R: Sample and theoretical Moment and Centile Skewness and...

momentSK {gamlss.dist} | R Documentation

## Sample and theoretical Moment and Centile Skewness and Kurtosis Functions

### Description

The functions `momentSK()`, `centileSK()`, `centileSkew()` and `centileKurt()`, calculate sample statistics related to skewness and kurtosis. The function `theoCentileSK()` calculates the theoretical centile statistics from a given `gamlss.family` distribution. The `plotCentileSK()` plots the theoretical centile skewness and kurtosis against `p` (see below).

The function `checkMomentSK()` can be use to check (a) whether the moment skewness and kurtosis of a fitted model are modelled adequantly (the residuals of the model are used). (b) whether a given sample display skewness or kurtosis.

### Usage

    momentSK(x,  weights=NULL)
    centileSK(x, cent = c(1, 25), weights=NULL)
    centileSkew(x, cent = 1, weights=NULL)
    centileKurt(x, cent = 1, weights=NULL)

    theoCentileSK(fam = "NO", p = 0.01, ...)
    plotCentileSK(fam = "NO", plotting = c("skew", "kurt", "standKurt"),
                 add = FALSE, col = 1, lty = 1, lwd = 1, ylim = NULL, ...)

    checkMomentSK(x, weights=NULL, add = FALSE, bootstrap = TRUE, no.bootstrap = 99,
                   col.bootstrap = "lightblue", pch.bootstrap = 21,
                   asCharacter = TRUE, col.point = "black", pch.point = 4,
                   lwd.point = 2,  text.to.show = NULL, cex.text = 1.5,
                   col.text = "black", show.legend = TRUE)

    checkCentileSK(x,weights=NULL, type = c("central", "tail"), add = FALSE,
                  bootstrap = TRUE, no.bootstrap = 99,
                  col.bootstrap = "lightblue", pch.bootstrap = 21,
                  asCharacter = TRUE, col.point = "black", pch.point = 4,
                  lwd.point = 2,  text.to.show = NULL, cex.text = 1.5,
                  col.text = "black", show.legend = TRUE)

### Arguments

`x` | data vector or gamlss model
`weights` | prior weights for the x
`cent` | the centile required
`type` | For centile skewness and kurtosis only whether "central" (default) or "tail")
`fam` | A gamlss distribution family
`plotting` | what to plot
`add` | whether to add the line to the existing plot
`col` | the colour of the line
`lty` | the type of the line
`lwd` | the width of the line
`ylim` | the y limit of the graph
`p` | the value determiming the centile skewness or kurtosis
`...` | additional arguments pass to `theoCentileSK()` function i.e. the values of the distribution parameters
`bootstrap` | whether a plot of the bootstrap skewness and kurtosis measures should be added in the plot
`no.bootstrap` | the number of boostrap skewness and kurtosis measures
`col.bootstrap` | the coloue for boostraps
`pch.bootstrap` | the point type of boostraps
`asCharacter` | whether to plot the estimated skewness and kurtosis measure as character or as point
`col.point` | the colour of the skewness and kurtosis measure
`pch.point` | the point type of the skewness and kurtosis measure
`lwd.point` | the width of the plotted point
`text.to.show` | to display text different from variable or model
`cex.text` | the size of the text
`col.text` | the colour of the text
`show.legend` | whether to show the legent

### Details

Those function calculate sample moment and centile skewness and kurtosis statistics and theoretical centile values for a specific distribution.

### Value

Different functions produce different output: The function `momentSK()` produce:

`mom.skew:` | sample moment skewness
`trans.mom.skew:` | sample transformed moment skewness
`mom.kurt:` | sample moment kurtosis
`excess.mom.kurt:` | sample excess moment kurtosis
`trans.mom.kurt:` | sample ransformed moment excess kurtosis
`jarque.bera.test:` | the value of the Jarque-bera test for testing whether skewness and excess kurtosis are zero or not

The function `centileSK()` produces:

`S0.25:` | sample centile central skewness
`S0.01:` | sample centile tail skewness
`trans.S0.25:` | sample centile transformed central skewness
`trans.S0.01:` | sample centile transformed tail skewness
`K0.01:` | sample centile kurtosis
`standK0.01:` | standardise centile kurtosis, (`K0.01/3.449`)
`exc.K0.01:` | excess centile kurtosis, (`K0.01-3.449`)
`trans.K0.01:` | transfored excess centile kurtosis, (exc.K0.01/(1+abs(exc.K0.01))

The function `centileSkew()` for a given argument `p` produces:

`p:` | the value determiming the centile skewness
`Sp:` | sample centile skewness at `p`
`tSp:` | sample transformed centile skewness at `p`

The function `centileKurt()` for a given argument `p` produces:

`p` | the value determiming the centile kurtosis
`Kp` | sample centile kurtosis at `p`
`sKp` | sample standardise centile kurtosis at `p`
`ex.Kp:` | sample excess centile kurtosis at `p`
`teKp:` | sample transformed excess centile kurtosis at `p`

The function `theoCentileSK` for a given `gamlss.family` produces:

`IR` | the interquartile range of the distribution
`SIR` | the semi interquartile range of the distribution
`S_0.25` | the central skewness of the distribution
`S_0.01:` | the tail skewness of the distribution
`K_0.01:` | the centile kurtosis of the distribution
`sK_0.01:` | the standardise centile kurtosis of the distribution

### Author(s)

Mikis Stasinopoulos, Bobert Rigby, Gillain Heller and Fernanda De Bastiani.

### References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, https://www.jstatsoft.org/v23/i07/.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.

(see also https://www.gamlss.com/).

### See Also

`gamlss.family`

### Examples

    Y <- rSEP3(1000)
    momentSK(Y)
    centileSK(Y)
    centileSkew(Y, cent=20)
    centileKurt(Y, cent=30)

    theoCentileSK("BCCG", mu=2, sigma=.2, nu=2)
    plotCentileSK(fam="BCCG",  mu=2, sigma=.2, nu=2)

    checkMomentSK(Y)
    checkCentileSK(Y)
    checkCentileSK(Y, type="tail")

[Package gamlss.dist version 5.3-2 Index]