poly_skurt_check {SimMultiCorrData}

Headrick's Fifth-Order Transformation Lagrangean Constraints for Lower Boundary of Standardized Kurtosis

Description

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

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

+ \lambda_{2} * [\gamma_{1} - h(c1, ..., c5)] + \lambda_{3} * [\gamma_{3} - i(c1, ..., c5)]

+ \lambda_{4} * [\gamma_{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, \lambda_{1}, ..., \lambda_{4} are the Lagrangean multipliers, \gamma_{1}, \gamma_{3}, \gamma_{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.

Usage

poly_skurt_check(c, a)

Arguments

c

a vector of constants c1, ..., c5, lambda1, ..., lambda4

a

a vector of skew, fifth standardized cumulant, sixth standardized cumulant

Value

A list with components:

dF/d\lambda_{1} = 1 - g(c1, ..., c5)

dF/d\lambda_{2} = \gamma_{1} - h(c1, ..., c5)

dF/d\lambda_{3} = \gamma_{3} - i(c1, ..., c5)

dF/d\lambda_{4} = \gamma_{4} - j(c1, ..., c5)

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

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

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

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

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

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

References

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.

See Also

calc_lower_skurt