# 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()`. 1.1 ```r 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 $F_i(x_i) = P(X_i <= x_i)$ $$ F_i(x_i) = \int_{a_1}^{b_1} \ldots \int_{a_{i-1}}^{b_{i-1}} \int_{a_{i}}^{x_i} \int_{a_{i+1}}^{b_{i+1}} \ldots \int_{a_k}^{b_k} f(x) dx= \alpha^{-1} \Phi_k(a, u, \mu, \Sigma) $$ where $u = (b_1,...,b_{i-1},x_i,b_{i+1},...,b_k)'$ is the upper integration bound and $\Phi_k$ is the k-dimensional normal probability (i.e. functions `pmvnorm()` and `pmvt()` in R package `mvtnorm`). ## Returns Returns a vector of the same length as xn with probabilities. ## Author(s) Stefan Wilhelm ## Examples ```r ## 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") ```

ptmvnorm.marginal function