ZeroModifiedBinomial {actuar} | R Documentation |

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

and `prob`

, and probability at zero
`p0`

.

dzmbinom(x, size, prob, p0, log = FALSE) pzmbinom(q, size, prob, p0, lower.tail = TRUE, log.p = FALSE) qzmbinom(p, size, prob, p0, lower.tail = TRUE, log.p = FALSE) rzmbinom(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` |
number of trials (strictly positive integer). |

`prob` |
probability of success on each trial. |

`p0` |
probability mass at zero. |

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

`lower.tail` |
logical; if |

The zero-modified binomial distribution with `size`

*= n*,
`prob`

*= p* and `p0`

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

*
p(x) = (1-p0)/[1 - (1-p)^n] f(x)*

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

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

The special case `p0 == 0`

is the zero-truncated binomial.

If an element of `x`

is not integer, the result of
`dzmbinom`

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.

`dzmbinom`

gives the probability mass function,
`pzmbinom`

gives the distribution function,
`qzmbinom`

gives the quantile function, and
`rzmbinom`

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
`rzmbinom`

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

Functions `{d,p,q}zmbinom`

use `{d,p,q}binom`

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.

`dbinom`

for the binomial distribution.

`dztbinom`

for the zero-truncated binomial distribution.

dzmbinom(1:5, size = 5, prob = 0.4, p0 = 0.2) (1-0.2) * dbinom(1:5, 5, 0.4)/pbinom(0, 5, 0.4, lower = FALSE) # same ## simple relation between survival functions pzmbinom(0:5, 5, 0.4, p0 = 0.2, lower = FALSE) (1-0.2) * pbinom(0:5, 5, 0.4, lower = FALSE) / pbinom(0, 5, 0.4, lower = FALSE) # same qzmbinom(pzmbinom(1:10, 10, 0.6, p0 = 0.1), 10, 0.6, p0 = 0.1) n <- 8; p <- 0.3; p0 <- 0.025 x <- 0:n title <- paste("ZM Binomial(", n, ", ", p, ", p0 = ", p0, ") and Binomial(", n, ", ", p,") PDF", sep = "") plot(x, dzmbinom(x, n, p, p0), type = "h", lwd = 2, ylab = "p(x)", main = title) points(x, dbinom(x, n, p), pch = 19, col = "red") legend("topright", c("ZT binomial probabilities", "Binomial probabilities"), col = c("black", "red"), lty = c(1, 0), lwd = 2, pch = c(NA, 19))

[Package *actuar* version 3.1-4 Index]