covTtTs: Covariance matrix of derivatives of sample cumulant...

In PlotNormTest: Graphical Univariate/Multivariate Assessments for Normality Assumption

Covariance matrix of derivatives of sample cumulant generating function (CGF).

Description

Stacking third/fourth derivatives of sample CGF together to obtain a vector, which (under normality assumption on data) approaches a normally distributed vector with zero mean and a covariance matrix. More specifically, covTsTs computes covariance between any two points as the form t = t^*1_p and s = s^*1_p.

Usage

mt3_covTtTs(bigt, p = 1, pos.matrix = NULL)

mt4_covTtTs(bigt, p = 1, pos.matrix = NULL)

Arguments

bigt	array contains value of t^*.
p	dimension of multivariate random vector which data are collected.
pos.matrix	matrix containing information of position of any derivatives. Default is NULL, the function will call mt3_pos() or mt4_pos().

Details

Number of distinct third derivatives is l_{T_3}= p + 2 \times \begin{pmatrix} p\\2 \end{pmatrix} + \begin{pmatrix} p \\ 3 \end{pmatrix} Number of distinct fourth derivatives is l_{T_4} = p + 3 \times \begin{pmatrix} p\\2 \end{pmatrix} + 3 \times \begin{pmatrix} p \\ 3 \end{pmatrix} + \begin{pmatrix} p \\ 4 \end{pmatrix} For each pairs of (t^*, s^*), covTsTt results a covariance matrix of size l_{T_3} \times l_{T_3} or l_{T_4} \times l_{T_4}.

Value

A 2 dimensional upper triangular array, with size equals to length of bigt. Each element contains a covariance matrix of derivatives sequences between any two points t = t^* 1_p and s = s^*1_p. mt3_covTsTt returns the resulting third derivatives.

mt4_covTsTt returns the resulting forth derivatives.

Examples

bigt <- seq(-1, 1, .5)
p <- 2
# Third derivatives
mt3_pos.matrix <- mt3_pos(p)
sTsTt3 <- mt3_covTtTs(bigt = bigt, p = p, pos.matrix = mt3_pos.matrix)
dim(sTsTt3)
sTsTt3[1:5, 1:5]
# Fourth derivatives
mt4_pos.matrix <- mt4_pos(p)
sTsTt4 <- mt4_covTtTs(bigt = bigt, p = p, pos.matrix = mt4_pos.matrix)
dim(sTsTt4)
sTsTt4[1:5, 1:5]

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