fleish_skurt_check {SimMultiCorrData}

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

Description

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, doi: 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 (see deriv). 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.

Usage

fleish_skurt_check(c, a)

Arguments

c

a vector of constants c1, c3, lambda

a

skew value

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.

References

Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43, 521-532. doi: 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. doi: 10.3102/10769986025004417.

See Also

fleish_Hessian, calc_lower_skurt