Estimating the mean and variance from highly skewed marine data

Lo et aI. , 1992; McConnaughey and Conquest, 1992; Conquest et aI. , 1996; Stefansson, in press). For marine data, the distribution of the nonzero values is often well approximated by a lognormal distribution (e.g. Pennington, 1983; Smith, 1988; McConnaughey and Conquest, 1992; Conquest et aI., 1996). Myers and Pepin (1990 I found that of the 69 marine data sets they examined, only 5 differed significantly from the lognormal distribution. Thus the lognormal model has been used as a basis for developing survey abundance estimators (e.g. Pennington, 1983,1986; Lo et aI., 1992; McConnaughey and Conquest, 1992) and for estimating commercial catch (Conquest et aI., 1996l. It is not surprising that marine abundance data often appear to follow a lognormal distribution. The factors that determine abundance over a region seem to have a multiplicative effect. When this is the case, survey data will be approximately lognormally distributed by the central limit theorem (see, e.g. Aitchison and Brown, 1957). More generally, the lognormal model has been useful for analyzing a wide range of ecological data. As Dennis and PatH (1988) put it: "Ecological abundance data are intrinsically positive, with a few enormously high data points typically arising in every study. The lognormal distribution is an ideal descriptor ofsuch tiibution of abundance data gcncr= ated by marine surveys has a large variance, is highly skewed to the right, and contains a substantial proportion of zeros. Because of this large variability, the sample mean has a low level of precision even for relatively intensive surveys (Grosslein, 1971; God0, 1994; Pennington and God0, 1995 I. A common problem in the analyses and interpretation of skewed survey data, is that a single immense catch may account for 50% or more of the total catch during a survey (Sissenwine, 1978; Dew, 1990; McConnaughey and Conquest, 1992; Bowering and Brodie, 1994). These extreme values not only greatly affect the estimate of the mean but also of the variance (Otto, 1986). As McConnaughey and Conquest (1992) observed, although these large values cause much uncertainty for management, they reflect the spatial distribution of the species and are not outliers that should be discarded. In practice, the use ofmore efficient sampling schemes or estimators is the only realistic way to increase survey precision; the total number of samples that can be taken is limited by the high cost of sampling at sea (Gunderson, 1993). One possible way to increase the precision of survey estimates is to model the observed distribution of catches and exploit the model's properties to develop more efficient estimators of population paramtimates for data sets that contain a very large catch. The properties and efficiency of the ll. -distribution estimators are examined and the techniques are applied to various marine data sets. Abstract. """'l'he spatial distribution of marine organisms is highly patchy. Because of this patchy distribution, data from marine abundance surveys are highly skewed and have a large variance. Compounding the problem of estimating the mean abundance from such data, is that occasionally a relatively huge catch will occur. These large catches are not "outliers· but do dominate the estimates of the mean and variance. Alognormal model ofthe nonzero survey values la ~-distribution)is used to model survey data. The estimators, based on the lognormal model, appear to be much more efficient for marine data than the usual sample estimators. In particular, the lognormalbased estimators provide reasonable es-