A useful decomposition of the resultant length for samples from von Mises-Fisher distributions

Efficient Simulation of the von Mises Distribution

“…However, because much of the statistical theory associated with the distribution is analytically intractable recent studies, such as those by Fisher and Willcox (1978) on the distribution of test statistics for circular data, and by Collett (1978) on problems relating to outliers, have relied on generation of pseudo-random observations from the von Mises distribution.…”

Goodness‐of‐fit and Discordancy Tests for Samples From the Watson Distribution on the Sphere

“…Each method is illustrated using samples of data from Structural Geology. The discussion parallels the material for the Fisher distribution in Lewis & Fisher (1982), Fisher, Lewis & Willcox (1981), and Fisher & Best (1984).…”

“…The Watson distribution is a special case of the general Bingham distribution for axial data (Bingham 1964(Bingham , 1974. It was first studied as a model for axial data by Watson (1965), although prior to that it had been suggested by a number of writers (see Fisher, Lewis & Embleton (1986) for details).…”

Tests of Discordancy for Samples from Fisher's Distribution on the Sphere

“…232) for large K. As expected, the percentiles of X; and y" did not exhibit noticeable dependence on A, and accorded well with the tabulated values for Ea 1 and Ea 2 respectively (Barnett and Lewis, 1978). It is known from elsewhere (Fisher and Willcox, 1978) that the distribution of the random variable (n -2)(1+ R~i~1-Rn)/(n -1-R~~I) is closely approximated by that of an F 2. 2n-4-variate for K > 2'5; with the use of Bonferroni inequalities the following approximation to the tail probability of En (see Appendix for derivation) was obtained, and found to agree satisfactorily with the simulated percentiles for K~10:…”

“…Because the distribution of Cn depends heavily on K, extensive simulations would be required to establish the distribution of the corresponding ·tatistic, C~) say, for detection of t outliers. However, the extension of En to E~>' say, is feasible from the point of view of tabulating the distribution, because the distribution of the random variable [(n-t-l)/t](t+R n _ t-R.)/(n-t-R n _ t ) is well-approximated by that of an F-variate with 2t and 2(n-t -1) degrees of freedom (Fisher and Willcox, 1978)providing K> 2·5 and n is not too small. As a rough guide to the significance of a given value e, of E~), one might use the first-order Bonferroni bound P(E~» et):::;;(~) P(F2t.2(n-t-1» e.), Some upper critical values of E~), t = 2,3, for selected sample sizes, have also been obtained by simulation (in the manner described in Section 2), using K = 100.…”