# Var: Generic Functions for the Computation of Functionals In distrEx: Extensions of Package 'distr'

## Description

Generic functions for the computation of functionals on distributions.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230``` ```IQR(x, ...) ## S4 method for signature 'UnivariateDistribution' IQR(x) ## S4 method for signature 'UnivariateCondDistribution' IQR(x,cond) ## S4 method for signature 'AffLinDistribution' IQR(x) ## S4 method for signature 'DiscreteDistribution' IQR(x) ## S4 method for signature 'Arcsine' IQR(x) ## S4 method for signature 'Cauchy' IQR(x) ## S4 method for signature 'Dirac' IQR(x) ## S4 method for signature 'DExp' IQR(x) ## S4 method for signature 'Exp' IQR(x) ## S4 method for signature 'Geom' IQR(x) ## S4 method for signature 'Logis' IQR(x) ## S4 method for signature 'Norm' IQR(x) ## S4 method for signature 'Unif' IQR(x) median(x, ...) ## S4 method for signature 'UnivariateDistribution' median(x) ## S4 method for signature 'UnivariateCondDistribution' median(x,cond) ## S4 method for signature 'AffLinDistribution' median(x) ## S4 method for signature 'Arcsine' median(x) ## S4 method for signature 'Cauchy' median(x) ## S4 method for signature 'Dirac' median(x) ## S4 method for signature 'DExp' median(x) ## S4 method for signature 'Exp' median(x) ## S4 method for signature 'Geom' median(x) ## S4 method for signature 'Logis' median(x) ## S4 method for signature 'Lnorm' median(x) ## S4 method for signature 'Norm' median(x) ## S4 method for signature 'Unif' median(x) mad(x, ...) ## S4 method for signature 'UnivariateDistribution' mad(x) ## S4 method for signature 'AffLinDistribution' mad(x) ## S4 method for signature 'Cauchy' mad(x) ## S4 method for signature 'Dirac' mad(x) ## S4 method for signature 'DExp' mad(x) ## S4 method for signature 'Exp' mad(x) ## S4 method for signature 'Geom' mad(x) ## S4 method for signature 'Logis' mad(x) ## S4 method for signature 'Norm' mad(x) ## S4 method for signature 'Unif' mad(x) ## S4 method for signature 'Arcsine' mad(x) sd(x, ...) ## S4 method for signature 'UnivariateDistribution' sd(x, fun, cond, withCond, useApply, ...) ## S4 method for signature 'Norm' sd(x, fun, cond, withCond = FALSE, useApply = TRUE, ...) var(x, ...) ## S4 method for signature 'UnivariateDistribution' var(x, fun, cond, withCond, useApply, ...) ## S4 method for signature 'AffLinDistribution' var(x, fun, cond, withCond, useApply, ...) ## S4 method for signature 'CompoundDistribution' var(x, ...) ## S4 method for signature 'Arcsine' var(x, ...) ## S4 method for signature 'Binom' var(x, ...) ## S4 method for signature 'Beta' var(x, ...) ## S4 method for signature 'Cauchy' var(x, ...) ## S4 method for signature 'Chisq' var(x, ...) ## S4 method for signature 'Dirac' var(x, ...) ## S4 method for signature 'DExp' var(x, ...) ## S4 method for signature 'Exp' var(x, ...) ## S4 method for signature 'Fd' var(x, ...) ## S4 method for signature 'Gammad' var(x, ...) ## S4 method for signature 'Geom' var(x, ...) ## S4 method for signature 'Hyper' var(x, ...) ## S4 method for signature 'Logis' var(x, ...) ## S4 method for signature 'Lnorm' var(x, ...) ## S4 method for signature 'Nbinom' var(x, ...) ## S4 method for signature 'Norm' var(x, ...) ## S4 method for signature 'Pois' var(x, ...) ## S4 method for signature 'Td' var(x, ...) ## S4 method for signature 'Unif' var(x, ...) ## S4 method for signature 'Weibull' var(x, ...) skewness(x, ...) ## S4 method for signature 'UnivariateDistribution' skewness(x, fun, cond, withCond, useApply, ...) ## S4 method for signature 'AffLinDistribution' skewness(x, fun, cond, withCond, useApply, ...) ## S4 method for signature 'Arcsine' skewness(x, ...) ## S4 method for signature 'Binom' skewness(x, ...) ## S4 method for signature 'Beta' skewness(x, ...) ## S4 method for signature 'Cauchy' skewness(x, ...) ## S4 method for signature 'Chisq' skewness(x, ...) ## S4 method for signature 'Dirac' skewness(x, ...) ## S4 method for signature 'DExp' skewness(x, ...) ## S4 method for signature 'Exp' skewness(x, ...) ## S4 method for signature 'Fd' skewness(x, ...) ## S4 method for signature 'Gammad' skewness(x, ...) ## S4 method for signature 'Geom' skewness(x, ...) ## S4 method for signature 'Hyper' skewness(x, ...) ## S4 method for signature 'Logis' skewness(x, ...) ## S4 method for signature 'Lnorm' skewness(x, ...) ## S4 method for signature 'Nbinom' skewness(x, ...) ## S4 method for signature 'Norm' skewness(x, ...) ## S4 method for signature 'Pois' skewness(x, ...) ## S4 method for signature 'Td' skewness(x, ...) ## S4 method for signature 'Unif' skewness(x, ...) ## S4 method for signature 'Weibull' skewness(x, ...) kurtosis(x, ...) ## S4 method for signature 'UnivariateDistribution' kurtosis(x, fun, cond, withCond, useApply, ...) ## S4 method for signature 'AffLinDistribution' kurtosis(x, fun, cond, withCond, useApply, ...) ## S4 method for signature 'Arcsine' kurtosis(x, ...) ## S4 method for signature 'Binom' kurtosis(x, ...) ## S4 method for signature 'Beta' kurtosis(x, ...) ## S4 method for signature 'Cauchy' kurtosis(x, ...) ## S4 method for signature 'Chisq' kurtosis(x, ...) ## S4 method for signature 'Dirac' kurtosis(x, ...) ## S4 method for signature 'DExp' kurtosis(x, ...) ## S4 method for signature 'Exp' kurtosis(x, ...) ## S4 method for signature 'Fd' kurtosis(x, ...) ## S4 method for signature 'Gammad' kurtosis(x, ...) ## S4 method for signature 'Geom' kurtosis(x, ...) ## S4 method for signature 'Hyper' kurtosis(x, ...) ## S4 method for signature 'Logis' kurtosis(x, ...) ## S4 method for signature 'Lnorm' kurtosis(x, ...) ## S4 method for signature 'Nbinom' kurtosis(x, ...) ## S4 method for signature 'Norm' kurtosis(x, ...) ## S4 method for signature 'Pois' kurtosis(x, ...) ## S4 method for signature 'Td' kurtosis(x, ...) ## S4 method for signature 'Unif' kurtosis(x, ...) ## S4 method for signature 'Weibull' kurtosis(x, ...) ```

