ZeroModifiedGeometric {actuar} R Documentation

## The Zero-Modified Geometric Distribution

### Description

Density function, distribution function, quantile function and random generation for the Zero-Modified Geometric distribution with parameter `prob` and arbitrary probability at zero `p0`.

### Usage

```dzmgeom(x, prob, p0, log = FALSE)
pzmgeom(q, prob, p0, lower.tail = TRUE, log.p = FALSE)
qzmgeom(p, prob, p0, lower.tail = TRUE, log.p = FALSE)
rzmgeom(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 geometric distribution with `prob` = p and `p0` = p0 is a discrete mixture between a degenerate distribution at zero and a (standard) geometric. The probability mass function is p(0) = p0 and

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

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

P(x) = p0 + (1 - p0) [F(x) - F(0)]/[1 - F(0)].

The mean is (1-p0)m and the variance is (1-p0)v + p0(1-p0)m^2, where m and v are the mean and variance of the zero-truncated geometric.

In the terminology of Klugman et al. (2012), the zero-modified geometric is a member of the (a, b, 1) class of distributions with a = 1-p and b = 0.

The special case `p0 == 0` is the zero-truncated geometric.

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

The quantile is defined as the smallest value x such that P(x) ≥ p, where P is the distribution function.

### Value

`dzmgeom` gives the (log) probability mass function, `pzmgeom` gives the (log) distribution function, `qzmgeom` gives the quantile function, and `rzmgeom` 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 `rzmgeom`, and is the maximum of the lengths of the numerical arguments for the other functions.

### Note

Functions `{d,p,q}zmgeom` use `{d,p,q}geom` 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

`dgeom` for the geometric distribution.

`dztgeom` for the zero-truncated geometric distribution.

`dzmnbinom` for the zero-modified negative binomial, of which the zero-modified geometric is a special case.

### Examples

```p <- 1/(1 + 0.5)
dzmgeom(1:5, prob = p, p0 = 0.6)
(1-0.6) * dgeom(1:5, p)/pgeom(0, p, lower = FALSE) # same

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

qzmgeom(pzmgeom(0:10, 0.3, p0 = 0.6), 0.3, p0 = 0.6)
```

[Package actuar version 3.1-4 Index]