R: Inverse of A1

A1inv {circular} | R Documentation

## 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)

### Arguments

`x` | numeric value in the interval between 0 and 1.

### 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).

### Value

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

### Author(s)

Claudio Agostinelli

### 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)))

[Package circular version 0.4-93 Index]