mvevd                  package:evd                  R Documentation

_P_a_r_a_m_e_t_r_i_c _M_u_l_t_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
     the multivariate logistic and multivariate asymmetric logistic
     models.

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

     pmvevd(q, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0),
         lower.tail = TRUE)
     rmvevd(n, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0))
     dmvevd(x, dep, asy, model = c("log", "alog"), d = 2, mar = c(0,1,0),
         log = FALSE) 

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

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

       n: Number of observations.

     dep: The dependence parameter(s). For the logistic model, should
          be a single value. For the asymmetric logistic model, should
          be a vector of length 2^d-d-1, or a single value, in which
          case the value is used for each of the 2^d-d-1 parameters
          (see *Details*).

     asy: The asymmetry parameters for the asymmetric logistic model.
          Should be a list with 2^d-1 vector elements containing the
          asymmetry parameters for each separate component (see
          *Details*).

   model: The specified model; a character string. Must be either
          '"log"' (the default) or '"alog"' (or any unique partial
          match), for the logistic and asymmetric logistic models
          respectively.

       d: The dimension.

     mar: A vector of length three containing marginal parameters for
          every univariate margin, or a matrix with three columns where
          each column represents a vector of values to be passed to the
          corresponding marginal parameter. It can also be a list with
          'd' elements, such that each element is either a vector of
          length three or a matrix with three columns, in which case
          the ith element represents the marginal parameters on the ith
          margin.

     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,...,d, where the marginal
     parameters are given by (ai,bi,si), bi > 0. If si = 0 then yi is
     defined by continuity. Let z = (z1,z2,...,zd). In each of the
     multivariate distributions functions G(z) given below, the
     univariate margins are generalized extreme value, so that G(zi) =
     exp(-yi) for i = 1,...,d. If 1+si(zi-ai)/bi <= 0 for some i =
     1,...,d, 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 'd' dimensional multivariate logistic distribution function
     with parameter 'dep' = r is

                G(z) = exp{-[sum_{i=1}^d yi^(1/r)]^r}

     where 0 < r <= 1. This is a special case of the multivariate
     asymmetric logistic model.

     'model = "alog"' (Tawn, 1990)

     Let B be the set of all non-empty subsets of {1,...,d}, let B1={b
     in B:|b|=1}, where |b| denotes the number of elements in the set
     b, and let B(i)={b in B:i in b}. The 'd' dimensional multivariate
     asymmetric logistic distribution function is

   G(z) = exp{-sum{b in B} [sum{i in b}(t{i,b}yi)^(1/r{b})]^r{b}},

     where the dependence parameters r{b} in (0,1] for all b in BB1,
     and the asymmetry parameters t{i,b} in [0,1] for all b in B and i
     in b. The constraints sum{b in B(i)} t{i,b}=1 for i = 1,...,d
     ensure that the marginal distributions are generalized extreme
     value. Further constraints arise from the possible redundancy of
     asymmetry parameters in the expansion of the distribution form.
     Let b_{-i0} = {i in b:i is not i_0}. If r{b} = 1 for some b in BB1
     then t{i,b} = 0 for all i in b. Furthermore, if for some b in BB1,
     t{i,b} = 0 for all i in b_{-i0}, then t{i0,b} = 0.

     'dep' should be a vector of length 2^d-d-1 which contains {r{b}:b
     in BB1}, with the order defined by the natural set ordering on the
     index. For example, for the trivariate model, 'dep' =
     (r{12},r{13},r{23},r{123}). 'asy' should be a list with 2^d-1
     elements. Each element is a vector which corresponds to a set b in
     B, containing t{i,b} for every integer i in b. The elements should
     be given using the natural set ordering on the b in B, so that the
     first d elements are vectors of length one corresponding to the
     sets {1},...,{d}, and the last element is a a vector of length d,
     corresponding to the set {1,...,d}. 'asy' must be constructed to
     ensure that all constraints are satisfied or an error will occur.

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

     'pmvevd' gives the distribution function, 'dmvevd' gives the
     density function and 'rmvevd' generates random deviates, for the
     multivariate logistic or multivariate asymmetric logistic model.

_N_o_t_e:

     Multivariate extensions of other bivariate models are more
     complex. A multivariate extension of the Husler-Reiss model
     exists, involving a multidimensional integral and one parameter
     for each bivariate margin. Multivariate extensions for the
     negative logistic model can be derived but are considerably more
     complex and appear to be less flexible. The ``multivariate
     negative logistic model'' often presented in the literature (e.g.
     Kotz _et al_, 2000) is not a valid distribution function and
     should not be used.

     The logistic and asymmetric logistic models respectively are
     simulated using Algorithms 2.1 and 2.2 in Stephenson(2003b).

     The density function of the logistic model is evaluated using the
     representation of Shi(1995). The density function of the
     asymmetric logistic model is evaluated using the representation
     given in Stephenson(2003a).

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

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

     Kotz, S. and Balakrishnan, N. and Johnson, N. L. (2000)
     _Continuous Multivariate Distributions_, vol. 1. New York: John
     Wiley & Sons, 2nd edn.

     Shi, D. (1995) Fisher information for a multivariate extreme value
     distribution. _Biometrika_, *82*(3), 644-649.

     Stephenson, A. G. (2003a) _Extreme Value Distributions and their
     Application_. Ph.D. Thesis, Lancaster University, Lancaster, UK.

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

     Tawn, J. A. (1990) Modelling multivariate extreme value
     distributions. _Biometrika_, *77*, 245-253.

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

     'rbvevd', 'rgev'

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

     pmvevd(matrix(rep(0:4,5), ncol=5), dep = .7, model = "log", d = 5)
     pmvevd(rep(4,5), dep = .7, model = "log", d = 5)
     rmvevd(10, dep = .7, model = "log", d = 5)
     dmvevd(rep(-1,20), dep = .7, model = "log", d = 20, log = TRUE)

     asy <- list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2))
     pmvevd(rep(2,3), dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3)
     asy <- list(.4, .0, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.4,.2))
     rmvevd(10, dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3)
     dmvevd(rep(0,3), dep = c(.6,.5,.8,.3), asy = asy, model = "alog", d = 3)

     asy <- list(0, 0, 0, 0, c(0,0), c(0,0), c(0,0), c(0,0), c(0,0), c(0,0),
       c(.2,.1,.2), c(.1,.1,.2), c(.3,.4,.1), c(.2,.2,.2), c(.4,.6,.2,.5))
     rmvevd(10, dep = .7, asy = asy, model = "alog", d = 4)
     rmvevd(10, dep = c(rep(1,6), rep(.7,5)), asy = asy, model = "alog", d = 4)

