lmomco (version 1.7.3)

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

$$\lambda_r = \frac{1}{r} \sum^{r-1}_{k=0}{(-1)^k {r-1 \choose k} \frac{r!\:I_r}{(r-k-1)!k!} } \mbox{, in which }$$

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

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,minF=0,maxF=1)

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.

minF

The end point of nonexceedance probability in which to perform the integration. Try setting to non-zero (but small) if you have a divergent integral.

maxF

The end point of nonexceedance probability in which to perform the integration. Try setting to non-unity (but close) if you have a divergent integral.

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