stable                package:stable                R Documentation

_T_h_e _S_t_a_b_l_e _D_i_s_t_r_i_b_u_t_i_o_n

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

     These functions provide information about the stable distribution
     with the location, the dispersion, the skewness and the tail
     thickness respectively modelled by the parameters `loc', `disp',
     `skew' and  `tail'.

     `dstable', `pstable', `qstable' and `hstable' compute the density,
     the distribution, the quantile and the hazard functions of a
     stable variate. `rstable' generates random deviates with  the
     prescribed stable distribution.

     `loc' is a location parameter in the same way as the mean in the
     normal distribution: it can take any real value. 

     `disp' is a dispersion parameter in the same way as the standard 
     deviation in the normal distribution: it can take any positive
     value. 

     `skew' is a skewness parameter: it can take any value in (-1,1).
     The distribution is right-skewed, symmetric and left-skewed when
     `skew' is negative, null or positive respectively. 

     `tail' is a tail parameter (often named the characteristic
     exponent): it can take any value in (0,2) (with `tail=1' and
     `tail=2' yielding the Cauchy and the normal distributions
     respectively when symmetry holds). 

     If `loc' or `disp' or `skew' or `tail' are not specified they
     assume the default values of 0, 1/sqrt(2), 0 and 2 respectively.
     This corresponds to a normal  variate with mean=0 and variance=1=2
     disp^2.

     The stable characteristic function is given by

  phi(t) = i loc t - disp |t|^tail [1+i skew sign(t) omega(t,tail)]

     where

                    omega(t,tail) = (2/pi) log|t|

     when `tail=1', and

                  omega(t,tail) = tan(pi alpha / 2)

     otherwise.

     The characteristic function can be inverted using Fourier's
     transform to obtain the corresponding stable density. This
     inversion requires the numerical evaluation of an integral from 0
     to infinity. Two algorithms are proposed to do this. The default
     is the Romberg's method  (`integration'="Romberg") which is used
     to evaluate the integral with an error bounded by `eps'. The
     alternative method is Simpson's integration 
     (`integration'="Simpson"): it approximates the integral from 0 to
     infinity by an integral  from 0 to `up' with `npt' points
     subdividing (O, up).  These three extra arguments - namely
     `integration', `up' and `npt' - are only available when using
     `dstable'. The other functions are all based on Romberg's
     algorithm.

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

     dstable(x, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1e-6)
     pstable(q, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1e-6)
     qstable(p, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1e-6)
     hstable(x, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1e-6)
     rstable(n, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1e-6)

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

     y,q: vector of quantiles.

       p: vector of probabilites.

       n: number of observations.

     loc: vector of (real) location parameters.

    disp: vector of (positive) dispersion parameters.

    skew: vector of skewness parameters (in [-1,1]).

    tail: vector of parameters (in [0,2]) related to the tail
          thickness.

     eps: scalar giving the required precision in computation.

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

     Philippe Lambert (University of Liege, Belgium,
     plambert@ulg.ac.be) and Jim Lindsey.

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

     `stableglm' to fit generalized linear models for the stable 
     distribution parameters. 
     `stable.mode' to compute the mode of a stable distribution.

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

     par(mfrow=c(2,2))
     x <- seq(-5,5,by=0.1)

     # Influence of loc (location)
     plot(x,dstable(x,loc=-2,disp=1/sqrt(2),skew=-0.8,tail=1.5),
       type="l",ylab="",title("Varying LOCation"))
     lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=-0.8,tail=1.5))
     lines(x,dstable(x,loc=2,disp=1/sqrt(2),skew=-0.8,tail=1.5))

     # Influence of disp (dispersion)
     plot(x,dstable(x,loc=0,disp=0.5,skew=0,tail=1.5),
       type="l",ylab="",title("Varying DISPersion"))
     lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=1.5))
     lines(x,dstable(x,loc=0,disp=0.9,skew=0,tail=1.5))

     # Influence of skew (skewness)
     plot(x,dstable(x,loc=0,disp=1/sqrt(2),skew=-0.8,tail=1.5),
       type="l",ylab="",title("Varying SKEWness"))
     lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=1.5))
     lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0.8,tail=1.5))

     # Influence of tail (tail)
     plot(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=0.8),
       type="l",ylab="",title("Varying TAIL thickness"))
     lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=1.5))
     lines(x,dstable(x,loc=0,disp=1/sqrt(2),skew=0,tail=2))

