pmvt                 package:mvtnorm                 R Documentation

_M_u_l_t_i_v_a_r_i_a_t_e _t _D_i_s_t_r_i_b_u_t_i_o_n

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

     Computes the the distribution function of the multivariate t
     distribution  for arbitrary limits, degrees of freedom and
     correlation matrices  based on algorithms by Genz and Bretz.

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

     pmvt(lower=-Inf, upper=Inf, delta=rep(0, length(lower)),
          df=1, corr=NULL, sigma=NULL, algorithm = GenzBretz(), ...)

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

   lower: the vector of lower limits of length n.

   upper: the vector of upper limits of length n.

   delta: the vector of noncentrality parameters of length n.

      df: degree of freedom as integer.

    corr: the correlation matrix of dimension n.

   sigma: the covariance matrix of dimension n. Either 'corr' or
          'sigma' can be specified. If 'sigma' is given, the   problem
          is standardized. If neither 'corr' nor 'sigma' is given, the
          identity matrix is used  for 'sigma'. 

algorithm: an object of class 'GenzBretz' defining the hyper parameters
          of this algorithm.

     ...: 

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

     This program involves the computation of central and noncentral
     multivariate t-probabilities with arbitrary correlation matrices.
     It involves both the computation of singular and nonsingular
     probabilities. The methodology is described in Genz and Bretz
     (1999, 2002).

     For a given correlation matrix 'corr', for short A, say,  (which
     has to be  positive semi-definite) and  degrees of freedom nu the
     following  values are numerically evaluated


 I = 2^{1-nu/2} / Gamma(nu/2) int_0^infty s^{nu-1} exp(-s^2/2) Phi(s cdot lower/sqrt{nu} - delta, s cdot upper/sqrt{nu} - delta) , ds


     where 


  Phi(a,b) = (det(A)(2pi)^m)^{-1/2} int_a^b exp(-x^prime Ax/2) , dx


     is the multivariate normal distribution and $m$ is the number of
     rows of A.

     Note that both '-Inf' and '+Inf' may be specified in  the lower
     and upper integral limits in order to compute one-sided
     probabilities. Randomized quasi-Monte Carlo methods are used for
     the  computations.

     Univariate problems are passed to 'pt'.  If 'df = 0', normal
     probabilities are returned.

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

     The evaluated distribution function is returned with attributes 

   error: estimated absolute error and

     msg: status messages.

_S_o_u_r_c_e:

     <URL: http://www.sci.wsu.edu/math/faculty/genz/homepage>

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

     Genz, A. and Bretz, F. (1999), Numerical computation of
     multivariate t-probabilities with application to power calculation
     of multiple contrasts. _Journal of Statistical Computation and
     Simulation_, *63*, 361-378.

     Genz, A. and Bretz, F. (2002), Methods for the computation of
     multivariate t-probabilities. _Journal of Computational and
     Graphical Statistics_, *11*, 950-971.

     Edwards D. and Berry, Jack J. (1987), The efficiency of
     simulation-based  multiple comparisons. _Biometrics_, *43*,
     913-928.

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

     'qmvt'

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

     n <- 5
     lower <- -1
     upper <- 3
     df <- 4
     corr <- diag(5)
     corr[lower.tri(corr)] <- 0.5
     delta <- rep(0, 5)
     prob <- pmvt(lower=lower, upper=upper, delta=delta, df=df, corr=corr)
     print(prob)

     pmvt(lower=-Inf, upper=3, df = 3, sigma = 1) == pt(3, 3)

     # Example from R News paper (original by Edwards and Berry, 1987)

     n <- c(26, 24, 20, 33, 32)
     V <- diag(1/n)
     df <- 130
     C <- c(1,1,1,0,0,-1,0,0,1,0,0,-1,0,0,1,0,0,0,-1,-1,0,0,-1,0,0)
     C <- matrix(C, ncol=5)
     ### covariance matrix
     cv <- C %*% V %*% t(C)
     ### correlation matrix
     dv <- t(1/sqrt(diag(cv)))
     cr <- cv * (t(dv) %*% dv)
     delta <- rep(0,5)

     myfct <- function(q, alpha) {
       lower <- rep(-q, ncol(cv))
       upper <- rep(q, ncol(cv))
       pmvt(lower=lower, upper=upper, delta=delta, df=df, 
            corr=cr, abseps=0.0001) - alpha
     }

     round(uniroot(myfct, lower=1, upper=5, alpha=0.95)$root, 3)

     # compare pmvt and pmvnorm for large df:

     a <- pmvnorm(lower=-Inf, upper=1, mean=rep(0, 5), corr=diag(5))
     b <- pmvt(lower=-Inf, upper=1, delta=rep(0, 5), df=rep(300,5),
               corr=diag(5))
     a
     b

     stopifnot(round(a, 2) == round(b, 2))

     # correlation and covariance matrix

     a <- pmvt(lower=-Inf, upper=2, delta=rep(0,5), df=3,
               sigma = diag(5)*2)
     b <- pmvt(lower=-Inf, upper=2/sqrt(2), delta=rep(0,5),
               df=3, corr=diag(5))  
     attributes(a) <- NULL
     attributes(b) <- NULL
     a
     b
     stopifnot(all.equal(round(a,3) , round(b, 3)))

     a <- pmvt(0, 1,df=10)
     attributes(a) <- NULL
     b <- pt(1, df=10) - pt(0, df=10)
     stopifnot(all.equal(round(a,10) , round(b, 10)))

