One-dimensional marginal CDF function for a Truncated Multivariate Normal and Student t distribution

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

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)))
ptmvt.marginal(xn, 
    n = 1, 
    mean = rep(0, nrow(sigma)), 
    sigma = diag(length(mean)), 
    df = 1, 
    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.

df

degrees of freedom parameter

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 \(F_i(x_i) = P(X_i \le x_i)\) $$F_i(x_i) = \int_{a_1}^{b_1} \ldots \int_{a_{i-1}}^{b_{i-1}} \int_{a_{i}}^{x_i} \int_{a_{i+1}}^{b_{i+1}} \ldots \int_{a_k}^{b_k} f(x) dx = \alpha^{-1} \Phi_k(a, u, \mu, \Sigma)$$ where \(u = (b_1,\ldots,b_{i-1},x_i,b_{i+1},\ldots,b_k)'\) is the upper integration bound and \(\Phi_k\) is the k-dimensional normal probability (i.e. functions pmvnorm() and pmvt() in R package mvtnorm).

Value

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

Author

Stefan Wilhelm <Stefan.Wilhelm@financial.com>

Examples

## Example 1: Truncated multi-normal
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=0.01)
Fx <- ptmvnorm.marginal(xn=x, n=1, mean=c(0,0,0), sigma=sigma, lower=lower, upper=upper) 

plot(ecdf(X[,1]), main="marginal CDF for truncated multi-normal")
lines(x, Fx, type="l", col="blue")


## Example 2: Truncated multi-t
X <- rtmvt(n=1000, mean=c(0,0,0), sigma=sigma, df=2, lower=lower, upper=upper)

x <- seq(-1, 1, by=0.01)
Fx <- ptmvt.marginal(xn=x, n=1, mean=c(0,0,0), sigma=sigma, lower=lower, upper=upper) 

plot(ecdf(X[,1]), main="marginal CDF for truncated multi-t")
lines(x, Fx, type="l", col="blue")