circular (version 0.5-1)

A1inv: Inverse of A1

Description

Inverse function of the ratio of the first and zeroth order Bessel functions of the first kind. This function is used to compute the maximum likelihood estimate of the concentration parameter of a von Mises distribution.

Usage

A1inv(x)

Value

Returns the value k, such that A1inv(x) = k, i.e. A1(k) = x.

Arguments

x

numeric value in the interval between 0 and 1.

Author

Claudio Agostinelli

Details

A1inv(0) = 0 and A1inv(1) = Inf. This function is useful in estimating the concentration parameter of data from a von Mises distribution. Our function use the results in Best and Fisher (1981). Tables use tabulated values by Gumbel, Greenwood and Durand (1953).

References

BEST, D.J. and FISHER, N.I. 1981. The bias of the maximum likelihood estimators for the von Mises-Fisher concentration parameters. Communications in Statistics, 10, 493-502.

GUMBEL, E.J., GREENWOOD, J.A. AND DURAND, D. 1953. The circular normal distribution: theory and tables. J. Amer. Statis. Assoc., 48, 131-152.

See Also

mle.vonmises, A1, besselI.

Examples

#Generate data from a von Mises distribution
data <- rvonmises(n=50, mu=circular(pi), kappa=4)
#Estimate the concentration parameter
s <- sum(sin(data))
c <- sum(cos(data))
mean.dir <- atan2(s, c)
kappa <- A1inv(mean(cos(data - mean.dir)))