mvNcdf function

Truncated multivariate normal cumulative distribution

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.

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

Returns

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$(l\<X\<u)$

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 $N(0, A\Sigma A^\top)$.

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.

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

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

Author(s)

Zdravko I. Botev