This function computes the B-type L-moments of the Generalized Pareto distribution
given the parameters ($\xi$, $\alpha$, and $\kappa$) from pargpaRC and the right-tail censoring fraction $\zeta$.
The B-type L-moments in terms of the parameters are$$\lambda^B_1 = \xi + \alpha m_1 \mbox{,}$$
$$\lambda^B_2 = \alpha (m_1 - m_2) \mbox{,}$$
$$\lambda^B_3 = \alpha (m_1 - 3m_2 + 2m_3)\mbox{,}$$
$$\lambda^B_4 = \alpha (m_1 - 6m_2 + 10m_3 - 5m_4)\mbox{, and}$$
$$\lambda^B_5 = \alpha (m_1 - 10m_2 + 30m_3 - 35m_4 + 14m_5)\mbox{,}$$
where $m_r = \lbrace 1-(1-\zeta)^{r+\kappa}\rbrace/(r+\kappa)$ and $\zeta$ is the right-tail censor fraction or the probability $\mathrm{Pr}\lbrace \rbrace$ that $x$ is less than the quantile at $\zeta$ nonexceedance probability: ($\mathrm{Pr}\lbrace x < X(\zeta) \rbrace$). Finally, the RC in the function name is to denote Right-tail Censoring.
lmomgpaRC(para)
- para
{The parameters of the distribution. Note that if the $\zeta$ part of the parameters (see pargpaRC) is not present then zeta=1 is assumed.}
An R list is returned.
- lambdas
{Vector of the L-moments. First element is
$\lambda_1$, second element is $\lambda_2$, and so on.}
- ratios
{Vector of the L-moment ratios. Second element is
$\tau$, third element is $\tau_3$ and so on.}
- source
{An attribute identifying the computational
source of the L-moments: lmomgpa2.}
- 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.}
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, vol. 52, p. 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.
[object Object]
pargpa, pargpaRC, lmomgpa, quagpa, cdfgpa
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.
distribution