ZeroModifiedLogarithmic {actuar} R Documentation

## The Zero-Modified Logarithmic Distribution

### Description

Density function, distribution function, quantile function and random generation for the Zero-Modified Logarithmic (or log-series) distribution with parameter `prob` and arbitrary probability at zero `p0`.

### Usage

```dzmlogarithmic(x, prob, p0, log = FALSE)
pzmlogarithmic(q, prob, p0, lower.tail = TRUE, log.p = FALSE)
qzmlogarithmic(p, prob, p0, lower.tail = TRUE, log.p = FALSE)
rzmlogarithmic(n, prob, p0)
```

### 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`. `p0` probability mass at zero. `0 <= p0 <= 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 zero-modified logarithmic distribution with `prob` = p and `p0` = p0 is a discrete mixture between a degenerate distribution at zero and a (standard) logarithmic. The probability mass function is p(0) = p0 and

p(x) = (1-p0) f(x)

for x = 1, 2, …, 0 < p < 1 and 0 ≤ p0 ≤ 1, where f(x) is the probability mass function of the logarithmic. The cumulative distribution function is

P(x) = p0 + (1 - p0) F(x).

The special case `p0 == 0` is the standard logarithmic.

The zero-modified logarithmic distribution is the limiting case of the zero-modified 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 `dzmlogarithmic` 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

`dzmlogarithmic` gives the probability mass function, `pzmlogarithmic` gives the distribution function, `qzmlogarithmic` gives the quantile function, and `rzmlogarithmic` generates random deviates.

Invalid `prob` or `p0` will result in return value `NaN`, with a warning.

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

### Note

Functions `{d,p,q}zmlogarithmic` use `{d,p,q}logarithmic` for all but the trivial input values and p(0).

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca

### References

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

### See Also

`dlogarithmic` for the logarithmic distribution.

`dztnbinom` for the zero modified negative binomial distribution.

### Examples

```p <- 1/(1 + 0.5)
dzmlogarithmic(1:5, prob = p, p0 = 0.6)
(1-0.6) * dlogarithmic(1:5, p)/plogarithmic(0, p, lower = FALSE) # same

## simple relation between survival functions
pzmlogarithmic(0:5, p, p0 = 0.2, lower = FALSE)
(1-0.2) * plogarithmic(0:5, p, lower = FALSE)/plogarithmic(0, p, lower = FALSE) # same

qzmlogarithmic(pzmlogarithmic(0:10, 0.3, p0 = 0.6), 0.3, p0 = 0.6)
```

[Package actuar version 3.1-4 Index]