 



s 2 Y 



s Y H h 



LL a = exp 



Y + — + S v 



Vn-1 





(3) 



(4) 



The quantities H^, and H„ [functions of a, (n-1) and 

 s Y ] are obtained from tables in Land ( 1975) for sample 

 sizes of n>3. 



The geometric mean (GM) The geometric mean, e^, 

 will be a biased estimate of |a (Appendix) but may be 

 more precise with respect to its population parameter 

 than will be the case for estimators of the population 



mean. (Actually, £(e ? ) = exp(u LN + - — - so the GM is 



2/JOln 



biased even for e^ L ' N ', but this bias decreases rapidly as 

 n increases.) When exponentiated, the population mean 

 of the transformed data, u LX . is the geometric mean 

 catch and, equivalently, the median catch for lognor- 

 mal data. It remains unaffected by skewness, a func- 

 tion of [exp(o 2 LN -D]. It is less affected by large values 

 of X, owing to the nature of a logarithmic transforma- 

 tion; hence, its sampling distribution is less skewed 



