remotes::install_github('tingtingzhan/TukeyGH77')
remotes::install_github('tingtingzhan/param2moment')param2moment
Introduction
This vignette of package param2moment (CRAN, Github) documents …
Prerequisite
New features are first implemented on Github.
And eventually make their way to CRAN.
utils::install.packages('TukeyGH77')
utils::install.packages('param2moment')Note to Users
Examples in this vignette require that the search path has
library(param2moment)
library(ggplot2)
library(geomtextpath)
library(TukeyGH77)
library(sn)Introduction
Moments
For random variable X,
Raw Moments
| Slot | Description |
|---|---|
@raw1 |
\mu = \text{E}(X) |
@raw2 |
\text{E}(X^2) |
@raw3 |
\text{E}(X^3) |
@raw4 |
\text{E}(X^4) |
Central Moments
According to Binomial theorem,
| Slot | Central Moment | Value | Derivation |
|---|---|---|---|
| 0 = \text{E}(X-\mu) | |||
@central2 |
\sigma^2 = \text{E}(X-\mu)^2 | \text{E}(X^2) - \mu^2 | \text{E}(X^2) - 2\mu\text{E}(X) + \mu^2 |
@central3 |
\text{E}(X-\mu)^3 | \text{E}(X^3) - 3\mu\text{E}(X^2) + 2\mu^3 | \text{E}(X^3) - 3\mu\text{E}(X^2) + 3\mu^2\text{E}(X) - \mu^3 |
@central4 |
\text{E}(X-\mu)^4 | \text{E}(X^4) - 4\mu\text{E}(X^3) + 6\mu^2\text{E}(X^2) - 3\mu^4 | \text{E}(X^4) - 4\mu\text{E}(X^3) + 6\mu^2 \text{E}(X^2) - 4\mu^3\text{E}(X) + \mu^4 |
Standardized Moments
| Slot | Standardized Moment | Description |
|---|---|---|
| 0 = \text{E}(X-\mu)/\sigma | ||
| 1 = \text{E}(X-\mu)^2/\sigma^2 | ||
@standardized3 |
\text{E}(X-\mu)^3/\sigma^3 | skewness |
@standardized4 |
\text{E}(X-\mu)^4/\sigma^4 | kurtosis |
Location-Scale Transformation
For Y = \text{location} + \text{scale} \cdot X,
\begin{cases} \mu_Y & = \text{location} + \text{scale}\cdot \mu_X\\ \sigma_Y & = \text{scale}\cdot \sigma_X \\ \text{skewness}_Y & = \text{skewness}_X\\ \text{kurtosis}_Y & = \text{kurtosis}_X \end{cases}
Conversion of Parameter and Moment
Normal Distribution
moment_norm(mean = 1.2, sd = .7)
#> Distribution: norm
#> Mean: 1.200
#> Standard Deviation: 0.700
#> Skewness: 0.000
#> (Excess) Kurtosis: 0.000Skew-Normal Distribution
moment_sn(xi = 2, omega = 1.3, alpha = 3)
#> Distribution: sn
#> Mean: 2.984
#> Standard Deviation: 0.850
#> Skewness: 0.667
#> (Excess) Kurtosis: 0.510moment2sn(skewness = .3)
#> xi omega alpha
#> -0.8874593 1.3369901 1.4991756Skew-t Distribution
moment_st(xi = 2, omega = 1.3, alpha = 3, nu = 6)
#> Distribution: st
#> Mean: 3.133
#> Standard Deviation: 1.119
#> Skewness: 1.666
#> (Excess) Kurtosis: 7.659moment2st(skewness = .2, kurtosis = .3)
#> xi omega alpha nu
#> -0.6627910 1.1615769 0.9752484 31.9777192moment2st(mean = 1, kurtosis = .3, alpha = 0)
#> xi omega nu
#> 1.0000017 0.9574701 24.0001063Tukey’s gh Distribution
Tukey’s g-distribution
moment_GH(A = 3, B = 1.5, g = 0, h = .05)
#> Mean: 3.000
#> Standard Deviation: 1.623
#> Skewness: 0.000
#> (Excess) Kurtosis: 0.821moment2GH(mean = 1, kurtosis = .3, g = 0)
#> A B h
#> 1.00001979 0.96685835 0.02199973Tukey’s h-distribution
