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 or bivariate L-moments (bilmoms) 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).
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, ...)
An R
list 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 R using the type as described, or (2) based on the fitted quantile function for the parameters of a distribution for the lmomco package.
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.
Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.
COP