SimMultiCorrData (version 0.2.2)

poly_skurt_check: 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 $$F(c1,...,c5,{\lambda }_1,...,{\lambda }_4)=f(c1,...,c5)+{\lambda }_1\ast [1-g(c1,...,c5)]$$ $$+{\lambda }_2\ast [{\gamma }_1-h(c1,...,c5)]+{\lambda }_3\ast [{\gamma }_3-i(c1,...,c5)]$$ $$+{\lambda }_4\ast [{\gamma }_4-j(c1,...,c5)]$$ 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, 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:

$dF/d{\lambda }_1=1-g(c1,...,c5)$

$dF/d{\lambda }_2={\gamma }_1-h(c1,...,c5)$

$dF/d{\lambda }_3={\gamma }_3-i(c1,...,c5)$

$dF/d{\lambda }_4={\gamma }_4-j(c1,...,c5)$

$dF/dc1=df/dc1-{\lambda }_1\ast dg/dc1-{\lambda }_2\ast dh/dc1-{\lambda }_3\ast di/dc1-{\lambda }_4\ast dj/dc1$

$dF/dc2=df/dc2-{\lambda }_1\ast dg/dc2-{\lambda }_2\ast dh/dc2-{\lambda }_3\ast di/dc2-{\lambda }_4\ast dj/dc2$

$dF/dc3=df/dc3-{\lambda }_1\ast dg/dc3-{\lambda }_2\ast dh/dc3-{\lambda }_3\ast di/dc3-{\lambda }_4\ast dj/dc3$

$dF/dc4=df/dc4-{\lambda }_1\ast dg/dc4-{\lambda }_2\ast dh/dc4-{\lambda }_3\ast di/dc4-{\lambda }_4\ast dj/dc4$

$dF/dc5=df/dc5-{\lambda }_1\ast dg/dc5-{\lambda }_2\ast dh/dc5-{\lambda }_3\ast di/dc5-{\lambda }_4\ast dj/dc5$

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. 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. 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. 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. 10.18637/jss.v019.i03.

See Also

calc_lower_skurt