public static double g01er(
	double t,
	double vk,
	out int ifail
)

Public Shared Function g01er ( _
	t As Double, _
	vk As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double g01er(
	double t, 
	double vk, 
	[OutAttribute] int% ifail
)

static member g01er : 
        t : float * 
        vk : float * 
        ifail : int byref -> float 

g01er returns the probability associated with the lower tail of the von Mises distribution between $-\pi$ and $\pi$ 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 ${\mu }_0=0$ and concentration parameter kappa, $\kappa$, can be written as where $\theta$ is reduced modulo $2\pi$ so that $-\pi \le \theta <\pi$ and $\kappa \ge 0$. Note that if $\theta =\pi$ then g01er returns a probability of $1$. For very small $\kappa$ the distribution is almost the uniform distribution, whereas for $\kappa \to \infty$ all the probability is concentrated at one point.

The method of calculation for small $\kappa$ involves backwards recursion through a series expansion in terms of modified Bessel functions, while for large $\kappa$ an asymptotic Normal approximation is used.

In the case of small $\kappa$ the series expansion of Pr($\Theta \le \theta$: $\kappa$) can be expressed as where $I_n\left(\kappa \right)$ is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio $R_n$, which is calculated using backwards recursion.

For large values of $\kappa$ (see [Accuracy]) an asymptotic Normal approximation is used. The angle $\Theta$ is transformed to the nearly Normally distributed variate $Z$, where and $b\left(\kappa \right)$ is computed from a continued fraction approximation. An approximation to order ${\kappa }^{-4}$ 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 $I_0$: The Von Mises distribution ACM Trans. Math. Software 3 279–284

Mardia K V (1972) Statistics of Directional Data Academic Press

Errors or warnings detected by the method:

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 $\mathbf{vk}\ge 6.5$, the other gives an accuracy of $12$ digits and uses the Normal approximation when $\mathbf{vk}\ge 50.0$.

None.

Using the series expansion for small $\kappa$ the time taken by g01er increases linearly with $\kappa$; for larger $\kappa$, for which the asymptotic Normal approximation is used, the time taken is much less.

If angles outside the region $-\pi \le \theta <\pi$ are used care has to be taken in evaluating the probability of being in a region ${\theta }_1\le \theta \le {\theta }_2$ if the region contains an odd multiple of $\pi$, $\left(2n+1\right)\pi$. The value of $F\left({\theta }_2\text{;}\kappa \right)-F\left({\theta }_1\text{;}\kappa \right)$ 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 $\kappa$. The probabilities are computed and printed.