cdf2lmom: Compute an L-moment from Cumulative Distribution Function

Description

Compute an L-moment from a cumulative distribution function. This function is sequentially called by the cdf2lmoms function to mimic the output structure seen by other L-moment computation functions in the lmomco package.

For $r = 1$, the quantile function is actually used for numerical integration to compute the mean. The expression for the mean is $$\lambda_1 = \int_0^1 x(F)\; \mathrm{d} F$$ for quantile function $x(F)$ and nonexceedance probability $F$. For $r \ge 2$, the L-moments can be computed by $$\lambda_r = \frac{1}{r}\sum_{j=0}^{r-2} (-1)^j {r-2 \choose j}{r \choose j+1} \int_{-\infty}^{\infty} \! [F(x)]^{r-j-1}\times [1 - F(x)]^{j+1}\; \mathrm{d}x$$ for cumulative distribution function $F(x)$. This equation is described by Asquith (2011, eq. 6.8), Hosking (1996), and Jones (2004).

Usage

cdf2lmom(r, para, fdepth=0, silent=TRUE, ...)

Arguments

The order of the L-moment.

para

The parameters from lmom2par or similar.

fdepth

The depth of the nonexceedance/exceedance probabilities to determine the lower and upper integration limits through a call to the par2qua function. The default of 0 implies the quantile for $F=0$ and quanti

silent

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.

Value

The value for the requested L-moment is returned ($\lambda_r$).

References

Asquith, W.H., 2011a, Distributional analysis with L-moment statistics using the R environment for statistical computing: First edition, CreateSpace Independent Publishing Platform, ISBN 978-1463508418, 344 p.

Hosking, J.R.M., 1996a, 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.

Examples

Run this code

para <- vec2par(c(.9,.4), type="nor")
cdf2lmom(4, para)