lmomco (version 0.88)

theoLmoms: The Theoretical L-moments and L-moment Ratios using Integration of the Quantile Function

Description

Compute the theoretrical L-moments for a vector. A theoretrical L-moment in integral form is

$${\lambda }_r=\frac{1}{r}\sum _{k=0}^{r-1}(-1{)}^k(\genfrac{}{}{0ex}{}{r-1}{k})\frac{r!I_r}{(r-k-1)!k!}\text{, in which }$$

$$I_r={\int }_0^{1}X(F)\times F^{r-k-1}(1-F{)}^k\mathrm{d}F\text{,}$$

where $X(F)$ is the quantile function of the random variable $X$ for nonexceedance probability $F$, and $r$ represents the order of the L-moments. This function actually dispatches to theoTLmoms with trim=0 argument.

Usage

theoLmoms(para,nmom=5,verbose=FALSE)

Arguments

para

A distribution parameter object of this package vec2par.

nmom

The number of moments to compute. Default is 5.

verbose

Toggle verbose output. Because the R function integrate is used to perform the numerical integration, it might be useful to see selected messages regarding the numerical integration.

Value

• An R list is returned.
• lambdas
  Vector of the TL-moments. First element is $\lambda_1$, second element is $\lambda_2$, and so on.
• ratios
  Vector of the L-moment ratios. Second element is $\tau_2$, third element is $\tau_3$ and so on.
• trim
  Level of symmetrical trimming used in the computation, which will equal zero (the ordinary L-moments).
• source
  An attribute identifying the computational source of the L-moments: theoTLmoms.

References

Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105--124.

See Also

theoTLmoms, par2qua, TLmoms, lmom.ub

Examples

para <- vec2par(c(0,1),type='nor') # standard normal
TL00 <- theoLmoms(para) # compute ordinary L-moments