lmomco (version 2.4.14)

cdftexp: Cumulative Distribution Function of the Truncated Exponential Distribution

Description

This function computes the cumulative probability or nonexceedance probability of the Truncated Exponential distribution given parameters (\(\psi\) and \(\alpha\)) computed by partexp. The parameter \(\psi\) is the right truncation of the distribution and \(\alpha\) is a scale parameter. The cumulative distribution function, letting \(\beta = 1/\alpha\) to match nomenclature of Vogel and others (2008), is $$F(x) = \frac{1-\mathrm{exp}(-\beta{t})}{1-\mathrm{exp}(-\beta\psi)}\mbox{,}$$ where \(F(x)\) is the nonexceedance probability 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 cumulative distribution function is simply the uniform distribution and \(F(x) = x/\psi\). 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 \(\beta\) (scale parameter \(\alpha = 1/\beta\)). These two limiting conditions are supported.

Usage

cdftexp(x, para)

Value

Nonexceedance probability (\(F\)) for \(x\).

Arguments

x

A real value vector.

para

The parameters from partexp or vec2par.

Author

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

pdftexp, quatexp, lmomtexp, partexp

Examples

cdftexp(50,partexp(vec2lmom(c(40,0.38), lscale=FALSE)))
if (FALSE) {
F <- seq(0,1,by=0.001)
A <- partexp(vec2lmom(c(100, 1/2), lscale=FALSE))
x <- quatexp(F, A)
plot(x, cdftexp(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, cdftexp(x, A), pch=16, col=rgb(reds[lcvs == lcv],0,0))
}

  # Vogel and others (2008) example sighting times for the bird
  # Eskimo Curlew, inspection shows that these are fairly uniform.
  # There is a sighting about every year to two.
  T <- c(1946, 1947, 1948, 1950, 1955, 1956, 1959, 1960, 1961,
         1962, 1963, 1964, 1968, 1970, 1972, 1973, 1974, 1976,
         1977, 1980, 1981, 1982, 1982, 1983, 1985)
  R <- 1945 # beginning of record
  S <- T - R
  lmr <- lmoms(S)
  PARcurlew <- partexp(lmr)
  # read the warning message and then force the texp to the
  # stationary process model (min(tau_2) = 1/3).
  lmr$ratios[2] <- 1/3
  lmr$lambdas[2] <- lmr$lambdas[1]*lmr$ratios[2]
  PARcurlew <- partexp(lmr)
  Xmax <- quatexp(1, PARcurlew)
  X <- seq(0,Xmax, by=.1)
  plot(X, cdftexp(X,PARcurlew), type="l")
  # or use the MVUE estimator
  TE <- max(S)*((length(S)+1)/length(S)) # Time of Extinction
  lines(X, punif(X, min=0, max=TE), col=2)}