moment-class {param2moment} | R Documentation |
Up to 4th raw \text{E}(Y^n)
, central \text{E}[(Y-\mu)^n]
and
standardized moments \text{E}[(Y-\mu)^n/\sigma^n]
of the random variable
Y = (X - \text{location})/\text{scale}
Also, the mean, standard deviation, skewness and excess kurtosis of the random variable X
.
For Y = (X - \text{location})/\text{scale}
,
let \mu = \text{E}(Y)
, then, according to
Binomial theorem,
the 2nd to 4th central moments of Y
are,
\text{E}[(Y-\mu)^2] = \text{E}(Y^2) - 2\mu \text{E}(Y) + \mu^2 = \text{E}(Y^2) - \mu^2
\text{E}[(Y-\mu)^3] = \text{E}(Y^3) - 3\mu \text{E}(Y^2) + 3\mu^2 \text{E}(Y) - \mu^3 = \text{E}(Y^3) - 3\mu \text{E}(Y^2) + 2\mu^3
\text{E}[(Y-\mu)^4] = \text{E}(Y^4) - 4\mu \text{E}(Y^3) + 6\mu^2 \text{E}(Y^2) - 4\mu^3 \text{E}(Y) + \mu^4 = \text{E}(Y^4) - 4\mu \text{E}(Y^3) + 6\mu^2 \text{E}(Y^2) - 3\mu^4
The distribution characteristics of Y
are,
\mu_Y = \mu
\sigma_Y = \sqrt{\text{E}[(Y-\mu)^2]}
\text{skewness}_Y = \text{E}[(Y-\mu)^3] / \sigma^3_Y
\text{kurtosis}_Y = \text{E}[(Y-\mu)^4] / \sigma^4_Y - 3
The distribution characteristics of X
are
\mu_X = \text{location} + \text{scale}\cdot \mu_Y
,
\sigma_X = \text{scale}\cdot \sigma_Y
,
\text{skewness}_X = \text{skewness}_Y
, and
\text{kurtosis}_X = \text{kurtosis}_Y
.
distname
character scalar, name of distribution,
e.g., 'norm'
for normal, 'sn'
for skew-normal, 'st'
for skew-t
,
and 'GH'
for Tukey g
-&-h
distribution,
following the nomenclature of dnorm, dsn, dst and QuantileGH::dGH
location,scale
mu
numeric scalar or vector,
1st raw moment \mu = \text{E}(Y)
.
Note that the 1st central moment \text{E}(Y-\mu)
and
standardized moment \text{E}(Y-\mu)/\sigma
are both 0.
raw2,raw3,raw4
numeric scalars or vectors,
2nd or higher raw moments \text{E}(Y^n)
, n\geq 2
central2,central3,central4
numeric scalars or vectors,
2nd or higher central moments, \sigma^2 = \text{E}[(Y-\mu)^2]
and
\text{E}[(Y-\mu)^n]
, n\geq 3
standardized3,standardized4
numeric scalars or vectors,
3rd or higher standardized moments,
skewness \text{E}[(Y-\mu)^3]/\sigma^3
and
kurtosis \text{E}[(Y-\mu)^4]/\sigma^4
.
Note that the 2nd standardized moment is 1
Potential name clash with function e1071::moment
.