fleish_Hessian {SimMultiCorrData}

Fleishman's Third-Order Transformation Hessian Calculation for Lower Boundary of Standardized Kurtosis in Asymmetric Distributions

Description

This function gives the second-order conditions necessary to verify that a kurtosis is a global minimum. A kurtosis solution from fleish_skurt_check is a global minimum if and only if the determinant of the bordered Hessian, H, is less than zero (see Headrick & Sawilowsky, 2002, doi: 10.3102/10769986025004417), where

|\bar{H}| = matrix(c(0, dg(c1, c3)/dc1, dg(c1, c3)/dc3,

dg(c1, c3)/dc1, d^2 F(c1, c3, \lambda)/dc1^2, d^2 F(c1, c3, \lambda)/(dc3 dc1),

dg(c1, c3)/dc3, d^2 F(c1, c3, \lambda)/(dc1 dc3), d^2 F(c1, c3, \lambda)/dc3^2), 3, 3, byrow = TRUE)

Here, F(c1, c3, \lambda) = f(c1, c3) + \lambda * [\gamma_{1} - g(c1, c3)] is the Fleishman Transformation Lagrangean expression (see fleish_skurt_check). Headrick & Sawilowsky (2002) gave equations for the second-order derivatives d^2 F/dc1^2, d^2 F/dc3^2, and d^2 F/(dc1 dc3). These were verified and dg/dc1 and dg/dc3 were calculated using D (see deriv). This function would not ordinarily be called by the user.

Usage

fleish_Hessian(c)

Arguments

c

a vector of constants c1, c3, lambda

Value

A list with components:

Hessian the Hessian matrix H

H_det the determinant of H

References

Please see references for fleish_skurt_check.

See Also

fleish_skurt_check, calc_lower_skurt