bvevd                  package:evd                  R Documentation

_P_a_r_a_m_e_t_r_i_c _B_i_v_a_r_i_a_t_e _E_x_t_r_e_m_e _V_a_l_u_e _D_i_s_t_r_i_b_u_t_i_o_n_s

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

     Density function, distribution function and random generation for
     eight parametric bivariate extreme value models.

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

     dbvevd(x, dep, asy = c(1, 1), alpha, beta, model = "log",
         mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) 
     pbvevd(q, dep, asy = c(1, 1), alpha, beta, model = "log",
         mar1 = c(0, 1, 0), mar2 = mar1, lower.tail = TRUE) 
     rbvevd(n, dep, asy = c(1, 1), alpha, beta, model = "log",
         mar1 = c(0, 1, 0), mar2 = mar1) 

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

    x, q: A vector of length two or a matrix with two columns, in which
          case the density/distribution is evaluated across the rows.

       n: Number of observations.

     dep: Dependence parameter for the logistic, asymmetric logistic,
          Husler-Reiss, negative logistic and asymmetric negative
          logistic models.

     asy: A vector of length two, containing the two asymmetry
          parameters for the asymmetric logistic and asymmetric
          negative logistic models.

alpha, beta: Alpha and beta parameters for the bilogistic, negative
          bilogistic and Coles-Tawn models.

   model: The specified model; a character string. Must be either
          '"log"' (the default), '"alog"', '"hr"', '"neglog"',
          '"aneglog"', '"bilog"', '"negbilog"' or '"ct"' (or any unique
          partial match), for the logistic, asymmetric logistic,
          Husler-Reiss, negative logistic, asymmetric negative
          logistic, bilogistic, negative bilogistic and Coles-Tawn
          models respectively. If parameter arguments are given that do
          not correspond to the specified model those arguments are
          ignored, with a warning.

mar1, mar2: Vectors of length three containing marginal parameters, or
          matrices with three columns where each column represents a
          vector of values to be passed to the corresponding marginal
          parameter.

     log: Logical; if 'TRUE', the log density is returned.

lower.tail: Logical; if 'TRUE' (default), the distribution function is
          returned; the survivor function is returned otherwise.

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

     Define

                yi = yi(zi) = {1+si(zi-ai)/bi}^(-1/si)

     for 1+si(zi-ai)/bi > 0 and i = 1,2, where the marginal parameters
     are given by 'mari' = (ai,bi,si), bi > 0. If si = 0 then yi is
     defined by continuity.

     In each of the bivariate distributions functions G(z1,z2) given
     below, the univariate margins are generalized extreme value, so
     that G(zi) = exp(-yi) for i = 1,2. If 1+si(zi-ai)/bi <= 0 for some
     i = 1,2, the value zi is either greater than the upper end point
     (if si < 0), or less than the lower end point (if si > 0), of the
     ith univariate marginal distribution.

     'model = "log"' (Gumbel, 1960)

     The bivariate logistic distribution function with parameter 'dep'
     = r is

                G(z1,z2) = exp{-[y1^(1/r)+y2^(1/r)]^r}

     where 0 < r <= 1. This is a special case of the bivariate
     asymmetric logistic model. Complete dependence is obtained in the
     limit as r approaches zero. Independence is obtained when r = 1.

     'model = "alog"' (Tawn, 1988)

     The bivariate asymmetric logistic distribution function with
     parameters 'dep' = r and 'asy' = (t1,t2) is

   G(z1,z2) = exp{-(1-t1)y1-(1-t2)y2-[(t1y1)^(1/r)+(t2y2)^(1/r)]^r}

     where 0 < r <= 1 and 0 <= t1,t2 <= 1. When t1 = t2 = 1 the
     asymmetric logistic model is equivalent to the logistic model.
     Independence is obtained when either r = 1, t1 = 0 or t2 = 0.
     Complete dependence is obtained in the limit when t1 = t2 = 1 and
     r approaches zero. Different limits occur when t1 and t2 are fixed
     and r approaches zero.

