# theoLmoms: The Theoretical L-moments and L-moment Ratios using... In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

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

λ_r = \frac{1}{r} ∑^{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

 1 theoLmoms(para, nmom=5, verbose=FALSE, minF=0, maxF=1) 

## Arguments

 para A distribution parameter object such as from 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 very small) if the integral is divergent. maxF The end point of nonexceedance probability in which to perform the integration. Try setting to non-unity (but still very close [perhaps 1 - minF]) if the integral is divergent.

## Value

An R list is returned.

 lambdas Vector of the TL-moments. First element is λ_1, second element is λ_2, and so on. ratios Vector of the L-moment ratios. Second element is τ_2, third element is τ_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”.

## Note

The actual function used is theoTLmoms(para,nmom=nmom,trim=0,verbose=verbose).

W.H. Asquith

## 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, v. 52, pp. 105–124.

theoTLmoms
 1 2 para <- vec2par(c(0,1),type='nor') # standard normal TL00 <- theoLmoms(para) # compute ordinary L-moments