The Uniformly Most Powerful Invariant Test for two Models of Detection Function in Point Transect Sampling

Estimating population abundance is of primary interest in wildlife population studies. Point transect sampling is a well established methodology for this purpose. The usual approach for estimating the density or the size of the population of interest is to assume a particular model for the detection function (the conditional probability of detecting an animal given that it is at a certain distance from the observer). Two popular models for this function are the half-normal model and the negative exponential model. However, it appears that the estimates are extremely sensitive to the shape of the detection function, particularly to the so-called shoulder condition, which ensures that an animal is nearly certain to be detected if it is at a small distance from the observer. The half-normal model satisfies this condition whereas the negative exponential does not. Testing whether such a hypothesis is consistent with the data at hand should be a primary concern. Given that the problem of testing the shoulder condition of a detection function is invariant under the group of scale transformations, in this paper we propose the uniformly most powerful test in the class of the scale invariant tests for the half-normal against the negative exponential model. The asymptotic distribution of the test statistic is calculated by utilising both the two models while the critical values and the power are tabulated via Monte Carlo simulations for small samples. Finally, the procedure is applied to two datasets of chipping sparrows collected at the Rocky Mountain Bird Observatory, Colorado.