Sequential Estimation of Log(Abundance)

Abstract

I discuss the estimation of the abundance of a biological population, its logarithm, and the variances of these estimates, from a sequential sampling scheme with minimum and maximum sample sizes. Observations are counts of organisms in randomly chosen such as cores, branches, bushes, and so forth. For preassigned values m, ni and n2, samples are taken until (a) at least ni packets and (b) either m positive packets or a total of n2 packets have been observed. Abundance estimates are based on an estimate of the fraction of positive packets given by Kremers (1987, Technometrics 29, 109-112), with a modification to avoid estimates of zero. Estimates of log abundance are given by log(estimated abundance) with an adjustment for bias due to the concavity of the log function. Two adjustments are considered, one based on Taylor series expansion (the delta method) and the other on the bootstrap. These techniques are also used to estimate the variance of the estimate of log(abundance). Simulations suggest that both methods are better than not adjusting, though the gain is small compared to the standard deviation of the estimates. The bootstrap estimates are less biased than the Taylor series estimates but have larger variances, so that the Taylor series estimates have smaller mean squared errors. The variances of the sequential estimates of log(abundance) tend to be only weakly dependent on the true abundance.