A1inv                package:circular                R Documentation

_I_n_v_e_r_s_e _o_f _A_1

_D_e_s_c_r_i_p_t_i_o_n:

     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.

_U_s_a_g_e:

     A1inv(x)

_A_r_g_u_m_e_n_t_s:

       x: numeric value in the interval between 0 and 1.

_D_e_t_a_i_l_s:

     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).

_V_a_l_u_e:

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

_A_u_t_h_o_r(_s):

     Claudio Agostinelli

_R_e_f_e_r_e_n_c_e_s:

     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.

_S_e_e _A_l_s_o:

     'mle.vonmises', 'A1', 'besselI'.

_E_x_a_m_p_l_e_s:

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

