Nonparametric comparison of mean directions or mean axes
Geological data often contains directions or axes measured in three dimensions. Examples are directions of remanent magnetization of lava cores, axes normal to geological folding planes, or positions on the surface of the earth. Biological measurements may include directions or axes in two dimensions. Instances are the directions in which birds or insects fly after release and time-of-day viewed as a circular variable. In econometrics, season or month-of-the-year are discrete circular variables that correspond to angular portions of the earth's orbit around the sun. Section 1.1 recalls the definitions of random direction or random axis and of mean direction or mean axis. This paper develops nonparametric methods for comparing all pairs of mean directions (or mean axes) of \(s\) independent samples of directions (or axes). No shape assumptions are imposed upon the unknown distribution of the observations in each sample. In practice, a mean direction or a mean axis summarizes a distribution most effectively when that distribution is unimodal. Central to our methodology is the representation of the rotational difference between two mean directions (or two mean axes) as a direction (or axis). This representation has two important features: it enables us to apply nonparametric methods for inference about one mean direction or one mean axis, and it suggests plots for rotational differences in two or three dimensions. Details of the representation are the subject of Section 1.2. Section 2 constructs confidence sets for the rotational difference between the mean directions (or mean axes) of two samples. Simultaneous confidence sets for all pairwise rotational differences among the mean directions (or mean axes) of \(s\) samples are developed in Section 3. Section 4 illustrates our methodology on data.
scientific article; zbMATH DE number 1359522