vcdExtra (version 0.8-5)

logseries: The Logarithmic Series Distribution

Description

The logarithmic series distribution is a long-tailed distribution introduced by Fisher etal. (1943) in connection with data on the abundance of individuals classified by species.

These functions provide the density, distribution function, quantile function and random generation for the logarithmic series distribution with parameter prob.

Usage

dlogseries(x, prob = 0.5, log = FALSE)
plogseries(q, prob = 0.5, lower.tail = TRUE, log.p = FALSE)
qlogseries(p, prob = 0.5, lower.tail = TRUE, log.p = FALSE, max.value = 10000)
rlogseries(n, prob = 0.5)

Value

dlogseries gives the density, plogseries gives the distribution function, qlogseries gives the quantile function, and rlogseries generates random deviates.

Arguments

x, q

vector of quantiles representing the number of events.

prob

parameter for the distribution, 0 < prob < 1

log, log.p

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

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

p

vector of probabilities

max.value

maximum value returned by qlogseries

n

number of observations for rlogseries

Author

Michael Friendly, using original code modified from the gmlss.dist package by Mikis Stasinopoulos.

Details

The logarithmic series distribution with prob = \(p\) has density $$ p ( x ) = \alpha p^x / x $$ for \(x = 1, 2, \dots\), where \(\alpha= -1 / \log(1 - p)\) and \(0 < p <1\). Note that counts x==2 cannot occur.

References

https://en.wikipedia.org/wiki/Logarithmic_distribution

Fisher, R. A. and Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals Journal of Animal Ecology, 12, 42-58.

See Also

Distributions, ~~~

Examples

XL <-expand.grid(x=1:5, p=c(0.33, 0.66, 0.99))
lgs.df <- data.frame(XL, prob=dlogseries(XL[,"x"], XL[,"p"]))
lgs.df$p = factor(lgs.df$p)
str(lgs.df)

require(lattice)
mycol <- palette()[2:4]
xyplot( prob ~ x, data=lgs.df, groups=p,
	xlab=list('Number of events (k)', cex=1.25),
	ylab=list('Probability',  cex=1.25),
	type='b', pch=15:17, lwd=2, cex=1.25, col=mycol,
	key = list(
					title = 'p',
					points = list(pch=15:17, col=mycol, cex=1.25),
					lines = list(lwd=2, col=mycol),
					text = list(levels(lgs.df$p)),
					x=0.9, y=0.98, corner=c(x=1, y=1)
					)
	)


# random numbers
hist(rlogseries(200, prob=.4), xlab='x')
hist(rlogseries(200, prob=.8), xlab='x')