# ptmvnorm.marginal: One-dimensional marginal CDF function for a Truncated... In tmvtnorm: Truncated Multivariate Normal and Student t Distribution

## Description

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().

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 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 <= 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 = α^{-1} Φ_k(a, u, μ, Σ)

where u = (b_1,...,b_{i-1},x_i,b_{i+1},...,b_k)' is the upper integration bound and Φ_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(s)

Stefan Wilhelm <[email protected]>

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ## 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") 

tmvtnorm documentation built on May 29, 2017, 9:36 p.m.