Goodness-of-fit tests of the Fisher distribution

Goodness-of-fit tests of the Fisher distribution

Usage

fisher_goodness_of_fit(x, level = 0.95)

Source

modified after the python package apsg (v.2.0.6) by O. Lexa

Arguments

x

object of class "Spherical"

level

numeric. Confidence level (\(1-\alpha\)). Default is 0.95

Value

object of class "Test"

Details

The data are rotated to their mean direction. Under the Fisher model the colatitudes \(\theta'\) give \(1 - \cos(\theta')\) approximately exponential with rate \(k = (n - 1) / \sum(1 - \cos(\theta'))\) and the longitudes are uniform.

The colatitude test $$M_E = (D_n - 0.2/n)(\sqrt(n) + 0.26 + 0.5/\sqrt(n))$$ uses the Kolmogorov statistic \(D_n\) against the exponential distribution, while the longitude test $$M_U = V_n(\sqrt(n) - 0.467 + 1.623/\sqrt(n))$$ uses the Kuiper statistic \(V_n\) against the uniform distribution (FLE eq. 5.15-5.16).

Critical values are known for the levels 0.90, 0.95 and 0.99.

Note

The constants are FLE's, which are the original ones of Stephens (1974) and Fisher & Best (1984). Mardia & Jupp (2000, eq. 12.3.1, 12.3.4) misquote them.

See also

Other tests: bingham_test(), fisher_ftest(), fisher_kent_test(), rayleigh_test3(), watson_symmetry_test(), wellner()

Examples

set.seed(20250411)
x <- rvmf(100)
fisher_goodness_of_fit(x)
#> $colatitude
#> 
#> 	Fisher colatitude M_E
#> 
#> statistic = 0.69128
#> critical value (95% confidence level) = 1.094
#> null hypothesis rejected at the 95% confidence level: no
#> further statistics:
#>          D          k 
#> 0.06904965 4.95238057 
#> source: Fisher, Lewis & Embleton (1987), eq. 5.15-5.16 
#> 
#> 
#> $longitude
#> 
#> 	Fisher longitude M_U
#> 
#> statistic = 1.0834
#> critical value (95% confidence level) = 1.207
#> null hypothesis rejected at the 95% confidence level: no
#> further statistics:
#>         V 
#> 0.1117444 
#> source: Fisher, Lewis & Embleton (1987), eq. 5.15-5.16 
#> 
#>