ZeroModifiedNegativeBinomial {actuar} | R Documentation |

Density function, distribution function, quantile function and random
generation for the Zero-Modified Negative Binomial distribution with
parameters `size`

and `prob`

, and arbitrary probability at
zero `p0`

.

dzmnbinom(x, size, prob, p0, log = FALSE) pzmnbinom(q, size, prob, p0, lower.tail = TRUE, log.p = FALSE) qzmnbinom(p, size, prob, p0, lower.tail = TRUE, log.p = FALSE) rzmnbinom(n, size, prob, p0)

`x` |
vector of (strictly positive integer) quantiles. |

`q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`size` |
target for number of successful trials, or dispersion parameter. Must be positive, need not be integer. |

`prob` |
parameter. |

`p0` |
probability mass at zero. |

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

`lower.tail` |
logical; if |

The zero-modified negative binomial distribution with `size`

*= r*, `prob`

*= p* and `p0`

*= p0* is a
discrete mixture between a degenerate distribution at zero and a
(standard) negative binomial. The probability mass function is
*p(0) = p0* and

*
p(x) = (1-p0)/(1-p^r) f(x)*

for *x = 1, 2, …*, *r ≥ 0*, *0 < p < 1* and *0 ≤ p0 ≤ 1*, where *f(x)* is the probability mass
function of the negative binomial.
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 negative binomial.

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

The special case `p0 == 0`

is the zero-truncated negative
binomial.

The limiting case `size == 0`

is the zero-modified logarithmic
distribution with parameters `1 - prob`

and `p0`

.

Unlike the standard negative binomial functions, parametrization
through the mean `mu`

is not supported to avoid ambiguity as
to whether `mu`

is the mean of the underlying negative binomial
or the mean of the zero-modified distribution.

If an element of `x`

is not integer, the result of
`dzmnbinom`

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.

`dzmnbinom`

gives the (log) probability mass function,
`pzmnbinom`

gives the (log) distribution function,
`qzmnbinom`

gives the quantile function, and
`rzmnbinom`

generates random deviates.

Invalid `size`

, `prob`

or `p0`

will result in return
value `NaN`

, with a warning.

The length of the result is determined by `n`

for
`rzmnbinom`

, and is the maximum of the lengths of the
numerical arguments for the other functions.

Functions `{d,p,q}zmnbinom`

use `{d,p,q}nbinom`

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.

`dnbinom`

for the negative binomial distribution.

`dztnbinom`

for the zero-truncated negative binomial
distribution.

`dzmgeom`

for the zero-modified geometric and
`dzmlogarithmic`

for the zero-modified logarithmic, which
are special cases of the zero-modified negative binomial.

## Example 6.3 of Klugman et al. (2012) p <- 1/(1 + 0.5) dzmnbinom(1:5, size = 2.5, prob = p, p0 = 0.6) (1-0.6) * dnbinom(1:5, 2.5, p)/pnbinom(0, 2.5, p, lower = FALSE) # same ## simple relation between survival functions pzmnbinom(0:5, 2.5, p, p0 = 0.2, lower = FALSE) (1-0.2) * pnbinom(0:5, 2.5, p, lower = FALSE) / pnbinom(0, 2.5, p, lower = FALSE) # same qzmnbinom(pzmnbinom(0:10, 2.5, 0.3, p0 = 0.1), 2.5, 0.3, p0 = 0.1)

[Package *actuar* version 3.1-4 Index]