# Logarithmic: The Logarithmic Distribution In actuar: Actuarial Functions and Heavy Tailed Distributions

 Logarithmic R Documentation

## 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 ≤ x], otherwise, P[X > x].

### Details

The logarithmic (or log-series) distribution with parameter `prob` = p has probability mass function

p(x) = a p^x / x,

with a = -1/log(1-p) and for x = 1, 2, …, 0 ≤ p < 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) ≥ 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 vincent.goulet@act.ulaval.ca

### 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

`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 July 16, 2022, 9:05 a.m.