ZeroModifiedPoisson | R Documentation |
Density function, distribution function, quantile function, random
generation for the Zero-Modified Poisson distribution with parameter
lambda
and arbitrary probability at zero p0
.
dzmpois(x, lambda, p0, log = FALSE)
pzmpois(q, lambda, p0, lower.tail = TRUE, log.p = FALSE)
qzmpois(p, lambda, p0, lower.tail = TRUE, log.p = FALSE)
rzmpois(n, lambda, p0)
x |
vector of (strictly positive integer) quantiles. |
q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of values to return. |
lambda |
vector of (non negative) means. |
p0 |
probability mass at zero. |
log, log.p |
logical; if |
lower.tail |
logical; if |
The zero-modified Poisson distribution is a discrete mixture between a
degenerate distribution at zero and a (standard) Poisson. The
probability mass function is p(0) = p_0
and
%
p(x) = \frac{(1-p_0)}{(1-e^{-\lambda})} f(x)
for x = 1, 2, ...
, \lambda > 0
and 0 \le
p_0 \le 1
, where f(x)
is the probability mass
function of the Poisson.
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 Poisson.
In the terminology of Klugman et al. (2012), the zero-modified
Poisson is a member of the (a, b, 1)
class of distributions
with a = 0
and b = \lambda
.
The special case p0 == 0
is the zero-truncated Poisson.
If an element of x
is not integer, the result of
dzmpois
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.
dzmpois
gives the (log) probability mass function,
pzmpois
gives the (log) distribution function,
qzmpois
gives the quantile function, and
rzmpois
generates random deviates.
Invalid lambda
or p0
will result in return value
NaN
, with a warning.
The length of the result is determined by n
for
rzmpois
, and is the maximum of the lengths of the
numerical arguments for the other functions.
Functions {d,p,q}zmpois
use {d,p,q}pois
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.
dpois
for the standard Poisson distribution.
dztpois
for the zero-truncated Poisson distribution.
dzmpois(0:5, lambda = 1, p0 = 0.2)
(1-0.2) * dpois(0:5, lambda = 1)/ppois(0, 1, lower = FALSE) # same
## simple relation between survival functions
pzmpois(0:5, 1, p0 = 0.2, lower = FALSE)
(1-0.2) * ppois(0:5, 1, lower = FALSE) /
ppois(0, 1, lower = FALSE) # same
qzmpois(pzmpois(0:10, 1, p0 = 0.7), 1, p0 = 0.7)
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