lmomco (version 2.0.1)

pdftexp: Probability Density Function of the Truncated Exponential Distribution

Description

This function computes the probability density 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 probability density function, letting $\beta = 1/\alpha$ to match nomenclature of Vogel and others (2008), is

$$f(x) = \frac{\beta\mathrm{exp}(-\beta{t})}{1 - \mathrm{exp}(-\beta\psi)}\mbox{,}$$

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 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.

Usage

pdftexp(x, para)

Arguments

x

A real value.

para

The parameters from partexp or similar.

Value

• Probability density ($F$) for $x$.

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

cdftexp, quatexp, partexp

Examples

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

F <- seq(0,1,by=0.001)
A <- partexp(vec2lmom(c(100, 1/2), lscale=FALSE))
x <- quatexp(F, A)
plot(x, pdftexp(x, 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))
    x <- quatexp(F, A)
    lines(x, pdftexp(x, A),
          pch=16, col=rgb(reds[lcvs == lcv],0,0))
}