This function gives the first-order conditions of the Fleishman Transformation Lagrangean expression
\(F(c1, c3, \lambda) = f(c1, c3) + \lambda * [\gamma_{1} - g(c1, c3)]\) used to find the lower kurtosis boundary for a given non-zero skewness
in calc_lower_skurt (see Headrick & Sawilowsky, 2002, 10.3102/10769986025004417). Here, \(f(c1, c3)\) is the equation for
standardized kurtosis expressed in terms of c1 and c3 only,
\(\lambda\) is the Lagrangean multiplier, \(\gamma_{1}\) is skewness, and \(g(c1, c3)\) is the equation for skewness expressed
in terms of c1 and c3 only. It should be noted that these equations are for \(\gamma_{1} > 0\). Negative skew values are handled within
calc_lower_skurt. Headrick & Sawilowsky (2002) gave equations for the first-order derivatives \(dF/dc1\)
and \(dF/dc3\). These were verified and \(dF/d\lambda\) was calculated using D. The second-order conditions to
verify that the kurtosis is a global minimum are in fleish_Hessian.
This function would not ordinarily be called by the user.
fleish_skurt_check(c, a)a vector of constants c1, c3, lambda
skew value
A list with components:
\(dF(c1, c3, \lambda)/d\lambda = \gamma_{1} - g(c1, c3)\)
\(dF(c1, c3, \lambda)/dc1 = df(c1, c3)/dc1 - \lambda * dg(c1, c3)/dc1\)
\(dF(c1, c3, \lambda)/dc3 = df(c1, c3)/dc3 - \lambda * dg(c1, c3)/dc3\)
If the suppled values for c and skew satisfy the Lagrangean expression, it will return 0 for each component.
Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43, 521-532. 10.1007/BF02293811.
Headrick TC, Sawilowsky SS (2002). Weighted Simplex Procedures for Determining Boundary Points and Constants for the Univariate and Multivariate Power Methods. Journal of Educational and Behavioral Statistics, 25, 417-436. 10.3102/10769986025004417.