mvNcdf: Truncated multivariate normal cumulative distribution

In lbelzile/TruncatedNormal: Truncated Multivariate Normal and Student Distributions

Truncated multivariate normal cumulative distribution

Description

Computes an estimate and a deterministic upper bound of the probability Pr(l<X<u), where X is a zero-mean multivariate normal vector with covariance matrix \Sigma, that is, X is drawn from N(0,\Sigma). Infinite values for vectors u and l are accepted. The Monte Carlo method uses sample size n; the larger n, the smaller the relative error of the estimator.

Usage

mvNcdf(l, u, Sig, n = 1e+05)

Arguments

l	lower truncation limit
u	upper truncation limit
Sig	covariance matrix of N(0,\Sigma)
n	number of Monte Carlo simulations

Details

Suppose you wish to estimate Pr(l<AX<u), where A is a full rank matrix and X is drawn from N(\mu,\Sigma), then you simply compute Pr(l-A\mu<AY<u-A\mu), where Y is drawn from N(0, A\Sigma A^\top).

Value

a list with components

• prob: estimated value of probability Pr(l<X<u)

• relErr: estimated relative error of estimator

• upbnd: theoretical upper bound on true Pr(l<X<u)

Note

For small dimensions, say d<50, better accuracy may be obtained by using the (usually slower) quasi-Monte Carlo version mvNqmc of this algorithm.

Author(s)

Zdravko I. Botev

References

Z. I. Botev (2017), The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting, Journal of the Royal Statistical Society, Series B, 79 (1), pp. 1–24.

See Also

mvNqmc, mvrandn

Examples

d <- 15; l <- 1:d; u <- rep(Inf, d);
Sig <- matrix(rnorm(d^2), d, d)*2; Sig=Sig %*% t(Sig)
mvNcdf(l, u, Sig, 1e4) # compute the probability 

lbelzile/TruncatedNormal documentation built on March 4, 2024, 5:50 p.m.