Precision of Acoustic Fish Stock Estimates

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Abstract
We derive efficient estimators of Q, the total fish quantity in an area sampled by acoustic survey, and the variance of Q. The estimators are valid when autocorrelation is insignificant, the nonzero samples have stationary statistics, and the distribution of the nonzero samples is normalised by a power transformation. Zero samples are treated separately, thus avoiding one source of nonstationarity. The Box–Cox test is applied to the nonzero data to determine the best λ to normalize the data through the transformation Z = (xλ − 1)/λ. Estimation formulae are presented for λ = 1/2, 1/3, 1/4, 1/6 and the log transform Z = log x which corresponds to λ = 0. Our methods have been applied to data from the 1984 and 1985 North Sea herring surveys. The sampling error is close to 25% in both years. Both the time between shoal encounters and the shoal size can be described by three-parameter Weibull distributions. Theoretical expressions for the mean fish quantity, and the variance of this mean, are presented as functions of the Weibull parameters. Computer simulations based on these distributions confirm the validity of the proposed estimation procedure. The estimators suggested by the Box–Cox test have the least bias. Other estimators exhibit severe bias. In particular, if log transforms are used when the data are not log-normally distributed and Box–Cox suggests λ>0, the mean fish quantity and its variance are both likely to be overestimated.