moment_GH(A = 3, B = 1.5, g = .7, h = 0)
#> Mean: 3.595
#> Standard Deviation: 2.177
#> Skewness: 2.888
#> (Excess) Kurtosis: 17.791moment2GH(skewness = .2, h = 0)
#> A B g
#> -0.03317378 0.99668770 0.06650087Tukey’s g-and-h Distribution
moment_GH(A = 3, B = 1.5, g = .7, h = .05)
#> Mean: 3.647
#> Standard Deviation: 2.468
#> Skewness: 4.082
#> (Excess) Kurtosis: 47.756moment2GH(skewness = .2, kurtosis = .3)
#> A B g h
#> -0.03065172 0.97137360 0.06145358 0.01708765Gamma Distribution
Batch Visualization
skw = c(.2, .5, .8)
krt = c(.5, 1, 1.5)Kurtosis only
Student (1908)-t vs. Tukey’s h
R function
sn::dst(Azzalini 2023) scale \omega, degree-of-freedom \nu\mu = 0, \sigma = 1, Kurtosis=.5, 1, or 1.5
[R] Student-t and Tukey-h
p1 = moment2param(distname = 'st', MoreArgs = list(alpha = 0), kurtosis = krt) +
xlim(-1.5, 1.5) + labs(y = latex2exp::TeX('Student-$t$'))
p2 = moment2param(distname = 'GH', MoreArgs = list(g = 0), kurtosis = krt) +
xlim(-1.5, 1.5) + labs(y = latex2exp::TeX('Tukey\'s $h$'))
p1 + p2
Skewness only
Skew-Normal vs. Tukey’s g
R function
sn::dsnlocation \xi, scale \omega, slant \alpha\mu = 0, \sigma = 1, Skewness=.2, .5, or .8
[R] Skew-Normal and Tukey-g
p1 = moment2param(dist = 'sn', skewness = skw, hjust = .42) + xlim(-2, 2.5) + labs(y = 'Skew-Normal')
p2 = moment2param(dist = 'GH', MoreArgs = list(h = 0), skewness = skw, hjust = .42) +
xlim(-2, 2.5) + labs(y = latex2exp::TeX('Tukey\'s $g$'))
p1 + p2
Skewness & Kurtosis
Skew-t vs. Tukey’s g-&-h
R function
sn::dstlocation \xi, scale \omega, slant \alpha, degree-of-freedom \nu\mu = 0, \sigma = 1, (Skewness, Kurtosis)=(.2,.5), (.5,1), or (.8,1.5)
[R] Skew-t and Tukey-gh
p1 = moment2param(distname = 'st', skewness = skw, kurtosis = krt, hjust = .35) +
xlim(-2, 2.5) + labs(y = latex2exp::TeX('Skew-$t$'))
p2 = moment2param(distname = 'GH', skewness = skw, kurtosis = krt, hjust = .35) +
xlim(-2, 2.5) + labs(y = latex2exp::TeX('Tukey\'s $g$-&-$h$'))
p1 + p2
[R] Gamma
moment2param(distname = 'gamma', skewness = skw, kurtosis = 3, hjust = .4) +
xlim(0, 4) + labs(y = latex2exp::TeX('$gamma$'))
Appendix
Terms and Abbreviations
| Term / Abbreviation | Description |
|---|---|
|> |
Forward pipe operator introduced in R 4.1.0 |
all.equal.numeric |
Test of near equality |
CRAN, R |
The Comprehensive R Archive Network |
curve |
Function plots |
.lm.fit |
Least squares regression, the internal workhorse |
mad |
Median Absolute Deviation |
median |
Median |
dnorm, pnorm, qnorm, rnorm |
Normal Distribution |
optim, optimize |
Optimization |
sd |
Standard Deviation |
search |
Search path |
seed |
Random number generation seed |
vuniroot |
Vectorised one-dimensional root-finding, in package rstpm2 |
References
Azzalini, Azzalini A. 2023. The R Package
sn: The Skew-Normal and Related Distributions Such as the Skew-t and the SUN (Version 2.1.1). Università degli Studi di Padova, Italia. https://cran.r-project.org/package=sn.
Student. 1908. “The Probable Error of a Mean.” Biometrika 6 (1): 1–25.