lmomco (version 2.0.1)

lmomtexp: 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 rate parameter $\beta$. 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 will equal NULL until trimming support is made.
• leftrim
  Level of left-tail trimming used in the computation, which will equal NULL until trimming support is made.
• rightrim
  Level of right-tail trimming used in the computation, which will equal NULL until trimming support is made.
• source
  An attribute identifying the computational source of the L-moments: lmomtexp.

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, quatexp, cdftexp

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)