SkewStudentDistribution       package:fSeries       R Documentation

_S_y_m_m_e_t_r_i_c _a_n_d _S_k_e_w _S_t_u_d_e_n_t-_t _D_i_s_t_r_i_b_u_t_i_o_n

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

     A collection and description of functions to compute  density,
     distribution function, quantile function and  to generate random
     variates for the symmetric and skew  Sudent-t distribution with
     unit variance.  

     The functions are:

       '[dpqr]std'   Symmetric Student-t Distribution,
       '[dpqr]sstd'  Skew Student-t Distribution.

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

     dstd(x, mean = 0, sd = 1, nu = 5)
     pstd(q, mean = 0, sd = 1, nu = 5)
     qstd(p, mean = 0, sd = 1, nu = 5)
     rstd(n, mean = 0, sd = 1, nu = 5)

     dsstd(x, mean = 0, sd = 1, nu = 5, xi = 1.5)
     psstd(q, mean = 0, sd = 1, nu = 5, xi = 1.5)
     qsstd(p, mean = 0, sd = 1, nu = 5, xi = 1.5)
     rsstd(n, mean = 0, sd = 1, nu = 5, xi = 1.5)

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

mean, sd, nu, xi: location parameter 'mean', scale parameter 'sd',
          shape parameter 'nu', skewness parameter 'xi'. 

       n: number of observations. 

       p: a numeric vector of probabilities. 

    x, q: a numeric vector of quantiles. 

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

     *Symmetric Student-t Distibution:* 

      The functions for the symmetric Student-t distribution are 
     rescaled in such a way that they have unit variance in  contrast
     to the Student-t family 'dt', 'pt',  'qt' and 'rt' which are part
     of R's base package. 

     *Skew Student-t Distribution:* 

      The skew Student-t distribution functions are defined as
     described by Fernandez and Steel (2000). Note that the function
     have unit variance.

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

     All values are numeric vectors: 'd*' returns the density, 'p*'
     returns the distribution function, 'q*' returns the quantile
     function, and 'r*' generates random deviates.

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

     Diethelm Wuertz for the Rmetrics R-port.

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

     Fernandez C., Steel M.F.J. (2000);  _On Bayesian Modelling of Fat
     Tails and Skewness_, Preprint, 31 pages.

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

     'snormDistribution', 'sgedDistribution'.

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

     ## sstd -
        xmpSeries("\nStart: Skew Student-t Distribuion:  > ")
        par(mfrow = c(2, 2), cex = 0.75)
        set.seed(1953)
        r = rsstd(n = 1000, nu =4, xi = 1.5)
        # Print Variance:
        var(r)
        plot(r, type = "l", main = "sstd: xi = 1.5")
        # Plot empirical density and compare with true density:
        hist(r, n = 30, xlim = c(-5, 5), probability = TRUE, 
          border = "white", col = "steelblue4")
        x = seq(-5, 5, 0.1)
        lines(x, dsnorm(x = x, xi = 1.5))
        # Plot df and compare with true df:
        plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue4")
        lines(x, psstd(x, xi = 1.5))
        # Compute quantiles:
        qsstd(psstd(q = -5:5, xi = 1.5), xi = 1.5) 

