# lmomtexp: L-moments of the Truncated Exponential Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

This function estimates the L-moments of the Truncated Exponential distribution. The parameter ψ is the right truncation of the distribution and α is a scale parameter, letting β = 1/α to match nomenclature of Vogel and others (2008), the L-moments in terms of the parameters, letting η = \mathrm{exp}(-αψ), are

λ_1 = \frac{1}{β} - \frac{ψη}{1-η} \mbox{,}

λ_2 = \frac{1}{1-η}\biggl[\frac{1+η}{2β} - \frac{ψη}{1-η}\biggr] \mbox{,}

λ_3 = \frac{1}{(1-η)^2}\biggl[\frac{1+10η+η^2}{6α} - \frac{ψη(1+η)}{1-η}\biggr] \mbox{, and}

λ_4 = \frac{1}{(1-η)^3}\biggl[\frac{1+29η+29η^2+η^3}{12α} - \frac{ψη(1+3η+η^2)}{1-η}\biggr] \mbox{.}

The distribution is restricted to a narrow range of L-CV (τ_2 = λ_2/λ_1). If τ_2 = 1/3, the process represented is a stationary Poisson for which the probability density function is simply the uniform distribution and f(x) = 1/ψ. If τ_2 = 1/2, then the distribution is represented as the usual exponential distribution with a location parameter of zero and a scale parameter 1/β. Both of these limiting conditions are supported.

If the distribution shows to be Uniform (τ_2 = 1/3), then λ_1 = ψ/2, λ_2 = ψ/6, τ_3 = 0, and τ_4 = 0. If the distribution shows to be Exponential (τ_2 = 1/2), then λ_1 = α, λ_2 = α/2, τ_3 = 1/3 and τ_4 = 1/6.

## Usage

 1 lmomtexp(para) 

## Arguments

 para The parameters of the distribution.

## 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: “lmomtexp”.

W.H. Asquith

## References

Vogel, R.M., Hosking, J.R.M., Elphick, C.S., Roberts, D.L., and Reed, J.M., 2008, Goodness of fit of probability distributions for sightings as species approach extinction: Bulletin of Mathematical Biology, DOI 10.1007/s11538-008-9377-3, 19 p.

partexp, cdftexp, pdftexp, quatexp
  1 2 3 4 5 6 7 8 9 10 set.seed(1) # to get a suitable L-CV X <- rexp(1000, rate=.001) + 100 Y <- X[X <= 2000] lmr <- lmoms(Y) print(lmr$lambdas) print(lmomtexp(partexp(lmr))$lambdas) print(lmr$ratios) print(lmomtexp(partexp(lmr))$ratios)