The Zero-Modified Geometric Distribution
Density function, distribution function, quantile function and random
generation for the Zero-Modified Geometric distribution with
parameter prob and arbitrary probability at zero p0.
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)
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 geometric distribution with prob = p
and p0 = p0 is a discrete mixture between a
degenerate distribution at zero and a (standard) geometric. The
probability mass function is p(0) = p0 and
p(x) = (1-p0)/(1-p) f(x)
for x = 1, 2, …, 0 < p < 1 and 0 ≤ p0 ≤ 1, where f(x) is the probability mass function of the geometric. 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 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) ≥ p, where P is the distribution function.
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.
Functions {d,p,q}zmgeom use {d,p,q}geom 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.
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.
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)
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