stable                package:stable                R Documentation

_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', 'disp', '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 is 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 for this. The default is
     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 - '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(y, loc=0, disp=1/sqrt(2), skew=0, tail=2,
                     npt=501, up=10, eps=1.0e-6, integration="Romberg")
     pstable(y, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-6)
     qstable(q, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-6)
     hstable(y, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-6)
     rstable(n=1, loc=0, disp=1/sqrt(2), skew=0, tail=2, eps=1.0e-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 (Catholic University of Louvain, Belgium,
     phlambert@stat.ucl.ac.be) and Jim Lindsey.

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

     Lambert, P. and Lindsey, J.K. (1999) Analysing financial returns
     using regression models based on non-symmetric stable
     distributions. Applied Statistics, 48, 409-424.

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

     'stablereg' to fit generalized nonlinear regression 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="",main="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="",main="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="",main="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="",main="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))

