Nonparametric comparison of mean directions or mean axes. (English) Zbl 0934.62057

Beran, Rudolf; Fisher, Nicholas I.

Ann. Stat. 26, No. 2, 472-493 (1998).

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.

MSC:

62H11	Directional data; spatial statistics
62G10	Nonparametric hypothesis testing
62G15	Nonparametric tolerance and confidence regions

Keywords:

simultaneous confidence sets; simultaneous tests; bootstrap; rotational difference

References:

[1]	Beran, R. (1979). Exponential models for directional data. Ann. Statist. 7 1162-1178. · Zbl 0426.62030 · doi:10.1214/aos/1176344838
[2]	Beran, R. (1984). Bootstrap methods in statistics. Jahresber. Deutsch. Math. Vereini. 86 212-225. · Zbl 0547.62023
[3]	Beran, R. (1990). Refining bootstrap simultaneous confidence sets. J. Amer. Statist. Assoc. 85 417-426. JSTOR: · Zbl 0721.62032 · doi:10.2307/2289778
[4]	Fisher, N. I. (1985). Spherical medians. J. Roy. Statist. Soc. Ser. B 47 342-348. JSTOR: · Zbl 0605.62055
[5]	Fisher, N. I. (1995). Statistical Analy sis of Circular Data. Cambridge Univ. Press.
[6]	Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1993). Statistical Analy sis of Spherical Data. Cambridge Univ. Press. · Zbl 0782.62059
[7]	Fisher, N. I., Lunn, A. D. and Davies, S. J. (1993). Spherical median axes. J. Roy. Statist. Soc. Ser. B 55 117-124. JSTOR: · Zbl 0782.62058
[8]	Hall, P. (1986). On the number of bootstrap simulations required to construct a confidence interval. Ann. Statist. 14 1453-1462. · Zbl 0611.62048 · doi:10.1214/aos/1176350169
[9]	Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York. · Zbl 0744.62026
[10]	Kato, T. (1982). A Short Introduction to Perturbation Theory for Linear Operators. Springer, New York. · Zbl 0493.47008
[11]	Kent, J. T. (1992). New directions in shape analysis. In The Art of Statistical Science (K. V. Mardia, ed.) 115-127. Wiley, New York.
[12]	Lewis, T. and Fisher, N. I. (1995). Estimating the angle between the mean directions of two spherical distributions. Austral. J. Statist. 37 179-191. · Zbl 0838.62041 · doi:10.1111/j.1467-842X.1995.tb00652.x
[13]	Mardia, K. V. (1972). Statistics of Directional Data. Academic Press, London. · Zbl 0244.62005
[14]	Powell, C. McA., Cole, J. P. and Cudahy, T. J. (1985). Megakinking in the Lachlan Fold Belt. J. Struct. Geol. 7 281-300.
[15]	Watson, G. S. (1983). Statistics on Spheres. Wiley, New York. · Zbl 0646.62045