ptmvnorm.marginal {tmvtnorm}R Documentation

One-dimensional marginal CDF function from a Truncated Multivariate Normal distribution

Description

This function computes the one-dimensional marginal probability function from a Truncated Multivariate Normal density function using integration in pmvnorm().

Usage

ptmvnorm.marginal(xn, 
    n = 1, 
    mean = rep(0, nrow(sigma)), 
    sigma = diag(length(mean)), 
    lower = rep(-Inf, length = length(mean)), 
    upper = rep(Inf, length = length(mean)))

Arguments

xn Vector of quantiles to calculate the marginal probability for.
n Index position (1..k) within the random vector xn to calculate the one-dimensional marginal probability for.
mean the mean vector of length k.
sigma the covariance matrix of dimension k. 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.
lower Vector of lower truncation points,\ default is rep(-Inf, length = length(mean)).
upper Vector of upper truncation points,\ default is rep( Inf, length = length(mean)).

Details

The one-dimensional marginal probability for index i is F_i(x_i) = P(X_i \le x_i)

F_i(x_i) = \int_{a_1}^{b_1} ... \int_{a_{i-1}}^{b_{i-1}} \int_{a_{i}}^{x_i} \int_{a_{i+1}}^{b_{i+1}} ... \int_{a_k}^{b_k} f(x) dx = \alpha^{-1} \Phi_k(a, u, \mu, \Sigma)

where u = (b_1,...,b_{i-1},x_i,b_{i+1},...,b_k)' is the upper integration bound and \Phi_k is the k-dimensional normal probability (i.e. function pmvnorm() in R package mvtnorm).

Value

Returns a vector of the same length as xn with probabilities.

Author(s)

Stefan Wilhelm <Stefan.Wilhelm@financial.com>

Examples

##
lower=c(-1,-1,-1)
upper=c(1,1,1)
mean=c(0,0,0)
sigma=matrix(c(1, 0.8, 0.2, 
              0.8,  1, 0.1,
              0.2, 0.1,  1), 3, 3)

X = rtmvnorm(n=1000, mean=c(0,0,0), sigma=sigma, lower=lower, upper=upper)

x = seq(-1, 1, by=0.001)
Fx = ptmvnorm.marginal(xn=x, n=1, mean=c(0,0,0), sigma=sigma, lower=lower, upper=upper) 

plot(ecdf(X[,1]))
lines(x, Fx, type="l", col="blue")

[Package tmvtnorm version 0.8-3 Index]