Note on the Variance of Radial Sampling Strategies with Random Versus Purposively Located Pivot Points
Several radial sampling strategies have recently been proposed, including sector sampling, radial line sampling for estimation of area of coverage, and radial line sampling for planar area estimation. These strategies are based on either sector or line samples that emanate from a common interior point (pivot point) within the land area to be sampled. It has usually been assumed that the pivot point will be purposively located so that its location will not be random. However, random location of the pivot point within the sampling area is conceptually possible, so that it is of interest to compare the variance of estimators with purposively located pivot points with those with randomly located pivot points. When this is done, it can be determined that the variance of the estimator with a randomly located pivot point is the expected value of the conditional (on pivot point location) variance of the estimator with respect to the distribution of possible pivot point locations. Thus, for instance, if each point in the area to be sampled is equally likely to be chosen, then the variance under a random pivot point location is the mean or expected value of the conditional variance with respect to every possible pivot point location within the sample area.