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Last data update: 2014.03.03

R: truncated multivariate normal cumulative distribution

mvNcdf

R Documentation

truncated multivariate normal cumulative distribution

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

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

`$prob`	estimated value of probability Pr(l<X<u)
`$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)

Z. I. Botev, email: botev@unsw.edu.au 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)

Examples

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)