Quantiles of the Truncated Multivariate Normal Distribution in one dimension

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Description

Computes the equicoordinate quantile function of the truncated multivariate normal distribution for arbitrary correlation matrices based on an inversion of the algorithms by Genz and Bretz.

Usage

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qtmvnorm.marginal(p, 
        interval = c(-10, 10), 
        tail = c("lower.tail","upper.tail","both.tails"), 
        n=1, 
		mean=rep(0, nrow(sigma)), 
		sigma=diag(length(mean)), 
		lower=rep(-Inf, length = length(mean)), 
		upper=rep( Inf, length = length(mean)),
		...)

Arguments

p

probability.

interval

a vector containing the end-points of the interval to be searched by uniroot.

tail

specifies which quantiles should be computed. lower.tail gives the quantile x for which P[X <= x] = p, upper.tail gives x with P[X > x] = p and both.tails leads to x with P[-x ≤ X ≤ x] = p.

P[-x <= X <= x] = p

n

index (1..n) to calculate marginal quantile for

mean

the mean vector of length n.

sigma

the covariance matrix of dimension n. Either corr or sigma can be specified. If sigma is given, the problem is standardized. If neither corr nor sigma is given, the identity matrix is used for sigma.

lower

Vector of lower truncation points,\ default is rep(-Inf, length = length(mean)).

upper

Vector of upper truncation points,\ default is rep( Inf, length = length(mean)).

...

additional parameters to be passed to uniroot.

Details

Only equicoordinate quantiles are computed, i.e., the quantiles in each dimension coincide. Currently, the distribution function is inverted by using the uniroot function which may result in limited accuracy of the quantiles.

Value

A list with four components: quantile and f.quantile give the location of the quantile and the value of the function evaluated at that point. iter and estim.prec give the number of iterations used and an approximate estimated precision from uniroot.

See Also

ptmvnorm, pmvnorm

Examples

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# finite dimensional distribution of the Geometric Brownian Motion log-returns 
# with truncation

# volatility p.a.
sigma=0.4

# risk free rate
r = 0.05

# n=3 points in time
T <- c(0.5, 0.7, 1)

# covariance matrix of Geometric Brownian Motion returns
Sigma = sigma^2*outer(T,T,pmin)

# mean vector of the Geometric Brownian Motion returns
mu    = (r - sigma^2/2) * T

# lower truncation vector a (a<=x<=b)
a = rep(-Inf, 3)

# upper truncation vector b (a<=x<=b)
b = c(0, 0, Inf)

# quantile of the t_1 returns
qtmvnorm.marginal(p=0.95, interval = c(-10, 10), tail = "lower.tail", n=1, 
  mean  = mu, sigma = Sigma, lower=a, upper=b)