This function contains Headrick's fifth-order polynomial transformation equations
(2002, 10.1016/S0167-9473(02)00072-5). It is used in
find_constants to find the constants c1, c2, c3, c4, and c5 (\(c0 = -c2 - 3 * c4\))
that satisfy the equations given skewness, standardized kurtosis, and standardized fifth and sixth cumulant values.
It can be used to verify a set of constants satisfy the equations. Note that there exist solutions that yield
invalid power method pdfs (see power_norm_corr,
pdf_check). This function would not ordinarily be called by the user.
poly(c, a)a vector of constants c1, c2, c3, c4, c5; note that find_constants returns
c0, c1, c2, c3, c4, c5
a vector c(skewness, standardized kurtosis, standardized fifth cumulant, standardized sixth cumulant)
a list of length 5; if the constants satisfy the equations, returns 0 for all list elements
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.
# NOT RUN {
# Laplace Distribution
poly(c = c(0.727709, 0, 0.096303, 0, -0.002232), a = c(0, 3, 0, 30))
# }
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