actuar (version 3.0-0)

Logarithmic: The Logarithmic Distribution

Description

Density function, distribution function, quantile function and random generation for the Logarithmic (or log-series) distribution with parameter prob.

Usage

dlogarithmic(x, prob, log = FALSE)
plogarithmic(q, prob, lower.tail = TRUE, log.p = FALSE)
qlogarithmic(p, prob, lower.tail = TRUE, log.p = FALSE)
rlogarithmic(n, prob)

Arguments

x

vector of (strictly positive integer) quantiles.

q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

prob

parameter. 0 <= prob < 1.

log, log.p

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

lower.tail

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

Value

dlogarithmic gives the probability mass function, plogarithmic gives the distribution function, qlogarithmic gives the quantile function, and rlogarithmic generates random deviates.

Invalid prob will result in return value NaN, with a warning.

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

Details

The logarithmic (or log-series) distribution with parameter prob \(= \theta\) has probability mass function $$% p(x) = \frac{a \theta^x}{x},$$ with \(a = -1/\log(1 - \theta)\) and for \(x = 1, 2, \ldots\), \(0 \le \theta < 1\).

The logarithmic distribution is the limiting case of the zero-truncated negative binomial distribution with size parameter equal to \(0\). Note that in this context, parameter prob generally corresponds to the probability of failure of the zero-truncated negative binomial.

If an element of x is not integer, the result of dlogarithmic is zero, with a warning.

The quantile is defined as the smallest value \(x\) such that \(F(x) \ge p\), where \(F\) is the distribution function.

References

Johnson, N. L., Kemp, A. W. and Kotz, S. (2005), Univariate Discrete Distributions, Third Edition, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Kemp, A. W. (1981), “Efficient Generation of Logarithmically Distributed Pseudo-Random Variables”, Journal of the Royal Statistical Society, Series C, vol. 30, p. 249-253.

Devroye, L. (1986), Non-Uniform Random Variate Generation, Springer-Verlag. http://luc.devroye.org/rnbookindex.html

See Also

dztnbinom for the zero-truncated negative binomial distribution.

Examples

# NOT RUN {
## Table 1 of Kemp (1981) [also found in Johnson et al. (2005), chapter 7]
p <- c(0.1, 0.3, 0.5, 0.7, 0.8, 0.85, 0.9, 0.95, 0.99, 0.995, 0.999, 0.9999)
round(rbind(dlogarithmic(1, p),
            dlogarithmic(2, p),
            plogarithmic(9, p, lower.tail = FALSE),
            -p/((1 - p) * log(1 - p))), 2)

qlogarithmic(plogarithmic(1:10, 0.9), 0.9)

x <- rlogarithmic(1000, 0.8)
y <- sort(unique(x))
plot(y, table(x)/length(x), type = "h", lwd = 2,
     pch = 19, col = "black", xlab = "x", ylab = "p(x)",
     main = "Empirical vs theoretical probabilities")
points(y, dlogarithmic(y, prob = 0.8),
       pch = 19, col = "red")
legend("topright", c("empirical", "theoretical"),
       lty = c(1, NA), pch = c(NA, 19), col = c("black", "red"))
# }