## Arguments

 `x` object of class `"UnivariateDistribution"` `fun` if missing the (conditional) variance resp. standard deviation is computed else the (conditional) variance resp. standard deviation of `fun` is computed. `cond` if not missing the conditional variance resp. standard deviation given `cond` is computed. `...` additional arguments to `fun` or `E` `useApply` logical: should `sapply`, respectively `apply` be used to evaluate `fund`. `withCond` logical: is `cond` in the argument list of `fun`.

## Value

The value of the corresponding functional at the distribution in the argument is computed.

## Methods

`var`, `signature(x = "Any")`:

interface to the stats-function `var` — see `var` resp. `help(var,package="stats")`.

`var`, `signature(x = "UnivariateDistribution")`:

variance of univariate distributions using corresponding `E()`-method.

`var`, `signature(x = "AffLinDistribution")`:

if arguments `fun`, `cond` are missing: `[email protected]^2 * var([email protected])` else uses method for `signature(x = "UnivariateDistribution")`

`var`, `signature(x = "CompoundDistribution")`:

if we are in i.i.d. situation (i.e., slot `SummandsDistr` is of class `UnivariateDistribution`) the formula E[N]var[S]+E[S*S]var(N) for N the frequency distribution and S the summand distribution; else we coerce to `"UnivarLebDecDistribution"`.

`sd`, `signature(x = "Any")`:

interface to the stats-function `sd` — see `sd` resp. `help(sd,package="stats")`.

`sd`, `signature(x = "NormParameter")`:

returns the slot `sd` of the parameter of a normal distribution — see `sd` resp. `help(sd,package="distr")`.

`sd`, `signature(x = "Norm")`:

returns the slot `sd` of the parameter of a normal distribution — see `sd` resp. `help(sd,package="distr")`.

`sd`, `signature(x = "UnivariateDistribution")`:

standard deviation of univariate distributions using corresponding `E()`-method.

`IQR`, `signature(x = "Any")`:

interface to the stats-function `IQR` — see `IQR` resp. `help(IQR,package="stats")`.

`IQR`, `signature(x = "UnivariateDistribution")`:

interquartile range of univariate distributions using corresponding `q()`-method.

`IQR`, `signature(x = "UnivariateCondDistribution")`:

interquartile range of univariate conditional distributions using corresponding `q()`-method.

`IQR`, `signature(x = "DiscreteDistribution")`:

interquartile range of discrete distributions using corresponding `q()`-method but taking care that between upper and lower quartile there is 50% probability

`IQR`, `signature(x = "AffLinDistribution")`:

`abs([email protected]) * IQR([email protected])`

`median`, `signature(x = "Any")`:

interface to the stats-function `median` — see `median` resp. `help(var,package="stats")`.

`median`, `signature(x = "UnivariateDistribution")`:

median of univariate distributions using corresponding `q()`-method.

`median`, `signature(x = "UnivariateCondDistribution")`:

median of univariate conditional distributions using corresponding `q()`-method.

