CGF_transformations: Derivatives of empirical moment generating function (MGF).

In PlotNormTest: Graphical Univariate/Multivariate Assessments for Normality Assumption

Derivatives of empirical moment generating function (MGF).

Description

Given dimension p, returns a dataframe containing the position of all derivatives of estimator of moment generating function \hat{M}_X(t), upto third/fourth order.

Usage

mt3_rev_pos(j1, j2, j3, p)

mt3_pos(p)

mt4_pos(p)

Arguments

j1	Index of the first variables
j2	Index of the first variables, should be at least j1
j3	Index of the first variables, should be at least j2
p	Dimension

Details

The estimator of multivariate moment generating function is \hat{M}_X(t) = \dfrac{1}{n} \sum_{i = 1}^n \exp(t'X_i) The chain containing all derivatives up to the third order is

Z = \bigg(\hat{M}, \hat{M}^{001}, \dots \hat{M}^{00p}, \hat{M}^{011}, \hat{M}^{012}, \dots \hat{M}^{0pp}, \hat{M}^{111}, \hat{M}^{112}, \dots \hat{M}^{ppp}\bigg)'

and

\hat{M} = \hat{M}^{000}(t)= \hat{M}_X(t)

\hat{M}^{j_1j_2j_3}(t) = \dfrac{\partial^k}{\partial t_{j_1} t_{j_2} t_{j_3}} \hat{M}(t)

where k is the number of j_1, j_2, j_3 different from 0. Similar notation is applied when fourth derivatives is used.

Value

mt3_rev_pos returns the position of this particular derivative in the chain of all derivatives, up to third order.

mt3_pos an array contaning all position with respect to index of j_1, j_2, j_3.

mt4_pos an array contaning all position with respect to the index of j_1, j_2, j_3, j_4.

Examples

mt3_rev_pos(1, 2, 2, p = 3)
p <- 3
mt3_pos(p)
mt4_pos(p)

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