lmomco (version 1.6.1)

lmomrevgum: L-moments of the Reverse Gumbel Distribution

Description

This function estimates the L-moments of the Reverse Gumbel distribution given the parameters ($\xi$ and $\alpha$) from parrevgum. The type-B L-moments in terms of the parameters are

$${\lambda }_1^B=\xi -(0.5722\dots )\alpha -\alpha \{\mathrm{Ei}(-\log (1-\zeta ))\}\text{,}$$ $${\lambda }_2^B=\alpha \{\log (2)+\mathrm{Ei}(-2\log (1-\zeta ))-\mathrm{Ei}(-\log (1-\zeta ))\}\text{,}$$ $${\tau }_3=\text{,}$$ $${\tau }_4=\text{, and}$$ $${\tau }_5=\text{.}$$

where $\zeta$ is the right-tail censoring fraction of the sample or the nonexceedance probability of the right-tail censoring threshold, and $\mathrm{Ei}(x)$ is the exponential integral defined as

$$\mathrm{Ei}(X)={\int }_X^{\mathrm{\infty }}{x}^{-1}{e}^{-x}\mathrm{d}x\text{,}$$

where $\mathrm{Ei}(-\log(1-\zeta)) \rightarrow 0$ as $\zeta \rightarrow 1$ and $\mathrm{Ei}(-\log(1-\zeta))$ can not be evaluated as $\zeta \rightarrow 0$.

Usage

lmomrevgum(para)

Arguments

para

The parameters of the distribution.

Value

• An R list is returned.
• L1
  Arithmetic mean.
• L2
  L-scale---analogous to standard deviation.
• LCV
  coefficient of L-variation---analogous to coe. of variation.
• TAU3
  The third L-moment ratio or L-skew--analogous to skew.
• TAU4
  The fourth L-moment ratio or L-kurtosis---analogous to kurtosis.
• TAU5
  The fifth L-moment ratio.
• L3
  The third L-moment.
• L4
  The fourth L-moment.
• L5
  The fifth L-moment.
• zeta
  Number of samples observed (noncensored) divided by the total number of samples.
• source
  An attribute identifying the computational source of the L-moments: lmomrevgum.

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, 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.

See Also

parrevgum, quarevgum, cdfrevgum

Examples

lmr <- lmom.ub(c(123,34,4,654,37,78))
rev.para <- lmom2par(lmr,type='revgum')
lmomrevgum(rev.para)