Logarithmic: The Logarithmic Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

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].

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.

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.

Note

qlogarithmic is based on qbinom et al.; it uses the Cornish–Fisher Expansion to include a skewness correction to a normal approximation, followed by a search.

rlogarithmic is an implementation of the LS and LK algorithms of Kemp (1981) with automatic selection. As suggested by Devroye (1986), the LS algorithm is used when prob < 0.95, and the LK algorithm otherwise.

Author(s)

Vincent Goulet [e-mail]

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

## 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"))

actuar documentation built on Nov. 8, 2023, 9:06 a.m.