`median`, `signature(x = "AffLinDistribution")`:

`[email protected] * median([email protected]) + [email protected] `

`mad`, `signature(x = "Any")`:

interface to the stats-function `mad` — see `mad`.

`mad`, `signature(x = "UnivariateDistribution")`:

mad of univariate distributions using corresponding `q()`-method applied to `abs(x-median(x))`.

`mad`, `signature(x = "AffLinDistribution")`:

`abs([email protected]) * mad([email protected])`

`skewness`, `signature(x = "Any")`:

bias free estimation of skewness under normal distribution (default) as well as sample version (by argument `sample.version = TRUE`).

`skewness`, `signature(x = "UnivariateDistribution")`:

skewness of univariate distributions using corresponding `E()`-method.

`skewness`, `signature(x = "AffLinDistribution")`:

if arguments `fun`, `cond` are missing: `skewness([email protected])` else uses method for `signature(x = "UnivariateDistribution")`

`kurtosis`, `signature(x = "Any")`:

bias free estimation of kurtosis under normal distribution (default) as well as sample version (by argument `sample.version = TRUE`).

`kurtosis`, `signature(x = "UnivariateDistribution")`:

kurtosis of univariate distributions using corresponding `E()`-method.

`kurtosis`, `signature(x = "AffLinDistribution")`:

if arguments `fun`, `cond` are missing: `kurtosis([email protected])` else uses method for `signature(x = "UnivariateDistribution")`

`var`, `signature(x = "Arcsine")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Beta")`:

for noncentrality 0 exact evaluation using explicit expressions.

`var`, `signature(x = "Binom")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Cauchy")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Chisq")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Dirac")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "DExp")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Exp")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Fd")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Gammad")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Geom")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Hyper")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Logis")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Lnorm")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Nbinom")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Norm")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Pois")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Td")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Unif")`:

exact evaluation using explicit expressions.

`var`, `signature(x = "Weibull")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Arcsine")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Cauchy")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Dirac")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "DExp")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Exp")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Geom")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Logis")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Norm")`:

exact evaluation using explicit expressions.

`IQR`, `signature(x = "Unif")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Arcsine")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Cauchy")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Dirac")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "DExp")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Exp")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Geom")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Logis")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Lnorm")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Norm")`:

exact evaluation using explicit expressions.

`median`, `signature(x = "Unif")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Arcsine")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Cauchy")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Dirac")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "DExp")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Exp")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Geom")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Logis")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Norm")`:

exact evaluation using explicit expressions.

`mad`, `signature(x = "Unif")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Arcsine")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Beta")`:

for noncentrality 0 exact evaluation using explicit expressions.

`skewness`, `signature(x = "Binom")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Cauchy")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Chisq")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Dirac")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "DExp")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Exp")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Fd")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Gammad")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Geom")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Hyper")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Logis")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Lnorm")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Nbinom")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Norm")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Pois")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Td")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Unif")`:

exact evaluation using explicit expressions.

`skewness`, `signature(x = "Weibull")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Arcsine")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Beta")`:

for noncentrality 0 exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Binom")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Cauchy")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Chisq")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Dirac")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "DExp")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Exp")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Fd")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Gammad")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Geom")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Hyper")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Logis")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Lnorm")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Nbinom")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Norm")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Pois")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Td")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Unif")`:

exact evaluation using explicit expressions.

`kurtosis`, `signature(x = "Weibull")`:

exact evaluation using explicit expressions.

## Caveat

If any of the packages e1071, moments, fBasics is to be used together with distrEx the latter must be attached after any of the first mentioned. Otherwise `kurtosis()` and `skewness()` defined as methods in distrEx may get masked.
To re-mask, you may use `kurtosis <- distrEx::kurtosis; skewness <- distrEx::skewness`. See also `distrExMASK()`.

## Acknowledgement

G. Jay Kerns, [email protected], has provided a major contribution, in particular the functionals `skewness` and `kurtosis` are due to him.

## Author(s)

Peter Ruckdeschel [email protected]

`distrExIntegrate`, `m1df`, `m2df`, `Distribution-class`,
`sd`, `var`, `IQR`,
`median`, `mad`, `sd`,
`Sn`, `Qn`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```# Variance of Exp(1) distribution var(Exp()) #median(Exp()) IQR(Exp()) mad(Exp()) # Variance of N(1,4)^2 var(Norm(mean=1, sd=2), fun = function(x){x^2}) var(Norm(mean=1, sd=2), fun = function(x){x^2}, useApply = FALSE) ## sd -- may equivalently be replaced by var sd(Pois()) ## uses explicit terms sd(as(Pois(),"DiscreteDistribution")) ## uses sums sd(as(Pois(),"UnivariateDistribution")) ## uses simulations sd(Norm(mean=2), fun = function(x){2*x^2}) ## uses simulations # mad(sin(exp(Norm()+2*Pois()))) ## weird ```

distrEx documentation built on May 30, 2017, 3:09 a.m.