leftparetolognormal {distributionsrd}

The Left-Pareto Lognormal distribution

Description

Density, distribution function, quantile function and random generation for the Left-Pareto Lognormal distribution.

Usage

dleftparetolognormal(x, shape1 = 1.5, meanlog = -0.5, sdlog = 0.5, log = FALSE)

pleftparetolognormal(
  q,
  shape1 = 1.5,
  meanlog = -0.5,
  sdlog = 0.5,
  lower.tail = TRUE,
  log.p = FALSE
)

qleftparetolognormal(
  p,
  shape1 = 1.5,
  meanlog = -0.5,
  sdlog = 0.5,
  lower.tail = TRUE,
  log.p = FALSE
)

mleftparetolognormal(
  r = 0,
  truncation = 0,
  shape1 = 1.5,
  meanlog = -0.5,
  sdlog = 0.5,
  lower.tail = TRUE
)

rleftparetolognormal(n, shape1 = 1.5, meanlog = -0.5, sdlog = 0.5)

Arguments

x, q

vector of quantiles

shape1, meanlog, sdlog

Shape, mean and variance of the Left-Pareto Lognormal distribution respectively.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities (moments) are P[X \le x] (E[x^r|X \le y]), otherwise, P[X > x] (E[x^r|X > y])

p

vector of probabilities

r

rth raw moment of the Pareto distribution

truncation

lower truncation parameter, defaults to xmin

n

number of observations

Details

Probability and Cumulative Distribution Function as provided by (Reed and Jorgensen 2004):

f(x) = shape1\omega^{shape1-1}e^{-shape1 meanlog + \frac{shape1^2sdlog^2}{2}}\Phi^c(\frac{ln\omega - meanlog +shape1 sdlog^2}{sdlog}), \newline F_X(x) = \Phi(\frac{ln\omega - meanlog }{sdlog}) - \omega^{shape1}e^{-shape1 meanlog + \frac{shape1^2sdlog^2}{2}}\Phi^c(\frac{ln\omega - meanlog +shape1 sdlog^2}{sdlog})

The y-bounded r-th raw moment of the Let-Pareto Lognormal distribution equals:

meanlog^{r}_{y} = -shape1e^{-shape1 meanlog + \frac{shape1^2sdlog^2}{2}}\frac{y^{\sigma_s + shape1-1}}{\sigma_s + shape1 - 1}\Phi^c(\frac{lny - meanlog + shape1 sdlog^2}{sdlog}) \newline \qquad + \frac{shape1}{r+shape1} e^{\frac{ r^2sdlog^2 + 2meanlog r }{2}}\Phi^c(\frac{lny - rsdlog^2 + meanlog}{sdlog})

Value

dleftparetolognormal gives the density, pleftparetolognormal gives the distribution function, qleftparetolognormal gives the quantile function, mleftparetolognormal gives the rth moment of the distribution and rleftparetolognormal generates random deviates.

The length of the result is determined by n for rleftparetolognormal, and is the maximum of the lengths of the numerical arguments for the other functions.

References

Reed WJ, Jorgensen M (2004). “The Double Pareto-Lognormal Distribution–A New Parametric Model for Size Distributions.” Communications in Statistics - Theory and Methods, 33(8), 1733–1753.

## Left-Pareto Lognormal density plot(x=seq(0,5,length.out=100),y=dleftparetolognormal(x=seq(0,5,length.out=100))) plot(x=seq(0,5,length.out=100),y=dleftparetolognormal(x=seq(0,5,length.out=100),shape1=1))

## Left-Pareto Lognormal relates to the Lognormal if the shape parameter goes to infinity pleftparetolognormal(q=6,shape1=1e20,meanlog=-0.5,sdlog=0.5) plnorm(q=6,meanlog=-0.5,sdlog=0.5)

## Demonstration of log functionality for probability and quantile function qleftparetolognormal(pleftparetolognormal(2,log.p=TRUE),log.p=TRUE)

## The zeroth truncated moment is equivalent to the probability function pleftparetolognormal(2) mleftparetolognormal(truncation=2)

## The (truncated) first moment is equivalent to the mean of a (truncated) random sample, #for large enough samples. x = rleftparetolognormal(1e5)

mean(x) mleftparetolognormal(r=1,lower.tail=FALSE)

sum(x[x>quantile(x,0.1)])/length(x) mleftparetolognormal(r=1,truncation=quantile(x,0.1),lower.tail=FALSE)