On the uniformly most powerful invariant test for the shoulder condition in line transect sampling

In wildlife population studies one of the main goals is estimating the population density. Line transect sampling is a well established methodology for this purpose. The usual approach for estimating the density of the population of interest is to assume a particular model for the detection function. 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. For instance, the half-normal model satisfies this condition whereas the negative exponential does not. So, testing whether the shoulder condition is consistent with the data is a primary concern. Since the problem of testing such a hypothesis 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 model against the negative exponential model. The asymptotic distribution of the test statistic is derived. The critical values and the power are tabulated via Monte Carlo simulations for small samples.