g01er returns the probability associated with the lower tail of the von Mises distribution between -π and π through the function name.


public static double g01er(
	double t,
	double vk,
	out int ifail
Visual Basic
Public Shared Function g01er ( _
	t As Double, _
	vk As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double g01er(
	double t, 
	double vk, 
	[OutAttribute] int% ifail
static member g01er : 
        t : float * 
        vk : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: θ, the observed von Mises statistic measured in radians.
Type: System..::..Double
On entry: the concentration parameter κ, of the von Mises distribution.
Constraint: vk0.0.
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01er returns the probability associated with the lower tail of the von Mises distribution between -π and π through the function name.


The von Mises distribution is a symmetric distribution used in the analysis of circular data. The lower tail area of this distribution on the circle with mean direction μ0=0 and concentration parameter kappa, κ, can be written as
where θ is reduced modulo 2π so that -πθ<π and κ0. Note that if θ=π then g01er returns a probability of 1. For very small κ the distribution is almost the uniform distribution, whereas for κ all the probability is concentrated at one point.
The method of calculation for small κ involves backwards recursion through a series expansion in terms of modified Bessel functions, while for large κ an asymptotic Normal approximation is used.
In the case of small κ the series expansion of Pr(Θθ: κ) can be expressed as
where Inκ is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio Rn,
Rnκ=InκIn-1κ,  n=1,2,3,,
which is calculated using backwards recursion.
For large values of κ (see [Accuracy]) an asymptotic Normal approximation is used. The angle Θ is transformed to the nearly Normally distributed variate Z,
and bκ is computed from a continued fraction approximation. An approximation to order κ-4 of the asymptotic normalizing series for z is then used. Finally the Normal probability integral is evaluated.
For a more detailed analysis of the methods used see Hill (1977).


Hill G W (1977) Algorithm 518: Incomplete Bessel function I0: The Von Mises distribution ACM Trans. Math. Software 3 279–284
Mardia K V (1972) Statistics of Directional Data Academic Press

Error Indicators and Warnings

Errors or warnings detected by the method:
On entry,vk<0.0 and g01er returns 0.
An error occured, see message report.


g01er uses one of two sets of constants depending on the value of machine precision. One set gives an accuracy of six digits and uses the Normal approximation when vk6.5, the other gives an accuracy of 12 digits and uses the Normal approximation when vk50.0.

Parallelism and Performance


Further Comments

Using the series expansion for small κ the time taken by g01er increases linearly with κ; for larger κ, for which the asymptotic Normal approximation is used, the time taken is much less.
If angles outside the region -πθ<π are used care has to be taken in evaluating the probability of being in a region θ1θθ2 if the region contains an odd multiple of π, 2n+1π. The value of Fθ2;κ-Fθ1;κ will be negative and the correct probability should then be obtained by adding one to the value.


This example inputs four values from the von Mises distribution along with the values of the parameter κ. The probabilities are computed and printed.

Example program (C#): g01ere.cs

Example program data: g01ere.d

Example program results: g01ere.r

See Also