kmoments: 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)

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 u2
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.
......: .

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 coefk is of length 4. Type "K1" and "K7" may be provided but are ignored.
lengthx: integer. The length of the vector x used to calculate the parameters. See the details for matrix and lists.
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 $n$ exist only if $min(a, k, w) > n$. The mean m1 exists only if $min(a, k, w) > 1$. The standard deviation sd and the variance u2 exist only if $min(a, k, w) > 2$. The skewness sk exists only if $min(a, k, w) > 3$. The kurtosis ku and the excess of kurtosis ke exist only if $min(a, k, w) > 4$.

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 functions paramkienerX, fitkienerX, regkienerLX (via "reg$coefk") and conversion function pk2pk.

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.

Examples

Run this code


## 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() ; 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])