ZeroModifiedPoisson {actuar} R Documentation

## The Zero-Modified Poisson Distribution

### Description

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

### Usage

```dzmpois(x, lambda, p0, log = FALSE)
pzmpois(q, lambda, p0, lower.tail = TRUE, log.p = FALSE)
qzmpois(p, lambda, p0, lower.tail = TRUE, log.p = FALSE)
rzmpois(n, lambda, p0)
```

### Arguments

 `x` vector of (strictly positive integer) quantiles. `q` vector of quantiles. `p` vector of probabilities. `n` number of values to return. `lambda` vector of (non negative) means. `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 Poisson distribution is a discrete mixture between a degenerate distribution at zero and a (standard) Poisson. The probability mass function is p(0) = p0 and

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

for x = 1, 2, ..., λ > 0 and 0 ≤ p0 ≤ 1, where f(x) is the probability mass function of the Poisson. 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 Poisson.

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

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

If an element of `x` is not integer, the result of `dzmpois` 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

`dzmpois` gives the (log) probability mass function, `pzmpois` gives the (log) distribution function, `qzmpois` gives the quantile function, and `rzmpois` generates random deviates.

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

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

### Note

Functions `{d,p,q}zmpois` use `{d,p,q}pois` 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

`dpois` for the standard Poisson distribution.

`dztpois` for the zero-truncated Poisson distribution.

### Examples

```dzmpois(0:5, lambda = 1, p0 = 0.2)
(1-0.2) * dpois(0:5, lambda = 1)/ppois(0, 1, lower = FALSE) # same

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

qzmpois(pzmpois(0:10, 1, p0 = 0.7), 1, p0 = 0.7)
```

[Package actuar version 3.1-4 Index]