# Cumulative Distribution Function of the Truncated Exponential Distribution

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### Description

This function computes the cumulative probability or nonexceedance probability of the Truncated Exponential distribution given parameters (ψ and α) computed by partexp. The parameter ψ is the right truncation of the distribution and α is a scale parameter. The cumulative distribution function, letting β = 1/α to match nomenclature of Vogel and others (2008), is

F(x) = \frac{1-\mathrm{exp}(-β{t})}{1-\mathrm{exp}(-βψ)}\mbox{,}

where F(x) is the nonexceedance probability for the quantile 0 ≤ x ≤ ψ and ψ > 0 and α > 0. This distribution represents a nonstationary Poisson process.

The distribution is restricted to a narrow range of L-CV (τ_2 = λ_2/λ_1). If τ_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/ψ. If τ_2 = 1/2, then the distribution is represented as the usual exponential distribution with a location parameter of zero and a rate parameter β (scale parameter α = 1/β). These two limiting conditions are supported.

### Usage

 1 cdftexp(x, para)

### Arguments

 x A real value vector. para The parameters from partexp or vec2par.

### Value

Nonexceedance probability (F) for x.

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.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 cdftexp(50,partexp(vec2lmom(c(40,0.38), lscale=FALSE))) ## Not run: 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) ## End(Not run)