Trimmed L-moments of the Slash Distribution

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.

Author(s)

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.

See Also

parsla, cdfsla, pdfsla, quasla

Examples

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## 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)

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