The Logarithmic Series Distribution

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)

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

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.

Value

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

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.

Author

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

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)
#> 'data.frame':	15 obs. of  3 variables:
#>  $ x   : int  1 2 3 4 5 1 2 3 4 5 ...
#>  $ p   : Factor w/ 3 levels "0.33","0.66",..: 1 1 1 1 1 2 2 2 2 2 ...
#>  $ prob: num  0.82402 0.13596 0.02991 0.0074 0.00195 ...

require(lattice)
#> Loading required package: lattice
#> Warning: package 'lattice' was built under R version 4.2.3
#> 
#> Attaching package: 'lattice'
#> The following object is masked from 'package:seriation':
#> 
#>     panel.lines
#> The following object is masked from 'package:gnm':
#> 
#>     barley
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')