# lmomgpaRC: B-type L-moments of the Generalized Pareto Distribution with... In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

This function computes the “B”-type L-moments of the Generalized Pareto distribution given the parameters (ξ, α, and κ) from pargpaRC and the right-tail censoring fraction ζ. The B-type L-moments in terms of the parameters are

λ^B_1 = ξ + α m_1 \mbox{,}

λ^B_2 = α (m_1 - m_2) \mbox{,}

λ^B_3 = α (m_1 - 3m_2 + 2m_3)\mbox{,}

λ^B_4 = α (m_1 - 6m_2 + 10m_3 - 5m_4)\mbox{, and}

λ^B_5 = α (m_1 - 10m_2 + 30m_3 - 35m_4 + 14m_5)\mbox{,}

where m_r = \lbrace 1-(1-ζ)^{r+κ}\rbrace/(r+κ) and ζ is the right-tail censor fraction or the probability \mathrm{Pr}\lbrace \rbrace that x is less than the quantile at ζ nonexceedance probability: (\mathrm{Pr}\lbrace x < X(ζ) \rbrace). In other words, if ζ = 1, then there is no right-tail censoring. Finally, the RC in the function name is to denote Right-tail Censoring.

## Usage

 1 lmomgpaRC(para) 

## Arguments

 para The parameters of the distribution. Note that if the ζ part of the parameters (see pargpaRC) is not present then zeta=1 (no right-tail censoring) is assumed.

## Value

An R list is returned.

 lambdas Vector of the L-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, third element is τ_3 and so on. trim Level of symmetrical trimming used in the computation, which is 0. leftrim Level of left-tail trimming used in the computation, which is NULL. rightrim Level of right-tail trimming used in the computation, which is NULL. source An attribute identifying the computational source of the L-moments: “lmomgpaRC”. message For clarity, this function adds the unusual message to an L-moment object that the lambdas and ratios are B-type L-moments. zeta The censoring fraction. Assumed equal to unity if not present in the gpa parameter object.

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.

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546–560.

pargpa, pargpaRC, lmomgpa, cdfgpa, pdfgpa, quagpa
 1 2 3 4 5 6 7 8 para <- vec2par(c(1500,160,.3),type="gpa") # build a GPA parameter set lmorph(lmomgpa(para)) lmomgpaRC(para) # zeta = 1 is internally assumed if not available # The previous two commands should output the same parameter values from # independent code bases. # Now assume that we have the sample parameters, but the zeta is nonunity. para\$zeta = .8 lmomgpaRC(para) # The B-type L-moments.