EXP: Two parameter exponential distribution and L-moments

In nsRFA: Non-Supervised Regional Frequency Analysis

Two parameter exponential distribution and L-moments

Description

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

Usage

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

Definition

Parameters (2): \xi (lower endpoint of the distribution), \alpha (scale).

Range of x: \xi \le x < \infty.

Probability density function:

f(x) = \alpha^{-1} \exp\{-(x-\xi)/\alpha\}

Cumulative distribution function:

F(x) = 1 - \exp\{-(x-\xi)/\alpha\}

Quantile function:

x(F) = \xi - \alpha \log(1-F)

L-moments

\lambda_1 = \xi + \alpha

\lambda_2 = 1/2 \cdot \alpha

\tau_3 = 1/3

\tau_4 = 1/6

Parameters

If \xi is known, \alpha is given by \alpha = \lambda_1 - \xi and the L-moment, moment, and maximum-likelihood estimators are identical. If \xi is unknown, the parameters are given by

\alpha = 2 \lambda_2

\xi = \lambda_1 - \alpha

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 (\lambda_1, \lambda_2, \tau_3, \tau_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.

See Also

rnorm, runif, GENLOGIS, GENPAR, GEV, GUMBEL, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

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)

nsRFA documentation built on May 29, 2024, 1:14 a.m.