lmomco (version 2.5.3)

lmomcau: Trimmed L-moments of the Cauchy Distribution

Description

This function estimates the trimmed L-moments of the Cauchy distribution given the parameters (\(\xi\) and \(\alpha\)) from parcau. The trimmed L-moments in terms of the parameters are \(\lambda^{(1)}_1 = \xi\), \(\lambda^{(1)}_2 = 0.69782723\alpha\), \(\tau^{(1)}_{3, 5, \cdots} = 0\), \(\tau^{(1)}_4 = 0.34280842\), and \(\tau^{(1)}_6 = 0.20274358\). These TL-moments (trim=1) are symmetrical for the first L-moments defined because \(\mathrm{E}[X_{1:n}]\) and \(\mathrm{E}[X_{n:n}]\) undefined expectations for the Cauchy.

Usage

lmomcau(para)

Value

An R

list is returned.

lambdas

Vector of the trimmed L-moments. First element is \(\lambda^{(1)}_1\), second element is \(\lambda^{(1)}_2\), and so on.

ratios

Vector of the L-moment ratios. Second element is \(\tau^{(1)}\), third element is \(\tau^{(1)}_3\) and so on.

trim

Level of symmetrical trimming used in the computation, which is unity.

leftrim

Level of left-tail trimming used in the computation, which is unity.

rightrim

Level of right-tail trimming used in the computation, which is unity.

source

An attribute identifying the computational source of the L-moments: “lmomcau”.

Arguments

para

The parameters of the distribution.

Author

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8.

Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments: Computational Statistics and Data Analysis, v. 43, pp. 299--314.

See Also

parcau, cdfcau, pdfcau, quacau

Examples

X1 <- rcauchy(20)
lmomcau( parcau( TLmoms(X1, trim=1) ) )

alpha <- 30
tlmr <- theoTLmoms(vec2par(c(100, alpha), type="cau"), nmom=6, trim=1)
print( c(tlmr$lambdas[2] / alpha, tlmr$ratios[c(4,6)]), 8 )