ZeroModifiedPoisson {actuar} | R Documentation |

Density function, distribution function, quantile function, random
generation for the Zero-Modified Poisson distribution with parameter
`lambda`

and arbitrary probability at zero `p0`

.

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)

`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. |

`log, log.p` |
logical; if |

`lower.tail` |
logical; if |

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.

`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.

Functions `{d,p,q}zmpois`

use `{d,p,q}pois`

for all
but the trivial input values and *p(0)*.

Vincent Goulet vincent.goulet@act.ulaval.ca

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

`dpois`

for the standard Poisson distribution.

`dztpois`

for the zero-truncated Poisson distribution.

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]