nsRFA (version 0.5-8)

LOGNORM: Three parameter lognormal distribution and L-moments

Description

LOGNORM provides the link between L-moments of a sample and the three parameter log-normal distribution.

Usage

f.lognorm (x, xi, alfa, k)
F.lognorm (x, xi, alfa, k)
invF.lognorm (F, xi, alfa, k)
Lmom.lognorm (xi, alfa, k)
par.lognorm (lambda1, lambda2, tau3)
rand.lognorm (numerosita, xi, alfa, k)

Arguments

x

vector of quantiles

xi

vector of lognorm location parameters

alfa

vector of lognorm scale parameters

k

vector of lognorm shape parameters

F

vector of probabilities

lambda1

vector of sample means

lambda2

vector of L-variances

tau3

vector of L-CA (or L-skewness)

numerosita

numeric value indicating the length of the vector to be generated

Value

• f.lognorm gives the density $f$, F.lognorm gives the distribution function $F$, invFlognorm gives the quantile function $x$, Lmom.lognorm gives the L-moments ($\lambda_1$, $\lambda_2$, $\tau_3$, $\tau_4$), par.lognorm gives the parameters (xi, alfa, k), and rand.lognorm generates random deviates.

Details

See http://en.wikipedia.org/wiki/Log-normal_distribution for an introduction to the lognormal distribution.

Definition

Parameters (3): $\xi$ (location), $\alpha$ (scale), $k$ (shape).

Range of $x$: $-\infty < x \le \xi + \alpha / k$ if $k>0$; $-\infty < x < \infty$ if $k=0$; $\xi + \alpha / k \le x < \infty$ if $k<0$.< p="">

Probability density function: $$f(x)=\frac{e^{ky-y^2/2}}{\alpha \sqrt{2\pi }}$$ where $y = -k^{-1}\log{1 - k(x - \xi)/\alpha}$ if $k \ne 0$, $y = (x-\xi)/\alpha$ if $k=0$.

Cumulative distribution function: $$F(x)=\mathrm{\Phi }(x)$$ where $\Phi(x)=\int_{-\infty}^x \phi(t)dt$.

Quantile function: $x(F)$ has no explicit analytical form.

$k=0$ is the Normal distribution with parameters $\xi$ and $alpha$.

L-moments

L-moments are defined for all values of $k$.

$${\lambda }_1=\xi +\alpha (1-e^{k^2/2})/k$$ $${\lambda }_2=\alpha /k{e}^{k^2/2}[1-2\mathrm{\Phi }(-k/\sqrt{2})]$$

There are no simple expressions for the L-moment ratios $\tau_r$ with $r \ge 3$. Here we use the rational-function approximation given in Hosking and Wallis (1997, p. 199).

Parameters

The shape parameter $k$ is a function of $\tau_3$ alone. No explicit solution is possible. Here we use the approximation given in Hosking and Wallis (1997, p. 199).

Given $k$, the other parameters are given by $$\alpha =\frac{{\lambda }_{2}k{e}^{-k^2/2}}{1-2\mathrm{\Phi }(-k/\sqrt{2})}$$ $$\xi ={\lambda }_1-\frac{\alpha }{k}(1-e^{k^2/2})$$

References

Hosking, J.R.M. and Wallis, J.R. (1997) Regional Frequency Analysis: an approach based on L-moments, Cambridge University Press, Cambridge, UK.

See Also

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

Examples

data(hydroSIMN)
annualflows
summary(annualflows)
x <- annualflows["dato"][,]
fac <- factor(annualflows["cod"][,])
split(x,fac)

camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.lognorm(ll[1],ll[2],ll[4])
f.lognorm(1800,parameters$xi,parameters$alfa,parameters$k)
F.lognorm(1800,parameters$xi,parameters$alfa,parameters$k)
invF.lognorm(0.7529877,parameters$xi,parameters$alfa,parameters$k)
Lmom.lognorm(parameters$xi,parameters$alfa,parameters$k)
rand.lognorm(100,parameters$xi,parameters$alfa,parameters$k)

Rll <- regionalLmoments(x,fac); Rll
parameters <- par.lognorm(Rll[1],Rll[2],Rll[4])
Lmom.lognorm(parameters$xi,parameters$alfa,parameters$k)