sstd                 package:fGarch                 R Documentation

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

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

     Functions to compute density, distribution function,  quantile
     function and to generate random variates  for the standardized
     skew Student-t distribution. In addition maximum likelihood
     estimators are available to  fit the parameters of the
     distribution. 

     The functions are:

       '[dpqr]std'   Symmetric Student-t Distribution,
       '[dpqr]sstd'  Skew Student-t Distribution,
       'stdFit'      MLE parameter fit for a Sudent-t distribution,
       'sstdFit'     MLE parameter fit for a skew Sudent-t distribution,
       'sstdSlider'  Displays interactively skew GED 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)

     stdFit(x, ...)
     sstdFit(x, ...)

     sstdSlider(type = c("dist", "rand"))

_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: the number of observations. 

       p: a numeric vector of probabilities. 

    type: a character string denoting which interactive plot should be
          displayed. Either a distribution plot 'type="dist"', the
          default value, or a random variates plot, 'type="rand"'. 

    x, q: a numeric vector of quantiles. 

     ...: parameters parsed to the optimization function 'nlm'.  

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

     *Parameter Estimation:* 

      The function 'nlminb' is used to minimize the "negative"  maximum
     log-likelihood function. 'nlminb' carries out a  minimization
     using a Newton-type algorithm.

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

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

     '[s]stdFit' returns a list with the following components:  


     par: The best set of parameters found.  

objective: The value of objective corresponding to 'par'. 

convergence: An integer code. 0 indicates successful convergence. 

 message: A character string giving any additional information 
          returned by the optimizer, or NULL. For details, see  PORT
          documentation. 

iterations: Number of iterations performed. 

evaluations: Number of objective function and gradient function 
          evaluations. 

_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.

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

     ## sstd -
        par(mfrow = c(2, 2))
        set.seed(1953)
        r = rsstd(n = 1000)
        plot(r, type = "l", main = "sstd", col = "steelblue")
        
        # Plot empirical density and compare with true density:
        hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
        box()
        x = seq(min(r), max(r), length = 201)
        lines(x, dsstd(x), lwd = 2)
        
        # Plot df and compare with true df:
        plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
          ylab = "Probability")
        lines(x, psstd(x), lwd = 2)
        
        # Compute quantiles:
        round(qsstd(psstd(q = seq(-1, 5, by = 1))), digits = 6)
            
     ## sstdFit -
        sstdFit(r, print.level = 2)

