# EXP: Two parameter exponential distribution and L-moments In nsRFA: Non-supervised Regional Frequency Analysis

## Description

EXP provides the link between L-moments of a sample and the two parameter exponential distribution.

## Usage

 1 2 3 4 5 6 f.exp (x, xi, alfa) F.exp (x, xi, alfa) invF.exp (F, xi, alfa) Lmom.exp (xi, alfa) par.exp (lambda1, lambda2) rand.exp (numerosita, xi, alfa) 

## Arguments

 x vector of quantiles xi vector of exp location parameters alfa vector of exp scale parameters F vector of probabilities lambda1 vector of sample means lambda2 vector of L-variances numerosita numeric value indicating the length of the vector to be generated

## Details

See http://en.wikipedia.org/wiki/Exponential_distribution for a brief introduction on the Exponential distribution.

Definition

Parameters (2): ξ (lower endpoint of the distribution), α (scale).

Range of x: ξ ≤ x < ∞.

Probability density function:

f(x) = α^{-1} \exp\{-(x-ξ)/α\}

Cumulative distribution function:

F(x) = 1 - \exp\{-(x-ξ)/α\}

Quantile function:

x(F) = ξ - α \log(1-F)

L-moments

λ_1 = ξ + α

λ_2 = 1/2 \cdot α

τ_3 = 1/3

τ_4 = 1/6

Parameters

If ξ is known, α is given by α = λ_1 - ξ and the L-moment, moment, and maximum-likelihood estimators are identical. If ξ is unknown, the parameters are given by

α = 2 λ_2

ξ = λ_1 - α

For estimation based on a single sample these estimates are inefficient, but in regional frequency analysis they can give reasonable estimates of upper-tail quantiles.

Lmom.exp and par.exp accept input as vectors of equal length. In f.exp, F.exp, invF.exp and rand.exp parameters (xi, alfa) must be atomic.

## Value

f.exp gives the density f, F.exp gives the distribution function F, invFexp gives the quantile function x, Lmom.exp gives the L-moments (λ_1, λ_2, τ_3, τ_4), par.exp gives the parameters (xi, alfa), and rand.exp generates random deviates.

## Note

For information on the package and the Author, and for all the references, see nsRFA.

rnorm, runif, GENLOGIS, GENPAR, GEV, GUMBEL, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 data(hydroSIMN) annualflows summary(annualflows) x <- annualflows["dato"][,] fac <- factor(annualflows["cod"][,]) split(x,fac) camp <- split(x,fac)$"45" ll <- Lmoments(camp) parameters <- par.exp(ll[1],ll[2]) f.exp(1800,parameters$xi,parameters$alfa) F.exp(1800,parameters$xi,parameters$alfa) invF.exp(0.7870856,parameters$xi,parameters$alfa) Lmom.exp(parameters$xi,parameters$alfa) rand.exp(100,parameters$xi,parameters$alfa) Rll <- regionalLmoments(x,fac); Rll parameters <- par.exp(Rll[1],Rll[2]) Lmom.exp(parameters$xi,parameters\$alfa)