rstable              package:CircStats              R Documentation

_R_a_n_d_o_m _G_e_n_e_r_a_t_o_r _f_o_r _S_t_a_b_l_e _F_a_m_i_l_y _o_f _D_i_s_t_r_i_b_u_t_i_o_n_s

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

     Returns random deviates from the stable family of probability
     distributions.

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

     rstable(n, scale = 1, index = stop("no index arg"), skewness = 0)

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

       n: sample size

   index: number from the interval (0, 2]. An index of 2 corresponds to
          the normal, 1 to the Cauchy.  Smaller values mean longer
          tails.

skewness: number giving the modified skewness (see Chambers et al.,
          1976).  Negative values correspond to skewness to the left
          (the median is smaller than the mean, if it exists), and
          positive values correspond to skewness to the right (the
          median is larger than the mean). The absolute value of
          skewness should not exceed 1.

   scale: the scale of the distribution.

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

     This function returns a random variate from the Levy skew stable
     distribution with `index'=$alpha$, `scale'=$c$ and
     `skewness'=$beta$.  The `skewness' parameter must lie in the range
     [-1,1] while the `index' parameter must lie in the range (0,2]. 
     The Levy skew stable probability distribution is defined by a
     fourier transform,

     p(x) = {1 over 2 pi} int_{-infty}^{+infty} dt exp(-it x - |c
     t|^alpha (1-i beta sign(t) tan(pialpha/2)))

     When $alpha = 1$ the term $\tan(pi alpha/2)$ is replaced by
     $-(2/pi)\log|t|$.   For $alpha = 2$ the distribution reduces to a
     Gaussian distribution with $sigma = \sqrt{2} scale$ and the
     skewness parameter has no effect.   For $alpha < 1$ the tails of
     the distribution become extremely wide.  The symmetric
     distribution corresponds to $beta =0$.

     The Levy alpha-stable distributions have the property that if $N$
     alpha-stable variates are drawn from the distribution $p(c, alpha,
     beta)$ then the sum $Y = X_1 + X_2 + ... + X_N$ will also be
     distributed as an alpha-stable variate, $p(N^{1/alpha} c, alpha,
     beta)$.

     There is no explicit solution for the form of $p(x)$ and there are
     no density, probability or quantile functions supplied for this
     distribution.

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

     random sample from the specified stable distribution.

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

     Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976). A Method
     for Simulating Stable Random Variables. Journal of the American
     Statistical Association 71, 340-344.

     Lo\"{g}ae\"{v}e, M. (1977). Probability Theory I. (fourth edition)
     Springer-Verlag, New York.

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

     rnorm, rcauchy.

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

     hist(rstable(200, 1.5, .5)) #fairly long tails, skewed right

