covLtLs: Linear combinations of distinct derivatives of empirical...

In PlotNormTest: Graphical Univariate/Multivariate Assessments for Normality Assumption

Linear combinations of distinct derivatives of empirical cumulant generating function (CGF).

Description

Linear combination of third/fourth derivatives of CGF gives an asymptotically univariate Gaussian process with mean 0 and covariance between two points t \in \mathbb{R}^p and s \in \mathbb{R}^p is defined. We consider vector t and s as the form t = t^*1_p and s = s^*1_p.

Usage

mt3_covLtLs(l, p, bigt = seq(-1, 1, 0.05)/sqrt(p), sTtTs = NULL, seed = 1)

mt4_covLtLs(l, p, bigt = seq(-1, 1, 0.05)/sqrt(p), sTtTs = NULL, seed = 1)

Arguments

l	vector of linear combination of size equal to the number of distinct derivatives, see l_dhCGF().
p	dimension of multivariate random vector which data are collected.
bigt	array of value t^* and s^*.
sTtTs	Covariance matrix of derivatives vector, see covTtTs(). Default is NULL, when the algorithm will call mt3_covTtTs() or mt4_covTtTs().
seed	Random seed to get the estimate of the supremum of the univariate Gaussian process obtained from the linear combination.

Value

• sLtLs covariance matrix of the linear combination of distinct derivatives, which is a zero-mean Gaussian process.

• m.supLt Monte-Carlo estimates of supremum of this Gaussian process

mt3_covLtLs returns values related to the use of third derivatives. mt4_covLtLs returns values related to the use of fourth derivatives.

Examples

bigt <- seq(-1, 1, .5)
p <- 2
# Third derivatives
lT3 <- l_dhCGF(p)[[1]]
l3 <- rep(1/sqrt(lT3), lT3)
mt3_covLtLs(l = l3, p = p, bigt = bigt/sqrt(p), seed = 1)
#fourth derivatives
lT4 <- l_dhCGF(p)[[2]]
l4 <- rep(1/sqrt(lT4), lT4)
mt4_covLtLs(l = l4, p = p, bigt = bigt/sqrt(p), seed = 1)

PlotNormTest documentation built on April 12, 2025, 9:14 a.m.