copBasic (version 2.0.1)

uvlmoms: Bivariate Skewness after Joe (2014) or the Univariate L-moments of Combined U and V

Description

Joe (2014, pp. 65--66) suggests two quantile-based measures of bivariate skewness defined for uniform random variables $U$ and $V$ combined as either $\psi_{u+v-1} = u + v - 1$ or $\psi_{u-v} = u - v$ for which the $\mathrm{E}[u] = \mathrm{E}[v] = 0$. The bivariate skewness is the quantity $\eta$:

$$\eta(p; \psi) = \frac{x(1-p) - 2x(\frac{1}{2}) + x(p)}{x(1-p) - x(p)} \mbox{,}$$

where $0 < p < \frac{1}{2}$, $x(F)$ is the quantile function for nonexceedance probability $F$ for either the quantities $X = \psi_{u+v-1}$ or $X = \psi_{u-v}$ using either the empirical quantile function or a fitted distribution. Joe (2014, p. 66) reports that $p = 0.05$ to achieve some sensitivity to the tails. How these might be related (intuitively) to L-coskew (see function lcomoms2() of the lmomco package) of the L-comoments is unknown, but see the Examples section of joeskewCOP.

Structurally the above definition for $\eta$ based on quantiles is oft shown in comparative literature concerning L-moments. But why stop there? Why not compute the L-moments themselves to arbitrary order for $\eta$ by either definition (the uvlmoms variation)? Why not fit a distribution to the computed L-moments for estimation of $x(F)$? Or simply compute skewness according to the definition above (the uvskew variation).

Usage

uvlmoms(u,v=NULL, umv=TRUE, p=NA,   type="gno", getlmoms=TRUE,  ...)
uvskew( u,v=NULL, umv=TRUE, p=0.05, type=6,     getlmoms=FALSE, ...)

Arguments

u

Nonexceedance probability $u$ in the $X$ direction;

v

Nonexceedance probability $v$ in the $Y$ direction and if NULL then u is treated as a two column Rdata.frame;

umv

A logical controlling the computation of $\psi$: $\psi = u - v$ (umv = TRUE) or $\psi = u + v - 1$ (umv = FALSE). The "m" is to read minus;

p

A suggested $p$ value is p = 0.05. If is.na(NA), then getlmoms is set to TRUE (see below);

type

The type argument is mutable, and is a syntax match to the canoncial use in package lmomco. Variation from that package however is permitted. Either type is an integer between 1 and 9 selecting one of the nine quantile

getlmoms

A logical triggering whether the L-moments of either $\psi_{u+v-1}$ or $\psi_{u - v}$ are returned instead computing the above definition of skewness; and

...

Additional arguments to pass to the lmomco function lmoms, such as the number of L-moments nmoms.

Value

• An Rlist of the univariate L-moments of $\eta$ is returned (see documentation for lmoms in the lmomco package). Or the skewness of $\eta$ can be either (1) based on the empirical distribution based on plotting positions by the quantile function in Rusing the type as described, or (2) based on the fitted quantile function for the parameters of a distribution for the lmomco package.

encoding

utf8

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

See Also

COP

Examples

set.seed(234)
UV <- simCOP(n=100, cop=GHcop, para=1.5, graphics=FALSE)
lmr <- uvlmoms(UV); print(lmr) # L-kurtosis = 0.16568268
uvskew(UV, p=0.10)             # -0.1271723
uvskew(UV, p=0.10, type="gno") # -0.1467011

ps <- seq(0.01,0.49, by=0.01)
ETA <- sapply(1:length(ps), function(i) uvskew(UV, p=ps[i], type=5, uvm1=FALSE) )
plot(ps, ETA, type="l", xlab="P FACTOR", ylab="BIVARIATE SKEWNESS") #