PlotNormTest (version 1.0.1)

dCGF: Calculation of derivatives of empirical cumulant generating function (CGF).

Description

Get the third/fortth derivatives of sample CGF at a given point.

Usage

d3hCGF(myt, x)
d4hCGF(myt, x)
l_dhCGF(p)
dhCGF1D(t, x)

Value

d3hCGF returns the sequence of third derivatives of empirical CGF, ordered by index of \(j_1 \leq j_2 \leq j_3 \leq p\).

d4hCGF returns the sequence of fourth derivatives of empirical CGF ordered by index of \(j_1 \leq j_2 \leq j_3 \leq j_4 \leq p\).

l_dhCGF returns number of distinct third and fourth derivatives.

dhCGF1D returns third/fourth derivatives of univariate empirical CGF, which are d3hCGF and d4hCGF when \(p = 1\).

Arguments

myt, t

numeric vector of length p.

x

data matrix.

p

Dimension.

Details

Estimator of standardized cumulant function is $$ \log\hat{M}_X(t) = \log \left(\dfrac{1}{n} \sum_{i = 1}^n \exp(t'S^{\frac{-1}{2}}(X_i - \bar{X})) \right) $$ and its $$k^{th}$$ order derivatives is defined as $$ T_k(t) = \dfrac{\partial^k}{ \partial t_{j_1}t_{j_2} \dots t_{j_k}} \log(\hat{M}_X(t)), t \in \mathbb{R}^p $$ where \(t_{j_1}t_{j_2} \dots t_{j_k}\) are the corresponding components of vector \(t \in \mathbb{R}^p\).

Examples

p <- 3
# Number of distinct derivatives
l_dhCGF(p)
set.seed(1)
x <- MASS::mvrnorm(100, rep(0, p), diag(p))
myt <- rep(.2, p)
d3hCGF(myt = myt, x = x)
d4hCGF(myt = myt, x = x)
#Univariate data
set.seed(1)
x <- rnorm(100)
t <- .3
dhCGF1D(t, x)