pmvt {tlrmvnmvt}

Quasi-Monte Carlo method for Student-$t$ probabilities

Description

Compute multivariate Student-$t$ probabilities with the dense-matrix based Quasi-Monte Carlo method and the tile-low-rank-matrix based Quasi-Monte Carlo method.

Usage

pmvt(lower = -Inf, upper = Inf, delta = 0, df = 1, sigma = NULL, 
     uselog2 = FALSE, algorithm = GenzBretz(), 
     type = "Kshirsagar", ...)

Arguments

lower

lower integration limits, a numeric vector of length n

upper

upper integration limits, a numeric vector of length n

delta

the vector of noncentrality parameters of length n, for type = "shifted" delta specifies the mode

df

a positive numeric value denoting the degrees of freedom

sigma

the covariance matrix of dimension n

uselog2

whether return the result as the logarithm to the base 2

algorithm

an object of class GenzBretz or TLRQMC defining the hyper parameters of this algorithm

type

type of the noncentral multivariate $t$ distribution to be computed. 'type' = "Kshirsagar" corresponds to formula (1.4) in Genz and Bretz (2009). 'type' = "shifted" corresponds to the formula right before formula (1.4) in Genz and Bretz (2009)

...

additional parameters used to construct 'sigma' when it is not given:

• geom a matrix of dimension n-by-2, specifying n spatial locations in the 2D unit square

• kernelType the name of the covariance kernel, a string. Currently, only the Matern covariance function, e.g., "matern", is supported. Not case-sensitive. It should be given when 'sigma' is not given

• para the parameter for the covariance kernel, a numeric vector. When 'type' is "matern", the length of 'para' should be 4, representing the scale, range, smoothness, and nugget parameters of the covariance function. It should be given when 'sigma' is not given

Details

When 'algorithm' is of the class 'GenzBretz', the Quasi-Monte Carlo sampling described in Genz, A. (1992) is used. When 'algorithm' is of the class 'TLRQMC', the Quasi-Monte Carlo sampling with the tile-low-rank representation of the covariance matrix, described in Cao et al. (2020), is used. When 'sigma', is given, 'geom', 'kernelType', and 'para' are not used. Otherwise, a covariance matrix is created with the information from 'geom', 'kernelType', and 'para'.

Value

When 'uselog2' is set FALSE, the function returns the estimated probability with one attribute of the estimation error. When 'uselog2' is set TRUE, the function only returns the estimated log-probability to the base 2. This is useful when the estimated probability is smaller than the machine precision.

Author(s)

Jian Cao, Marc Genton, David Keyes, George Turkiyyah

References

Genz, A. (1992), "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics, 1, 141-149. Genz, A. and Bretz, F. (2009), Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg. Cao, J., Genton, M. G., Keyes, D. E., & Turkiyyah, G. M. (2022), "tlrmvnmvt: Computing High-Dimensional Multivariate Normal and Student-t Probabilities with Low-Rank Methods in R," Journal of Statistical Software, 101.4, 1-25.

Examples

  n = 225
  set.seed(0)
  a = rep(-10, n)
  b = rnorm(n, 3, 2)
  m = 15
  epsl = 1e-4
  vec1 = 1 : m
  vec2 = rep(1, m)
  geom = cbind(kronecker(vec1, vec2), kronecker(vec2, vec1))
  geom = geom / m
  beta = 0.3
  idx = zorder(geom)
  geom = geom[idx, ]
  a = a[idx]
  b = b[idx]
  distM = as.matrix(dist(geom))
  covM = exp(-distM / beta)
  df = 10
  pmvt(lower = a, upper = b, delta = 2, df = df,
       sigma = covM, uselog2 = FALSE, algorithm = GenzBretz(N = 521),
       type = "Kshirsagar")
  pmvt(lower = a, upper = b, delta = 2, df = df,
       uselog2 = TRUE, type = "shifted", geom = geom, 
       kernelType = "matern", para = c(1.0, 0.3, 0.5, 0.0))
  pmvt(lower = a, upper = b, delta = 2, df = df,
       sigma = covM, uselog2 = FALSE, 
       algorithm = TLRQMC(N = 521, m = m, epsl = epsl), type = "Kshirsagar")
  pmvt(lower = a, upper = b, delta = 2, df = df,
       uselog2 = TRUE, type = "shifted", geom = geom,
       algorithm = TLRQMC(N = 521, m = m, epsl = epsl),
       kernelType = "matern", para = c(1.0, 0.3, 0.5, 0.0))