# One-dimensional marginal density functions from a Truncated Multivariate Normal distribution This function computes the one-dimensional marginal density function from a Truncated Multivariate Normal density function using the algorithm given in Cartinhour (1990). ```r dtmvnorm.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)), log=FALSE) ``` ## Arguments - `xn`: Vector of quantiles to calculate the marginal density for. - `n`: Index position (1..k) within the random vector x to calculate the one-dimensional 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). ## Details The one-dimensional marginal density $f_i(x_i)$ of $x_i$ is $$ f_i(x_i) = \int_{a_1}^{b_1} \ldots \int_{a_{i-1}}^{b_{i-1}} \int_{a_{i+1}}^{b_{i+1}} \ldots \int_{a_k}^{b_k} f(x) dx_{-i} $$ Note that the one-dimensional marginal density is not truncated normal, but only conditional densities are truncated normal. ## Author(s) Stefan Wilhelm ## References Cartinhour, J. (1990). One-dimensional marginal density functions of a truncated multivariate normal density function. **Communications in Statistics - Theory and Methods**, 19 , 197--203 Arnold et al. (1993). The Nontruncated Marginal of a Truncated Bivariate Normal Distribution. **Psychometrika**, 58 , 471--488 ## Examples ```r ############################################# # # Example 1: truncated bivariate normal # ############################################# # parameters of the bivariate normal distribution sigma = matrix(c(1 , 0.95, 0.95, 1 ), 2, 2) mu = c(0,0) # sample from multivariate normal distribution X = rmvnorm(5000, mu, sigma) # tuncation in x2 with x2 <= 0 X.trunc = X[X[,2]<0,] # plot the realisations before and after truncation par(mfrow=c(2,2)) plot(X, col="gray", xlab=expression(x[1]), ylab=expression(x[2]), main="realisations from a\n truncated bivariate normal distribution") points(X.trunc) abline(h=0, lty=2, col="gray") #legend("topleft", col=c("gray", "black") # marginal density for x1 from realisations plot(density(X.trunc[,1]), main=expression("marginal density for "*x[1])) # one-dimensional marginal density for x1 using the formula x <- seq(-5, 5, by=0.01) fx <- dtmvnorm.marginal(x, n=1, mean=mu, sigma=sigma, lower=c(-Inf,-Inf), upper=c(Inf,0)) lines(x, fx, lwd=2, col="red") # marginal density for x2 plot(density(X.trunc[,2]), main=expression("marginal density for "*x[2])) # one-dimensional marginal density for x2 using the formula x <- seq(-5, 5, by=0.01) fx <- dtmvnorm.marginal(x, n=2, mean=mu, sigma=sigma, lower=c(-Inf,-Inf), upper=c(Inf,0)) lines(x, fx, lwd=2, col="blue") ############################################# # # Example 2 : truncated trivariate normal # ############################################# # parameters of the trivariate normal distribution sigma = outer(1:3,1:3,pmin) mu = c(0,0,0) # sample from multivariate normal distribution X = rmvnorm(2000, mu, sigma) # truncation in x2 and x3 : x2 <= 0, x3 <= 0 X.trunc = X[X[,2]<=0 & X[,3]<=0,] par(mfrow=c(2,3)) plot(X, col="gray", xlab=expression(x[1]), ylab=expression(x[2]), main="realisations from a\n truncated trivariate normal distribution") points(X.trunc, col="black") abline(h=0, lty=2, col="gray") plot(X[,2:3], col="gray", xlab=expression(x[2]), ylab=expression(x[3]), main="realisations from a\n truncated trivariate normal distribution") points(X.trunc[,2:3], col="black") abline(h=0, lty=2, col="gray") abline(v=0, lty=2, col="gray") plot(X[,c(1,3)], col="gray", xlab=expression(x[1]), ylab=expression(x[3]), main="realisations from a\n truncated trivariate normal distribution") points(X.trunc[,c(1,3)], col="black") abline(h=0, lty=2, col="gray") # one-dimensional marginal density for x1 from realisations and formula plot(density(X.trunc[,1]), main=expression("marginal density for "*x[1])) x <- seq(-5, 5, by=0.01) fx <- dtmvnorm.marginal(x, n=1, mean=mu, sigma=sigma, lower=c(-Inf,-Inf,-Inf), upper=c(Inf,0,0)) lines(x, fx, lwd=2, col="red") # one-dimensional marginal density for x2 from realisations and formula plot(density(X.trunc[,2]), main=expression("marginal density for "*x[2])) x <- seq(-5, 5, by=0.01) fx <- dtmvnorm.marginal(x, n=2, mean=mu, sigma=sigma, lower=c(-Inf,-Inf,-Inf), upper=c(Inf,0,0)) lines(x, fx, lwd=2, col="red") # one-dimensional marginal density for x3 from realisations and formula plot(density(X.trunc[,3]), main=expression("marginal density for "*x[3])) x <- seq(-5, 5, by=0.01) fx <- dtmvnorm.marginal(x, n=3, mean=mu, sigma=sigma, lower=c(-Inf,-Inf,-Inf), upper=c(Inf,0,0)) lines(x, fx, lwd=2, col="red") ```

dmvnorm.marginal function