LogSeries: Logarithmic series distribution

Description

Density, distribution function, quantile function and random generation for the logarithmic series distribution.

Usage

dlgser(x, theta, log = FALSE)
plgser(q, theta, lower.tail = TRUE, log.p = FALSE)
qlgser(p, theta, lower.tail = TRUE, log.p = FALSE)
rlgser(n, theta)

Arguments

x, q: vector of quantiles.
theta: vector; concentration parameter; (0 < theta < 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.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function $$ f(x) = \frac{-1}{\log(1-\theta)} \frac{\theta^x}{x} $$

Cumulative distribution function $$ F(x) = \frac{-1}{\log(1-\theta)} \sum_{k=1}^x \frac{\theta^x}{x} $$

Quantile function and random generation are computed using algorithm described in Krishnamoorthy (2006).

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

Examples

Run this code


x <- rlgser(1e5, 0.66)
xx <- seq(0, 100, by = 1)
plot(prop.table(table(x)), type = "h")
lines(xx, dlgser(xx, 0.66), col = "red")

# Notice: distribution of F(X) is far from uniform:
hist(plgser(x, 0.66), 50)

xx <- seq(0, 100, by = 0.01)
plot(ecdf(x))
lines(xx, plgser(xx, 0.66), col = "red", lwd = 2)

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