     'model = "hr"' (Husler and Reiss, 1989)

     The Husler-Reiss distribution function with parameter 'dep' = r is

 G(z1,z2) = exp(-y1 Phi{r^{-1}+r[log(y1/y2)]/2} - y2 Phi{r^{-1}+r[log(y2/y1)]/2}

     where Phi() is the standard normal distribution function and r >
     0. Independence is obtained in the limit as r approaches zero.
     Complete dependence is obtained as r tends to infinity.

     'model = "neglog"' (Galambos, 1975)

     The bivariate negative logistic distribution function with
     parameter 'dep' = r is

           G(z1,z2) = exp{-y1-y2+[y1^(-r)+y2^(-r)]^(-1/r)}

     where r > 0. This is a special case of the bivariate asymmetric
     negative logistic model. Independence is obtained in the limit as
     r approaches zero. Complete dependence is obtained as r tends to
     infinity. The earliest reference to this model appears to be
     Galambos (1975, Section 4).

     'model = "aneglog"' (Joe, 1990)

     The bivariate asymmetric negative logistic distribution function
     with parameters parameters 'dep' = r and 'asy' = (t1,t2) is

       G(z1,z2) = exp{-y1-y2+[(t1y1)^(-r)+(t2y2)^(-r)]^(-1/r)}

     where r > 0 and 0 < t1,t2 <= 1. When t1 = t2 = 1 the asymmetric
     negative logistic model is equivalent to the negative logistic
     model. Independence is obtained in the limit as either r, t1 or t2
     approaches zero. Complete dependence is obtained in the limit when
     t1 = t2 = 1 and r tends to infinity. Different limits occur when
     t1 and t2 are fixed and r tends to infinity. The earliest
     reference to this model appears to be Joe (1990), who introduces a
     multivariate extreme value distribution which reduces to G(z1,z2)
     in the bivariate case.

     'model = "bilog"' (Smith, 1990)

     The bilogistic distribution function with parameters 'alpha' =
     alpha and 'beta' = beta is

         G(z1,z2) = exp{- y1 q^(1-alpha) - y2 (1-q)^(1-beta)}

     where q = q(y1,y2;alpha,beta) is the root of the equation

          (1-alpha) y1 (1-q)^beta - (1-beta) y2 q^alpha = 0,

     0 < alpha,beta < 1. When alpha = beta the bilogistic model is
     equivalent to the logistic model with dependence parameter 'dep' =
     alpha = beta. Complete dependence is obtained in the limit as
     alpha = beta approaches zero. Independence is obtained as alpha =
     beta approaches one, and when one of alpha,beta is fixed and the
     other approaches one. Different limits occur when one of
     alpha,beta is fixed and the other approaches zero. A bilogistic
     model is fitted in Smith (1990), where it appears to have been
     first introduced.

     'model = "negbilog"' (Coles and Tawn, 1994)

     The negative bilogistic distribution function with parameters
     'alpha' = alpha and 'beta' = beta is

    G(z1,z2) = exp{- y1 - y2 + y1 q^(1+alpha) + y2 (1-q)^(1+beta)}

     where q = q(y1,y2;alpha,beta) is the root of the equation

          (1+alpha) y1 q^alpha - (1+beta) y2 (1-q)^beta = 0,

     alpha > 0 and beta > 0. When alpha = beta the negative bilogistic
     model is equivalent to the negative logistic model with dependence
     parameter 'dep' = 1/alpha = 1/beta. Complete dependence is
     obtained in the limit as alpha = beta approaches zero.
     Independence is obtained as alpha = beta tends to infinity, and
     when one of alpha,beta is fixed and the other tends to infinity.
     Different limits occur when one of alpha,beta is fixed and the
     other approaches zero.

