# mvNcdf: truncated multivariate normal cumulative distribution In TruncatedNormal: Truncated Multivariate Normal

## Description

computes an estimator and a deterministic upper bound of the probability Pr(l<X<u), where X is a zero-mean multivariate normal vector with covariance matrix Σ, that is, X is drawn from N(0,Σ) infinite values for vectors u and l are accepted; Monte Carlo method uses sample size n;

## Usage

 `1` ```mvNcdf(l, u, Sig, n) ```

## Arguments

 `l` lower truncation limit `u` upper truncation limit `Sig` covariance matrix of N(0,Σ) `n` Monte Carlo simulation effort — the larger the n, the smaller the relative error of the estimator.

## Details

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

## Value

`est` with structure

 `\$prob` estimated value of probability Pr(l

## 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)

Z. I. Botev, email: [email protected] and web page: http://web.maths.unsw.edu.au/~zdravkobotev/

## References

Z. I. Botev (2015), The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting, submitted to JRSS(B)

See also: `mvNqmc` and `mvrandn`
 ```1 2 3 4 5``` ```d=15;l=1:d;u=l+Inf; Sig=matrix(rnorm(d^2),d,d)*2;Sig=Sig%*%t(Sig) n=10^3 # size of simulation effort x=mvNcdf(l,u,Sig,10^4) # compute the probability print(x) ```