Truncated multivariate student cumulative distribution (QMC version)
Computes an estimator of the probability Pr(l<X<u),
where X is a zero-mean multivariate student vector
with scale matrix Sig and degrees of freedom df.
Infinite values for vectors u and l are accepted.
mvTqmc(l, u, Sig, df, n = 1e+05)
l |
lower bound for truncation (infinite values allowed) |
u |
upper bound for truncation |
Sig |
covariance matrix |
df |
degrees of freedom |
n |
sample size |
This version uses a Quasi Monte Carlo (QMC) pointset
of size ceiling(n/12) and estimates the relative error
using 12 independent randomized QMC estimators; QMC
is slower than ordinary Monte Carlo (see mvTcdf),
but is also likely to be more accurate when d<50.
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)
If you want to estimate Pr(l<Y<u),
where Y follows a Student distribution with df degrees of freedom,
location vector m and scale matrix Sig,
then use mvTqmc(Sig, l - m, u - m, nu, n).
Matlab code by Zdravko I. Botev, R port by Leo Belzile
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
Z. I. Botev and P. L'Ecuyer (2015), Efficient probability estimation and simulation of the truncated multivariate Student-t distribution, Proceedings of the 2015 Winter Simulation Conference, pp. 380-391
d <- 25; nu <- 30; l <- rep(1, d) * 5; u <- rep(Inf, d); Sig <- 0.5 * matrix(1, d, d) + 0.5 * diag(d); est <- mvTqmc(l, u, Sig, nu, n = 1e4) ## Not run: d <- 5 Sig <- solve(0.5*diag(d)+matrix(0.5, d,d)) ## mvtnorm::pmvt(lower = rep(-1,d), upper = rep(Inf, d), df = 10, sigma = Sig)[1] mvTqmc(rep(-1, d), u = rep(Inf, d), Sig = Sig, df = 10, n=1e4)$prob ## End(Not run)
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