EllipticalDependency        package:fCopulae        R Documentation

_B_i_v_a_r_i_a_t_e _E_l_l_i_p_t_i_c_a_l _C_o_p_u_l_a_e

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

     A collection and description of functions to investigate 
     bivariate elliptical copulae.  

     Elliptical Copulae Functions:

       'ellipticalTau'        Computes Kendall's tau for elliptical copulae,
       'ellipticalRho'        computes Spearman's rho for elliptical copulae,
       'ellipticalTailCoeff'  computes tail dependence for elliptical copulae,
       'ellipticalTailPlot'   plots tail dependence for elliptical copulae.

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

        
     ellipticalTau(rho)
     ellipticalRho(rho, param = NULL, type = ellipticalList(), subdivisions = 500)
         
     ellipticalTailCoeff(rho, param = NULL, type = c("norm", "cauchy", "t"))
     ellipticalTailPlot(param = NULL, type = c("norm", "cauchy", "t"), 
         tail = c("Lower", "Upper"))

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

     rho: [*ellipticalCopula] - 
           is the numeric value setting the correlation strength,
          ranging between minus one and one. 

   param: [*ellipticalCopula][gfunc] - 
           additional distributional parameters: for the Sudent-t
          distribution this is "nu", for the Kotz distribution this is
          "r", and for the Exponential Power distribution these are "r"
          and "s". If the argument 'param=NULL' then default values are
          taken. These are for the Student-t 'param=c(nu=4))', for the
          Kotz distribution 'param=c(r=1))', and for the exponential
          power distribution 'param=c(r=1,s=1)'. Note, that the Kotz
          and exponential power copulae are independent of 'r', and
          that 'r' only enters the generator, the density, the
          probability and the quantile  functions. 

subdivisions: [ellipticalRho] - 
           an integer value with the number of subdivisons in each
          direction on the two dimensional unit square to compute the
          mean value of Spearman's Rho. By default 500 subdivisions are
          used. 

    tail: [ellipticalTailPlot] - 
           a character string, either '"Upper"' or '"Lower"' denoting 
          which of the two tails should be displayed. By default the
          upper tail dependence will be considered. 

    type: [*ellipticalCopula][gfunc] - 
           the type of the elliptical copula. A character string
          selected  from: "norm", "cauchy", "t", "logistic", "laplace",
          "kotz",  or "epower". [*ellipticalSlider] - 
           a character string which indicates what kind of plot should
          be  displayed, either a perspective plot if 'type="persp"',
          the default value, or a contour plot if 'type="contour"'. 

     ...: [ellipticalCopulaFit] - 
           arguments passed to the optimization function 'nlminb'. 

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

     *Copula Functions:* 

     The functions '[rpd]ellipticalCopula' return a numeric vector of
     random variates, probabilities, or densities for the specified 
     copula computed at grid coordinates 'u'|'v'. 

     The functions '[rpd]ellipticalSlider' display an interactive graph
     of an perspective copula plot either for random variates,
     probabilities or densities. Alternatively, an image underlayed
     contour plot can be shown. 

     *Copula Dependence Measures:* 

     The functions 'ellipticalTau' and 'ellipticalRho' return a
     numericc value for Kendall's Tau and Spearman's Rho. 

     *Copula Tail Coefficient:* 

     The function 'ellipticalTailCoeff' returns the coefficient of tail
     dependence for a specified copula. The function 
     'ellipticalTailPlot' displays a whole plot for the upper or
     alternatively for the lower tail dependence as a function of  'u'
     for a set of nine 'rho' values. 

     *Copula Generator Function:* 

     The function 'gfunc' computes the generator function for the
     specified copula, by default the normal copula. If the argument
     'x' is missing, then the normalization constand lambda will be
     returned, otherwise if 'x' is specified the values for the
     function _g(x)_ will be freturned. The selected type of copula is
     added to the output as an attribute named '"control"'.  The
     function 'gfuncSlider' allows to display interactively  the
     generator function, the marginal density, the marginal 
     probability, and the contours of the the bivariate density. 

     *Copula Simulation and Parameter Fitting:* 

     The function 'ellipticalCopulaSim' returns a numeric two-column
     matrix with randomly generated variates for the specified copula. 

     The function 'ellipticalCopulaFit' returns a fit to empirical data
     for the specified copula. The returned object is a list with
     elements from the function 'nlminb'.

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

     Diethelm Wuertz for the Rmetrics R-port.

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

     ## Not run: 
     ## [rp]ellipticalCopula -
        # Default Normal Copula:
        rellipticalCopula(10)
        pellipticalCopula(10)

     ## [rp]ellipticalCopula -   
        # Student-t Copula Probability and Density:
        u = grid2d(x = (0:25)/25)
        pellipticalCopula(u, rho = 0.75, param = 4, 
          type = "t", output = "list")
        d = dellipticalCopula(u, rho = 0.75, param = 4, 
          type = "t", output = "list")   
        persp(d, theta = -40, phi = 30, col = "steelblue")
        
     ## ellipticalTau -
     ## ellipticalRho -
        # Dependence Measures:
        ellipticalTau(rho = -0.5)
        ellipticalRho(rho = 0.75, type = "logistic", subdivisions = 100)
        
     ## ellipticalTailCoeff -
        # Student-t Tail Coefficient:
        ellipticalTailCoeff(rho = 0.25, param = 3, type = "t")

     ## gfunc -
        # Generator Function:
        plot(gfunc(x = 0:10), main = "Generator Function")
        
     ## ellipticalCopulaSim -
     ## ellipticalCopulaSim -
        # Simualtion and Parameter Fitting:
        rv = ellipticalCopulaSim(n = 100, rho = 0.75)
        ellipticalCopulaFit(rv)
     ## End(Not run)  

