TruncatedNormal (version 2.3)

mvNcdf: 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)

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\((l<X<u)\)

Arguments

l

lower truncation limit

u

upper truncation limit

Sig

covariance matrix of \(N(0,\Sigma)\)

n

number of Monte Carlo simulations

Author

Zdravko I. Botev

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

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