kmoments {FatTailsR} | R Documentation |
Moments Associated To Kiener Distribution Parameters
Description
Non-central moments, central moments, mean, standard deviation, variance,
skewness, kurtosis, excess of kurtosis and cumulants associated to
the parameters of Kiener distributions K1, K2, K3 and K4.
All-in-one vectors kmoments
(estimated from the parameters)
and xmoments
(estimated from the vector of quantiles) are provided.
Usage
kmoments(coefk, model = "K2", lengthx = NA, dgts = NULL,
dimnames = FALSE)
xmoments(x, dgts = NULL, dimnames = FALSE)
kmoment(n, coefk, model = "K2", dgts = NULL)
kcmoment(n, coefk, model = "K2", dgts = NULL)
kmean(coefk, model = "K2", dgts = NULL)
kstandev(coefk, model = "K2", dgts = NULL)
kvariance(coefk, model = "K2", dgts = NULL)
kskewness(coefk, model = "K2", dgts = NULL)
kkurtosis(coefk, model = "K2", dgts = NULL)
kekurtosis(coefk, model = "K2", dgts = NULL)
Arguments
coefk |
vector. Parameters of the distribution of length 3 ("K1"), length 4 (model = K2, K3, K4) and length 7 ("K7"). |
model |
character. Model type, either "K2", "K3" or "K4" if |
lengthx |
integer. The length of the vector |
dgts |
integer. The rounding applied to the output. |
dimnames |
boolean. Display dimnames. |
x |
numeric. Vector of quantiles. |
n |
integer. The moment order. |
Details
The non-central moments m1,m2,m3,m4,..,mn
,
the central moments u1,u2,u3,u4,..,un
(where u stands for mu in Greek)
and the cumulants k1,k2,k3,k4,..,kn
(where k stands for kappa in Greek;
not to be confounded with tail parameter "k" and models "K1", "K2", "K3", "K4")
of order exist only if
.
The mean
m1
exists only if .
The standard deviation
sd
and the variance u2
exist only
if .
The skewness
sk
exists only if .
The kurtosis
ku
and the excess of kurtosis ke
exist only
if .
coefk
may take five different forms :
c(m, g, k)
of length 3 for distribution "K1".c(m, g, a, w)
of length 4 for distribution "K2".c(m, g, k, d)
of length 4 for distribution "K3".c(m, g, k, e)
of length 4 for distribution "K4".c(m, g, a, k, w, d, e)
of length 7 (sometimes referred as "K7") provided by estimation/regression functionsparamkienerX
,fitkienerX
,regkienerLX
(via"reg$coefk"
) and conversion functionpk2pk
.
Forms of length 3 and 7 are automatically recognized and do not require
model = "K1"
or "K7"
which are ignored.
Forms of length 4 require model = "K2"
, "K3"
or "K4"
.
Visit pk2pk
for details on the parameter conversion function
used within kmoments
.
xmoments
and kmoments
provide all-in-one vectors.
xmoments
is the traditional mean of squares, cubic and power 4 functions
of non-central and central values of x, from which NA values have been removed.
Therefore, length of x ignores NA values and may be different from the true length.
kmoments
calls every specialized functions from order 1 to order 4 and
uses the estimated parameters as inputs, not the initial dataset x
.
As it does not know a priori the length of x
, this latest can
be provided separately via lengthx = length(x)
, lengthx = nrow(x)
and lengthx = sapply(x, length)
if x
is a vector, a matrix or a list.
See the examples.
Value
Vectors kmoments
and xmoments
have the following structure
(with a third letter x
added to xmoments
):
ku |
Kurtosis. |
ke |
Excess of kurtosis. |
sk |
Skewness. |
sd |
Standard deviation. Square root of the variance |
m1 |
Mean. |
m2 |
Non-central moment of second order. |
m3 |
Non-central moment of third order. |
m4 |
Non-central moment of fourth order. |
u1 |
Central moment of first order. Should be 0. |
u2 |
Central moment of second order. Variance |
u3 |
Central moment of third order. |
u4 |
Central moment of fourth order. |
k1 |
Cumulant of first order. Should be 0. |
k2 |
Cumulant of second order. |
k3 |
Cumulant of third order. |
k4 |
Cumulant of fourth order. |
lh |
Length of x, from which NA values were removed. |
...... |
. |
See Also
pk2pk
, paramkienerX
, regkienerLX
.
Examples
## Example 1
kcmoment(2, c(-1, 1, 6, 9), model = "K2")
kcmoment(2, c(-1, 1, 7.2, -0.2/7.2), model = "K3")
kcmoment(2, c(-1, 1, 7.2, -0.2), model = "K4")
kcmoment(2, c(-1, 1, 6, 7.2, 9, -0.2/7.2, -0.2))
kvariance(c(-1, 1, 6, 9))
kmoments(c(-1, 1, 6, 9), dgts = 3)
## Example 2: "K2" and "K7" are preferred input formats for kmoments
## Moments fall at expected parameter values (=> NA).
## apply and direct calculation (= transpose)
(mat4 <- matrix(c(rep(0,4), rep(1,4), c(1.9,2.1,3.9,4.1), rep(5,4)),
nrow = 4, byrow = TRUE,
dimnames = list(c("m","g","a","w"), paste0("b",1:4))))
round(mat7 <- apply(mat4, 2, pk2pk), 2)
round(rbind(mat7, apply(mat7, 2, kmoments)[2:5,]), 2)
round(cbind(t(mat7), kmoments(t(mat7), dgts = 2)[,2:5]), 2)
## Example 3: Matrix, timeSeries, xts, zoo + apply
matret <- 100*diff(log((EuStockMarkets)))
(matcoefk <- apply(matret, 2, paramkienerX5, dgts = 2))
(matmomk <- apply(matcoefk, 2, kmoments, lengthx = nrow(matret), dgts = 2))
(matmomx <- apply(matret, 2, xmoments, dgts = 2))
rbind(matcoefk, matmomk[2:5,], matmomx[2:5,])
## Example 4: List + direct calculation = transpose
DS <- getDSdata() ; dimdim(DS) ; class(DS)
(pDS <- paramkienerX5(DS, dimnames = FALSE))
(kDS <- kmoments(pDS, lengthx = sapply(DS, length), dgts = 3))
(xDS <- xmoments( DS, dgts = 3))
cbind(pDS, kDS[,2:5], xDS[,2:5])