     'model = "ct"' (Coles and Tawn, 1991)

     The Coles-Tawn distribution function with parameters 'alpha' =
     alpha > 0 and 'beta' = beta > 0 is

 G(z1,z2) = exp{- y1 [1 - Be(q;alpha+1,beta)] - y2 Be(q;alpha,beta+1)}

     where q = alpha y2 / (alpha y2 + beta y1) and Be(q;alpha,beta) is
     the beta distribution function evaluated at q with 'shape1' =
     alpha and 'shape2' = beta. Complete dependence is obtained in the
     limit as alpha = beta tends to infinity. Independence is obtained
     as alpha = beta approaches zero, and when one of alpha,beta is
     fixed and the other approaches zero. Different limits occur when
     one of alpha,beta is fixed and the other tends to infinity.

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

     'dbvevd' gives the density function, 'pbvevd' gives the
     distribution function and 'rbvevd' generates random deviates, for
     one of eight parametric bivariate extreme value models.

_N_o_t_e:

     The logistic and asymmetric logistic models respectively are
     simulated using bivariate versions of Algorithms 1.1 and 1.2 in
     Stephenson(2003). All other models are simulated using a root
     finding algorithm to simulate from the conditional distributions.

     The simulation of the bilogistic and negative bilogistic models
     requires a root finding algorithm to evaluate q within the root
     finding algorithm used to simulate from the conditional
     distributions. The generation of bilogistic and negative
     bilogistic random deviates is therefore relatively slow (about 2.8
     seconds per 1000 random vectors on a 450MHz PIII, 512Mb RAM).

     The bilogistic and negative bilogistic models can be represented
     under a single model, using the integral of the maximum of two
     beta distributions (Joe, 1997).

     The Coles-Tawn model is called the Dirichelet model in Coles and
     Tawn (1991).

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

     Coles, S. G. and Tawn, J. A. (1991) Modelling extreme multivariate
     events. _J. Roy. Statist. Soc., B_, *53*, 377-392.

     Coles, S. G. and Tawn, J. A. (1994) Statistical methods for
     multivariate extremes: an application to structural design (with
     discussion). _Appl. Statist._, *43*, 1-48.

     Galambos, J. (1975) Order statistics of samples from multivariate
     distributions. _J. Amer. Statist. Assoc._, *70*, 674-680.

     Gumbel, E. J. (1960) Distributions des valeurs extremes en
     plusieurs dimensions. _Publ. Inst. Statist. Univ. Paris_, *9*,
     171-173.

     Husler, J. and Reiss, R.-D. (1989) Maxima of normal random
     vectors: between independence  and complete dependence. _Statist.
     Probab. Letters_, *7*, 283-286.

     Joe, H. (1990) Families of min-stable multivariate exponential and
     multivariate extreme value distributions. _Statist. Probab.
     Letters_, *9*, 75-81.

     Joe, H. (1997) _Multivariate Models and Dependence Concepts_,
     London: Chapman & Hall.

     Smith, R. L. (1990) Extreme value theory. In _Handbook of
     Applicable Mathematics_ (ed. W. Ledermann), vol. 7. Chichester:
     John Wiley, pp. 437-471.

     Stephenson, A. G. (2003) Simulating multivariate extreme value
     distributions of logistic type. _Extremes_, *6*(1), 49-60.

     Tawn, J. A. (1988) Bivariate extreme value theory: models and
     estimation. _Biometrika_, *75*, 397-415.

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

     'abvpar', 'rgev', 'rmvevd'

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

     pbvevd(matrix(rep(0:4,2), ncol=2), dep = 0.7, model = "log")
     pbvevd(c(2,2), dep = 0.7, asy = c(0.6,0.8), model = "alog")
     pbvevd(c(1,1), dep = 1.7, model = "hr")

     margins <- cbind(0, 1, seq(-0.5,0.5,0.1))
     rbvevd(11, dep = 1.7, model = "hr", mar1 = margins)
     rbvevd(10, dep = 1.2, model = "neglog", mar1 = c(10, 1, 1))
     rbvevd(10, alpha = 0.7, beta = 0.52, model = "bilog")

     dbvevd(c(0,0), dep = 1.2, asy = c(0.5,0.9), model = "aneglog")
     dbvevd(c(0,0), alpha = 0.75, beta = 0.5, model = "ct", log = TRUE)
     dbvevd(c(0,0), alpha = 0.7, beta = 1.52, model = "negbilog")

