# dtmvnorm.marginal2: Bivariate marginal density functions from a Truncated... In tmvtnorm: Truncated Multivariate Normal and Student t Distribution

## Description

This function computes the bivariate marginal density function f(x_q, x_r) from a k-dimensional Truncated Multivariate Normal density function (k>=2). The bivariate marginal density is obtained by integrating out (k-2) dimensions as proposed by Tallis (1961). This function is basically an extraction of the Leppard and Tallis (1989) Fortran code for moments calculation, but extended to the double truncated case.

## Usage

 1 2 3 4 5 6 dtmvnorm.marginal2(xq, xr, q, r, mean = rep(0, nrow(sigma)), sigma = diag(length(mean)), lower = rep(-Inf, length = length(mean)), upper = rep(Inf, length = length(mean)), log = FALSE, pmvnorm.algorithm=GenzBretz()) 

## Arguments

 xq Value x_q xr Value x_r q Index position for x_q within mean vector to calculate the bivariate marginal density for. r Index position for x_r within mean vector to calculate the bivariate marginal density for. mean Mean vector, default is rep(0, length = nrow(sigma)). sigma Covariance matrix, default is diag(length(mean)). 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)). log Logical; if TRUE, densities d are given as log(d). pmvnorm.algorithm Algorithm used for pmvnorm

## Details

The bivariate marginal density function f(x_q, x_r) for x \sim TN(μ, Σ, a, b) and q \ne r is defined as

F_{q,r}(x_q=c_q, x_r=c_r) = \int^{b_1}_{a_1}...\int^{b_{q-1}}_{a_{q-1}}\int^{b_{q+1}}_{a_{q+1}}...\int^{b_{r-1}}_{a_{r-1}}\int^{b_{r+1}}_{a_{r+1}}...\int^{b_{k}}_{a_{k}} \varphi{_{α}}_{Σ}(x_s, c_q, c_r) dx_s

## Author(s)

Stefan Wilhelm <[email protected]>, Manjunath B G <[email protected]>

## References

Tallis, G. M. (1961). The moment generating function of the truncated multinormal distribution. Journal of the Royal Statistical Society, Series B, 23, 223–229

Leppard, P. and Tallis, G. M. (1989). Evaluation of the Mean and Covariance of the Truncated Multinormal Applied Statistics, 38, 543–553

Manjunath B G and Wilhelm, S. (2009). Moments Calculation For the Double Truncated Multivariate Normal Density. Working Paper. Available at SSRN: http://ssrn.com/abstract=1472153

## 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 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  lower = c(-0.5, -1, -1) upper = c( 2.2, 2, 2) mean = c(0,0,0) sigma = matrix(c(2.0, -0.6, 0.7, -0.6, 1.0, -0.2, 0.7, -0.2, 1.0), 3, 3) # generate random samples from untruncated and truncated distribution Y = rmvnorm(10000, mean=mean, sigma=sigma) X = rtmvnorm(500, mean=mean, sigma=sigma, lower=lower, upper=upper, algorithm="gibbs") # compute bivariate marginal density of x1 and x2 xq <- seq(lower[1], upper[1], by=0.1) xr <- seq(lower[2], upper[2], by=0.1) grid <- matrix(NA, length(xq), length(xr)) for (i in 1:length(xq)) { for (j in 1:length(xr)) { grid[i,j] = dtmvnorm.marginal2(xq=xq[i], xr=xr[j], q=1, r=2, sigma=sigma, lower=lower, upper=upper) } } plot(Y[,1], Y[,2], xlim=c(-4, 4), ylim=c(-4, 4), main=expression("bivariate marginal density ("*x[1]*","*x[2]*")"), xlab=expression(x[1]), ylab=expression(x[2]), col="gray80") points(X[,1], X[,2], col="black") lines(x=c(lower[1], upper[1], upper[1], lower[1], lower[1]), y=c(lower[2],lower[2],upper[2],upper[2],lower[2]), lty=2, col="red") contour(xq, xr, grid, add=TRUE, nlevels = 8, col="red", lwd=2) # scatterplot matrices for untruncated and truncated points require(lattice) splom(Y) splom(X) 

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