SimMultiCorrData (version 0.2.2)

cdf_prob: Calculate Theoretical Cumulative Probability for Continuous Variables

Description

This function calculates a cumulative probability using the theoretical power method cdf \(F_p(Z)(p(z)) = F_p(Z)(p(z), F_Z(z))\) up to \(sigma * y + mu = delta\), where \(y = p(z)\), after using pdf_check. If the given constants do not produce a valid power method pdf, a warning is given. The formulas were obtained from Headrick & Kowalchuk (2007, 10.1080/10629360600605065).

Usage

cdf_prob(c, method = c("Fleishman", "Polynomial"), delta = 0.5, mu = 0,
  sigma = 1, lower = -1000000, upper = 1000000)

Arguments

c

a vector of constants c0, c1, c2, c3 (if method = "Fleishman") or c0, c1, c2, c3, c4, c5 (if method = "Polynomial"), like that returned by find_constants

method

the method used to find the constants. "Fleishman" uses a third-order polynomial transformation and "Polynomial" uses Headrick's fifth-order transformation.

delta

the value \(sigma * y + mu\), where \(y = p(z)\), at which to evaluate the cumulative probability

mu

mean for the continuous variable

sigma

standard deviation for the continuous variable

lower

lower bound for integration of the standard normal variable Z that generates the continuous variable

upper

upper bound for integration

Value

A list with components:

cumulative probability the theoretical cumulative probability up to delta

roots the roots z that make \(sigma * p(z) + mu = delta\)

References

Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43, 521-532. 10.1007/BF02293811.

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, Sawilowsky SS (1999). Simulating Correlated Non-normal Distributions: Extending the Fleishman Power Method. Psychometrika, 64, 25-35. 10.1007/BF02294317.

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

find_constants, pdf_check

Examples

# NOT RUN {
# Normal distribution with Headrick's fifth-order PMT:
cdf_prob(c = c(0, 1, 0, 0, 0, 0), "Polynomial", delta = qnorm(0.05))

# }
# NOT RUN {
# Beta(a = 4, b = 2) Distribution:
con <- find_constants(method = "Polynomial", skews = -0.467707, skurts = -0.375,
                      fifths = 1.403122, sixths = -0.426136)$constants
cdf_prob(c = con, method = "Polynomial", delta = 0.5)
# }