lmomco (version 2.0.1)

quatexp: Quantile Function of the Truncated Exponential Distribution

Description

This function computes the quantiles of the Truncated Exponential distribution given parameters ($\psi$ and $\alpha$) of the distribution computed by partexp. The parameter $\psi$ is the right truncation of the distribution, and $\alpha$ is a scale parameter. The quantile function, letting $\beta = 1/\alpha$ to match nomenclature of Vogel and others (2008), is

$$x(F) = \frac{\log(1-F[1-\mathrm{exp}(-\beta\psi)])}{-\beta}\mbox{,}$$

where $x(x)$ is the probability density for the quantile $0 \le x \le \psi$ and $\psi > 0$ and $\alpha > 0$. This distribution represents a nonstationary Poisson process.

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 quantile function is simply the uniform distribution and $x(F) = F\psi$. 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.

Usage

quatexp(f, para, paracheck=TRUE)

Arguments

f

Nonexceedance probability ($0 \le F \le 1$).

para

The parameters from partexp or similar.

paracheck

A logical controlling whether the parameters and checked for validity. Overriding of this check might be extremely important and needed for use of the distribution quantile function in the context of TL-moments with nonzero trimming.

Value

• Quantile value for nonexceedance probability $F$.

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

pdftexp, cdftexp, partexp

Examples

lmr <- vec2lmom(c(40,0.38), lscale=FALSE)
quatexp(0.5,partexp(lmr))

F <- seq(0,1,by=0.001)
A <- partexp(vec2lmom(c(100, 1/2), lscale=FALSE))
plot(qnorm(F), quatexp(F, A), pch=16, type='l')
by <- 0.01; lcvs <- c(1/3, seq(1/3+by, 1/2-by, by=by), 1/2)
reds <- (lcvs - 1/3)/max(lcvs - 1/3)
for(lcv in lcvs) {
    A <- partexp(vec2lmom(c(100, lcv), lscale=FALSE))
    lines(qnorm(F), quatexp(F, A),
    pch=16, col=rgb(reds[lcvs == lcv],0,0))
}