Daubenspeck and Ogden in a recent paper recommended the use of directional statistics in the analysis of response slopes, and their advice has been followed by other workers. Their method is not valid, since it does not follow directly from their model. An efficient estimator of the slope (i.e., an estimator with minimum variance) is well known and is given here with a confidence interval for the true slope. They were also concerned with the two-sample problem to compare the slopes from two different samples. The method for this is more complicated but is summarized here. The likelihood ratio test and point and interval estimates are given. We discuss Daubenspeck and Ogden's example and the reason why, despite its invalidity, their method gave good results using their own data. Their data are also used to illustrate the methods described in this paper, and examples are given to highlight the practical differences between the two methods. Step-by-step procedures are included in the appendixes to enable readers to use these methods.

In the analysis of astrophysical data assumed to lie on the celestial sphere, new directional data techniques are necessary when the monitoring station is on the earth's surface. Weighted spherical random variables are introduced and investigated for the cases when the true underlying distributions are either uniform, or of the Fisher or Dimroth-Watson forms. The uniform distribution on a rotating cap is examined and numerical examples are given with data for the arrival directions of ultra-high energy cosmic rays.

: Central Place Theory predicts a regular spatial pattern in the plane and it is observed that the Delaunay triangles will be equilateral under the theory. However, when the pattern is random, the asymptotic p.d.f. of the interior angles of a random Delaunay triangle are given. A von Mises-type model is proposed with a concentration parameter K; the larger the value of K, the closer one is to the Central Place Theory. The model can be approximated to the Miles' density for some value of K. The moment and maximum likelihood estimators of K are provided, and it is recognized that the areas of the Delaunay triangles play an important role. A test of departure from the random pattern is constructed with the alternative of Central Place Theory. As a numerical example, 44 Central Places in Iowa are analyzed where some evidence for the validity of Central Place Theory in that particular region is found.