

stable(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' gener-
     ates random deviates with the prescribed stable distri-
     bution.

     `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 posi-
     tive respectively.

     `tail' is a tail parameter (often named the character-
     istic exponent): it can take any value in (0,2) (with
     `tail=1' and `tail=2' yielding the Cauchy and the nor-
     mal distributions respectively when symmetry holds).

     If `loc' or `disp' or `skew' or `tail' are not speci-
     fied they assume the default values of 0, 1/sqrt(2), 0
     and 2 respectively. This corresponds to a normal vari-
     ate 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 evalua-
     tion 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 eval-
     uate the integral with an error bounded by `eps'.  The
     alternative method is Simpson's integration (`integra-
     tion'="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 argu-
     ments - 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 computa-
          tion.

_A_u_t_h_o_r_(_s_)_:

     Philippe Lambert (University of Liege, Belgium, plam-
     bert@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 distribu-
     tion.

_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))

