lmomtexp {lmomco}

L-moments of the Truncated Exponential Distribution

Description

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

\lambda_1 = \frac{1}{\beta} - \frac{\psi\eta}{1-\eta} \mbox{,}

\lambda_2 = \frac{1}{1-\eta}\biggl[\frac{1+\eta}{2\beta} - \frac{\psi\eta}{1-\eta}\biggr] \mbox{,}

\lambda_3 = \frac{1}{(1-\eta)^2}\biggl[\frac{1+10\eta+\eta^2}{6\alpha} - \frac{\psi\eta(1+\eta)}{1-\eta}\biggr] \mbox{, and}

\lambda_4 = \frac{1}{(1-\eta)^3}\biggl[\frac{1+29\eta+29\eta^2+\eta^3}{12\alpha} - \frac{\psi\eta(1+3\eta+\eta^2)}{1-\eta}\biggr] \mbox{.}

The distribution is restricted to a narrow range of L-CV (\tau_2 = \lambda_2/\lambda_1). If \tau_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/\psi. If \tau_2 = 1/2, then the distribution is represented as the usual exponential distribution with a location parameter of zero and a scale parameter 1/\beta. Both of these limiting conditions are supported.

If the distribution shows to be Uniform (\tau_2 = 1/3), then \lambda_1 = \psi/2, \lambda_2 = \psi/6, \tau_3 = 0, and \tau_4 = 0. If the distribution shows to be Exponential (\tau_2 = 1/2), then \lambda_1 = \alpha, \lambda_2 = \alpha/2, \tau_3 = 1/3 and \tau_4 = 1/6.

Usage

lmomtexp(para)

Arguments

para

The parameters of the distribution.

Value

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.

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”.

Author(s)

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.

See Also

partexp, cdftexp, pdftexp, quatexp

Examples

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)