poly_skurt_check: Headrick's Fifth-Order Transformation Lagrangean Constraints...
In SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types

Description Usage Arguments Value References See Also

This function gives the first-order conditions of the multi-constraint Lagrangean expression

F(c1, ..., c5, λ_{1}, ..., λ_{4}) = f(c1, ..., c5) + λ_{1} * [1 - g(c1, ..., c5)]

+ λ_{2} * [γ_{1} - h(c1, ..., c5)] + λ_{3} * [γ_{3} - i(c1, ..., c5)]

+ λ_{4} * [γ_{4} - j(c1, ..., c5)]

used to find the lower kurtosis boundary for a given skewness and standardized fifth and sixth cumulants in calc_lower_skurt. The partial derivatives are described in Headrick (2002, doi: 10.1016/S0167-9473(02)00072-5), but he does not provide the actual equations. The equations used here were found with D (see deriv). Here, λ_{1}, ..., λ_{4} are the Lagrangean multipliers, γ_{1}, γ_{3}, γ_{4} are the user-specified values of skewness, fifth cumulant, and sixth cumulant, and f, g, h, i, j are the equations for standardized kurtosis, variance, fifth cumulant, and sixth cumulant expressed in terms of the constants. This function would not ordinarily be called by the user.

1	poly_skurt_check(c, a)

`c`	a vector of constants c1, ..., c5, lambda1, ..., lambda4
`a`	a vector of skew, fifth standardized cumulant, sixth standardized cumulant

A list with components:

dF/dλ_{1} = 1 - g(c1, ..., c5)

dF/dλ_{2} = γ_{1} - h(c1, ..., c5)

dF/dλ_{3} = γ_{3} - i(c1, ..., c5)

dF/dλ_{4} = γ_{4} - j(c1, ..., c5)

dF/dc1 = df/dc1 - λ_{1} * dg/dc1 - λ_{2} * dh/dc1 - λ_{3} * di/dc1 - λ_{4} * dj/dc1

dF/dc2 = df/dc2 - λ_{1} * dg/dc2 - λ_{2} * dh/dc2 - λ_{3} * di/dc2 - λ_{4} * dj/dc2

dF/dc3 = df/dc3 - λ_{1} * dg/dc3 - λ_{2} * dh/dc3 - λ_{3} * di/dc3 - λ_{4} * dj/dc3

dF/dc4 = df/dc4 - λ_{1} * dg/dc4 - λ_{2} * dh/dc4 - λ_{3} * di/dc4 - λ_{4} * dj/dc4

dF/dc5 = df/dc5 - λ_{1} * dg/dc5 - λ_{2} * dh/dc5 - λ_{3} * di/dc5 - λ_{4} * dj/dc5

If the suppled values for c and a satisfy the Lagrangean expression, it will return 0 for each component.

Headrick TC (2002). Fast Fifth-order Polynomial Transforms for Generating Univariate and Multivariate Non-normal Distributions. Computational Statistics & Data Analysis, 40(4):685-711. doi: 10.1016/S0167-9473(02)00072-5. (ScienceDirect)

Headrick TC (2004). On Polynomial Transformations for Simulating Multivariate Nonnormal Distributions. Journal of Modern Applied Statistical Methods, 3(1), 65-71. doi: 10.22237/jmasm/1083370080.

Headrick TC, Kowalchuk RK (2007). The Power Method Transformation: Its Probability Density Function, Distribution Function, and Its Further Use for Fitting Data. Journal of Statistical Computation and Simulation, 77, 229-249. doi: 10.1080/10629360600605065.

Headrick TC, Sheng Y, & Hodis FA (2007). Numerical Computing and Graphics for the Power Method Transformation Using Mathematica. Journal of Statistical Software, 19(3), 1 - 17. doi: 10.18637/jss.v019.i03.

calc_lower_skurt

SimMultiCorrData documentation built on May 2, 2019, 9:50 a.m.