lmomsla: Trimmed L-moments of the Slash Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

Description

This function estimates the trimmed L-moments of the Slash distribution given the parameters (ξ and α) from parsla. The relation between the TL-moments (trim=1) and the parameters have been numerically determined and are λ^{(1)}_1 = ξ, λ^{(1)}_2 = 0.9368627α, τ^{(1)}_3 = 0, τ^{(1)}_4 = 0.3042045, τ^{(1)}_5 = 0, and τ^{(1)}_6 = 0.1890072. These TL-moments (trim=1) are symmetrical for the first L-moments defined because \mathrm{E}[X_{1:n}] and \mathrm{E}[X_{n:n}] are undefined expectations for the Slash.

Usage

 1 lmomsla(para)

Arguments

 para The parameters of the distribution.

Value

An R list is returned.

 lambdas Vector of the trimmed L-moments. First element is λ^{(1)}_1, second element is λ^{(1)}_2, and so on. ratios Vector of the L-moment ratios. Second element is τ^{(1)}, third element is τ^{(1)}_3 and so on. trim Level of symmetrical trimming used in the computation, which is 1. leftrim Level of left-tail trimming used in the computation, which is 1. rightrim Level of right-tail trimming used in the computation, which is 1. source An attribute identifying the computational source of the L-moments: “lmomsla” trim Level of symmetrical trimming used.

W.H. Asquith

References

Rogers, W.H., and Tukey, J.W., 1972, Understanding some long-tailed symmetrical distributions: Statistica Neerlandica, v. 26, no. 3, pp. 211–226.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 ## Not run: # This example was used to numerically back into the TL-moments and the # relation between \alpha and \lambda_2. "lmomtrim1" <- function(para) { bigF <- 0.999 minX <- para$para[1] - para$para[2]*qnorm(1 - bigF) / qunif(1 - bigF) maxX <- para$para[1] + para$para[2]*qnorm( bigF) / qunif(1 - bigF) minF <- cdfsla(minX, para); maxF <- cdfsla(maxX, para) lmr <- theoTLmoms(para, nmom = 6, leftrim = 1, rightrim = 1) } U <- -10; i <- 0 As <- seq(.1,abs(10),by=.2) L1s <- L2s <- T3s <- T4s <- T5s <- T6s <- vector(mode="numeric", length=length(As)) for(A in As) { i <- i + 1 lmr <- lmomtrim1(vec2par(c(U, A), type="sla")) L1s[i] <- lmr$lambdas[1]; L2s[i] <- lmr$lambdas[2] T3s[i] <- lmr$ratios[3]; T4s[i] <- lmr$ratios[4] T5s[i] <- lmr$ratios[5]; T6s[i] <- lmr$ratios[6] } print(summary(lm(L2s~As-1))\$coe) print(mean(T4s)) print(mean(T6s)) ## End(Not run)