ZeroModifiedGeometric: The Zero-Modified Geometric Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Zero-Modified Geometric Distribution

Description

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

Usage

dzmgeom(x, prob, p0, log = FALSE)
pzmgeom(q, prob, p0, lower.tail = TRUE, log.p = FALSE)
qzmgeom(p, prob, p0, lower.tail = TRUE, log.p = FALSE)
rzmgeom(n, prob, p0)

Arguments

x	vector of (strictly positive integer) quantiles.
q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
prob	parameter. 0 < prob <= 1.
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 \le x], otherwise, P[X > x].

Details

The zero-modified geometric distribution with prob = p and p0 = p_0 is a discrete mixture between a degenerate distribution at zero and a (standard) geometric. The probability mass function is p(0) = p_0 and

% p(x) = \frac{(1-p_0)}{(1-p)} f(x)

for x = 1, 2, \ldots, 0 < p < 1 and 0 \le p_0 \le 1, where f(x) is the probability mass function of the geometric. The cumulative distribution function is

P(x) = p_0 + (1 - p_0) \left(\frac{F(x) - F(0)}{1 - F(0)}\right)

The mean is (1-p_0) \mu and the variance is (1-p_0) \sigma^2 + p_0(1-p_0) \mu^2, where \mu and \sigma^2 are the mean and variance of the zero-truncated geometric.

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

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

If an element of x is not integer, the result of dzmgeom is zero, with a warning.

The quantile is defined as the smallest value x such that P(x) \ge p, where P is the distribution function.

Value

dzmgeom gives the (log) probability mass function, pzmgeom gives the (log) distribution function, qzmgeom gives the quantile function, and rzmgeom 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 rzmgeom, and is the maximum of the lengths of the numerical arguments for the other functions.

Note

Functions {d,p,q}zmgeom use {d,p,q}geom for all but the trivial input values and p(0).

Author(s)

Vincent Goulet [e-mail]

References

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

See Also

dgeom for the geometric distribution.

dztgeom for the zero-truncated geometric distribution.

dzmnbinom for the zero-modified negative binomial, of which the zero-modified geometric is a special case.

Examples

p <- 1/(1 + 0.5)
dzmgeom(1:5, prob = p, p0 = 0.6)
(1-0.6) * dgeom(1:5, p)/pgeom(0, p, lower = FALSE) # same

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

qzmgeom(pzmgeom(0:10, 0.3, p0 = 0.6), 0.3, p0 = 0.6)

actuar documentation built on Nov. 8, 2023, 9:06 a.m.