Simple estimators of salmonid escapement and its variance using a new area-under-the-curve method

If observed numbers of spawning salmon are plotted against sampling date, then the area under the curve (AUC) gives an estimate of spawner-days. Dividing AUC by spawner lifetime and adjusting for observer efficiency gives an estimate of spawner escapement. In particular, the trapezoidal form of AUC estimator has been widely used over the last two decades, despite the absence of a direct method for calculating its variance. For this reason, an alternative estimator of escapement was developed using a maximum likelihood (ML) model of spawner arrivals. However, this alternative has not been widely used, perhaps because of its complexity and concerns over validity of assumptions. Here, a simpler ML approach is used to estimate AUC by fitting a model directly to spawner numbers. It can be fitted using existing generalized linear modeling software and provides an explicit variance estimator for AUC and escapement. Simulations show that it has consistently higher coverage than the existing ML estimator. However, the trapezoidal AUC estimator was generally a slightly more precise estimator of escapement than either of the ML approaches.