ZeroModifiedBinomial | 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
= p_0
is a discrete
mixture between a degenerate distribution at zero and a (standard)
binomial. The probability mass function is p(0) = p_0
and
%
p(x) = \frac{(1-p_0)}{(1 - (1-p)^n)} f(x)
for x = 1, \ldots, n
, 0 < p \le 1
and 0 \le
p_0 \le 1
, where f(x)
is the probability mass
function of the binomial.
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 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) \ge 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))
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