Description Usage Arguments Value Author(s) References See Also Examples

This function estimates the parameters of the Truncated Exponential distribution given
the L-moments of the data in an L-moment object such as that returned by `lmoms`

. The parameter *ψ* is the right truncation of the distribution, and *α* is a scale parameter, letting *β = 1/α* to match nomenclature of Vogel and others (2008), and recalling the L-moments in terms of the parameters and letting *η = \exp(-βψ)* are

*λ_1 = \frac{1 - η + η\log(η)}{β(1-η)}\mbox{,}*

*λ_2 = \frac{1 + 2η\log(η) - η^2}{2β(1-η)^2}\mbox{, and}*

*τ_2 = \frac{λ_2}{λ_1} = \frac{1 + 2η\log(η) - η^2}{2(1-η)[1-η+η\log(η)]}\mbox{,}*

and *τ_2* is a monotonic function of *η* is decreasing from *τ_2 = 1/2* at *η = 0* to *τ_2 = 1/3* at *η = 1* the parameters are readily solved given *τ_2 = [1/3, 1/2]*, the **R** function `uniroot`

can be used to solve for *η* with a starting interval of *(0, 1)*, then the parameters in terms of the parameters are

*α = \frac{1 - η + η\log(η)}{(1 - η)λ_1}\mbox{, and}*

*ψ = -\log(η)/α\mbox{.}*

If the *η* is rooted as equaling zero, then it is assumed that *\hatτ_2 \equiv τ_2* and the exponential distribution triggered, or if the *η* is rooted as equaling unity, then it is assumed that *\hatτ_2 \equiv τ_2* and the uniform distribution triggered (see below).

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 probability density function is simply the uniform distribution and *f(x) = 1/ψ*. If *τ_2 = 1/2*, then the distribution is represented as the usual exponential distribution with a location parameter of zero and a scale parameter *1/β*. Both of these limiting conditions are supported.

If the distribution shows to be uniform (*τ_2 = 1/3*), then the third element in the returned parameter vector is used as the *ψ* parameter for the uniform distribution, and the first and second elements are `NA`

of the returned parameter vector.

If the distribution shows to be exponential (*τ_2 = 1/2*), then the second element in the returned parameter vector is the inverse of the rate parameter for the exponential distribution, and the first element is `NA`

and the third element is `0`

(a numeric `FALSE`

) of the returned parameter vector.

1 |

`lmom` |
An L-moment object created by |

`checklmom` |
Should the |

`...` |
Other arguments to pass. |

An **R** `list`

is returned.

`type` |
The type of distribution: |

`para` |
The parameters of the distribution. |

`ifail` |
A logical value expressed in numeric form indicating the failure or success state of the parameter estimation. A value of two indicates that the |

`ifail.message` |
Various messages for successful and failed parameter estimations are reported. In particular, there are two conditions for which each distributional boundary (uniform or exponential) can be obtained. First, for the uniform distribution, one message would indicate if the |

`eta` |
The value for |

`source` |
The source of the parameters: “partexp”. |

W.H. Asquith

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.

`lmomtexp`

, `cdftexp`

, `pdftexp`

, `quatexp`

1 2 3 4 5 6 | ```
# truncated exponential is a nonstationary poisson process
A <- partexp(vec2lmom(c(100, 1/2), lscale=FALSE)) # pure exponential
B <- partexp(vec2lmom(c(100, 0.499), lscale=FALSE)) # almost exponential
BB <- partexp(vec2lmom(c(100, 0.45), lscale=FALSE)) # truncated exponential
C <- partexp(vec2lmom(c(100, 1/3), lscale=FALSE)) # stationary poisson process
D <- partexp(vec2lmom(c(100, 40))) # truncated exponential
``` |

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