EXP {nsRFA} | R Documentation |
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)