ZeroModifiedLogarithmic {actuar} | R Documentation |

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`

.

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)

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

`q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`prob` |
parameter. |

`p0` |
probability mass at zero. |

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

`lower.tail` |
logical; if |

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.

`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.

Functions `{d,p,q}zmlogarithmic`

use
`{d,p,q}logarithmic`

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.

`dlogarithmic`

for the logarithmic distribution.

`dztnbinom`

for the zero modified negative binomial
distribution.

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]