calc_theory {SimMultiCorrData}

Find Theoretical Standardized Cumulants for Continuous Distributions

Description

This function calculates the theoretical mean, standard deviation, skewness, standardized kurtosis, and standardized fifth and sixth cumulants given either a Distribution name (plus up to 4 parameters) or a pdf (with specified lower and upper support bounds). The result can be used as input to find_constants or for data simulation.

Note: Due to the nature of the integration involved in calculating the standardized cumulants, the results are approximations. Greater accuracy can be achieved by increasing the number of subdivisions (sub) used in the integration process. However, the user may need to round the cumulants (i.e. using round(x, 8)) before using them in other functions (i.e. find_constants, calc_lower_skurt, nonnormvar1, rcorrvar, rcorrvar2) in order to achieve the desired results. For example, in order to ensure that skew is exactly 0 for symmetric distributions.

Usage

calc_theory(Dist = c("Benini", "Beta", "Beta-Normal", "Birnbaum-Saunders",
  "Chisq", "Dagum", "Exponential", "Exp-Geometric", "Exp-Logarithmic",
  "Exp-Poisson", "F", "Fisk", "Frechet", "Gamma", "Gaussian", "Gompertz",
  "Gumbel", "Kumaraswamy", "Laplace", "Lindley", "Logistic", "Loggamma",
  "Lognormal", "Lomax", "Makeham", "Maxwell", "Nakagami", "Paralogistic",
  "Pareto", "Perks", "Rayleigh", "Rice", "Singh-Maddala", "Skewnormal", "t",
  "Topp-Leone", "Triangular", "Uniform", "Weibull"), params = NULL,
  fx = NULL, lower = NULL, upper = NULL, sub = 1000)

Arguments

Dist

name of the distribution. The possible values are: "Benini", "Beta", "Beta-Normal", "Birnbaum-Saunders", "Chisq", "Exponential", "Exp-Geometric", "Exp-Logarithmic", "Exp-Poisson", "F", "Fisk", "Frechet", "Gamma", "Gaussian", "Gompertz", "Gumbel", "Kumaraswamy", "Laplace", "Lindley", "Logistic", "Loggamma", "Lognormal", "Lomax", "Makeham", "Maxwell", "Nakagami", "Paralogistic", "Pareto", "Perks", "Rayleigh", "Rice", "Singh-Maddala", "Skewnormal", "t", "Topp-Leone", "Triangular", "Uniform", "Weibull". Please refer to the documentation for each package (either stats-package, VGAM-package, or triangle) for information on appropriate parameter inputs.

params

a vector of parameters (up to 4) for the desired distribution (keep NULL if fx supplied instead)

fx

a pdf input as a function of x only, i.e. fx <- function(x) 0.5*(x-1)^2; must return a scalar (keep NULL if Dist supplied instead)

lower

the lower support bound for a supplied fx, else keep NULL

upper

the upper support bound for a supplied fx, else keep NULL

sub

the number of subdivisions to use in the integration; if no result, try increasing sub (requires longer computation time)

Value

A vector of the mean, standard deviation, skewness, standardized kurtosis, and standardized fifth and sixth cumulants

References

Headrick TC (2002). Fast Fifth-order Polynomial Transforms for Generating Univariate and Multivariate Non-normal Distributions. Computational Statistics & Data Analysis, 40(4):685-711. doi: 10.1016/S0167-9473(02)00072-5. (ScienceDirect)

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. doi: 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. doi: 10.18637/jss.v019.i03

Thomas W. Yee (2018). VGAM: Vector Generalized Linear and Additive Models. R package version 1.0-5. https://CRAN.R-project.org/package=VGAM

Rob Carnell (2017). triangle: Provides the Standard Distribution Functions for the Triangle Distribution. R package version 0.11. https://CRAN.R-project.org/package=triangle

See Also

calc_fisherk, calc_moments, find_constants

Examples

options(scipen = 999)

# Pareto Distribution: params = c(alpha = scale, theta = shape)
calc_theory(Dist = "Pareto", params = c(1, 10))

# Generalized Rayleigh Distribution: params = c(alpha = scale, mu/sqrt(pi/2) = shape)
calc_theory(Dist = "Rayleigh", params = c(0.5, 1))

# Laplace Distribution: params = c(location, scale)
calc_theory(Dist = "Laplace", params = c(0, 1))

# Triangle Distribution: params = c(a, b)
calc_theory(Dist = "Triangular", params = c(0, 1))