Compute a single L-moment from a cumulative distribution function. This function is sequentially called by cdf2lmoms to mimic the output structure for multiple L-moments seen by other L-moment computation functions in lmomco.
For \(r = 1\), the quantile function is actually used for numerical integration to compute the mean. The expression for the mean is $$ \lambda_1 = \int_0^1 x(F)\; \mathrm{d} F\mbox{,} $$ for quantile function \(x(F)\) and nonexceedance probability \(F\). For \(r \ge 2\), the L-moments can be computed from the cumulative distribution function \(F(x)\) by $$ \lambda_r = \frac{1}{r}\sum_{j=0}^{r-2} (-1)^j {r-2 \choose j}{r \choose j+1} \int_{-\infty}^{\infty} \! [F(x)]^{r-j-1}\times [1 - F(x)]^{j+1}\; \mathrm{d}x\mbox{.} $$ This equation is described by Asquith (2011, eq. 6.8), Hosking (1996), and Jones (2004).
cdf2lmom(r, para, fdepth=0, silent=TRUE, ...)The value for the requested L-moment is returned (\(\lambda_r\)).
The order of the L-moment.
The parameters from lmom2par or similar.
The depth of the nonexceedance/exceedance probabilities to determine the lower and upper integration limits for the integration involving \(F(x)\) through a call to the par2qua function. The default of 0 implies the quantile for \(F=0\) and quantile for \(F=1\) as the respective lower and upper limits.
A logical to be passed into cdf2lmom and then onto the try functions encompassing the integrate function calls.
Additional arguments to pass to par2qua and par2cdf.
W.H. Asquith
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.
Hosking, J.R.M., 1996, Some theoretical results concerning L-moments: Research Report RC14492, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York.
Jones, M.C., 2004, On some expressions for variance, covariance, skewness and L-moments: Journal of Statistical Planning and Inference, v. 126, pp. 97--106.
cdf2lmoms
para <- vec2par(c(.9,.4), type="nor")
cdf2lmom(4, para) # summarize the value
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