betabinomUC.RdDensity, distribution function, and random generation for the beta-binomial distribution and the inflated beta-binomial distribution.
dbetabinom(x, size, prob, rho = 0, log = FALSE)
pbetabinom(q, size, prob, rho = 0, log.p = FALSE)
rbetabinom(n, size, prob, rho = 0)
dbetabinom.ab(x, size, shape1, shape2, log = FALSE,
Inf.shape = exp(20), limit.prob = 0.5)
pbetabinom.ab(q, size, shape1, shape2, limit.prob = 0.5,
log.p = FALSE)
rbetabinom.ab(n, size, shape1, shape2, limit.prob = 0.5,
.dontuse.prob = NULL)
dzoibetabinom(x, size, prob, rho = 0, pstr0 = 0, pstrsize = 0,
log = FALSE)
pzoibetabinom(q, size, prob, rho, pstr0 = 0, pstrsize = 0,
lower.tail = TRUE, log.p = FALSE)
rzoibetabinom(n, size, prob, rho = 0, pstr0 = 0, pstrsize = 0)
dzoibetabinom.ab(x, size, shape1, shape2, pstr0 = 0, pstrsize = 0,
log = FALSE)
pzoibetabinom.ab(q, size, shape1, shape2, pstr0 = 0, pstrsize = 0,
lower.tail = TRUE, log.p = FALSE)
rzoibetabinom.ab(n, size, shape1, shape2, pstr0 = 0, pstrsize = 0)number of trials.
number of observations.
Same as runif.
the probability of success \(\mu\). Must be in the unit closed interval \([0,1]\).
the correlation parameter \(\rho\), which
should be in the interval \([0, 1)\).
The default value of 0 corresponds to the
usual binomial distribution with probability prob.
Setting rho = 1 would set both shape parameters equal
to 0, and the ratio 0/0, which is actually NaN,
is interpreted by Beta as 0.5. See the
warning below.
the two (positive) shape parameters of the standard
beta distribution. They are called a and b in
beta respectively.
Note that
shape1 = prob*(1-rho)/rho and
shape2 = (1-prob)*(1-rho)/rho
is an important relationship between the parameters,
so that the shape parameters are infinite by default because
rho = 0; hence limit.prob = prob is used to
obtain the behaviour of the
usual binomial distribution.
Same meaning as runif.
Numeric. A large value such that,
if shape1 or shape2 exceeds this, then
special measures are taken,
e.g., calling dbinom.
Also, if shape1 or shape2 is less than
its reciprocal, then special measures are also taken.
This feature/approximation is needed to avoid numerical
problem with catastrophic cancellation of multiple
lbeta calls.
Numerical vector; recycled if necessary.
If either shape parameters are Inf then the binomial
limit is
taken, with shape1 / (shape1 + shape2) as the
probability of success.
In the case where both are Inf this probability
will be a NaN = Inf/Inf, however,
the value limit.prob is used instead.
Hence the default
for dbetabinom.ab()
is to assume that
both shape parameters are equal as the limit is taken
(indeed, Beta uses 0.5).
Note that
for [dpr]betabinom(),
because rho = 0 by default, then
limit.prob = prob so that the beta-binomial distribution
behaves like the ordinary binomial distribution with respect
to arguments size and prob.
An argument that should be ignored and not used.
Probability of a structual zero
(i.e., ignoring the beta-binomial distribution).
The default value of pstr0 corresponds to the response
having a beta-binomial distribuion inflated only at size.
Probability of a structual maximum value size.
The default value of
pstrsize corresponds to the response having a
beta-binomial distribution inflated only at 0.
dbetabinom and dbetabinom.ab give the density,
pbetabinom and pbetabinom.ab give the
distribution function, and
rbetabinom and rbetabinom.ab generate random
deviates.
dzoibetabinom and dzoibetabinom.ab give the
inflated density,
pzoibetabinom and pzoibetabinom.ab give the
inflated distribution function, and
rzoibetabinom and rzoibetabinom.ab generate
random inflated deviates.
The beta-binomial distribution is a binomial distribution whose
probability of success is not a constant but it is generated
from a beta distribution with parameters shape1 and
shape2. Note that the mean of this beta distribution
is mu = shape1/(shape1+shape2), which therefore is the
mean or the probability of success.
See betabinomial and betabinomialff,
the VGAM family functions for
estimating the parameters, for the formula of the probability
density function and other details.
For the inflated beta-binomial distribution, the probability mass function is $$P(Y = y) = (1 - pstr0 - pstrsize) \times BB(y) + pstr0 \times I[y = 0] + pstrsize \times I[y = size]$$
where \(BB(y)\) is the probability mass function
of the beta-binomial distribution with the same shape parameters
(pbetabinom.ab),
pstr0 is the inflated probability at 0
and pstrsize is the inflated probability at 1.
The default values of pstr0 and pstrsize
mean that these functions behave like the ordinary
Betabinom when only the essential arguments
are inputted.
pzoibetabinom, pzoibetabinom.ab,
pbetabinom and pbetabinom.ab can be particularly
slow.
The functions here ending in .ab are called from those
functions which don't.
The simple transformations
\(\mu=\alpha / (\alpha + \beta)\) and
\(\rho=1/(1 + \alpha + \beta)\) are
used, where \(\alpha\) and \(\beta\) are the
two shape parameters.
Setting rho = 1 is not recommended,
however the code may be
modified in the future to handle this special case.
set.seed(1); rbetabinom(10, 100, prob = 0.5)
#> [1] 52 46 60 49 52 51 63 52 47 38
set.seed(1); rbinom(10, 100, prob = 0.5) # The same as rho = 0
#> [1] 52 46 60 49 52 51 63 52 47 38
if (FALSE) N <- 9; xx <- 0:N; s1 <- 2; s2 <- 3
#> Error: object 'N' not found
dy <- dbetabinom.ab(xx, size = N, shape1 = s1, shape2 = s2)
#> Error: object 'xx' not found
barplot(rbind(dy, dbinom(xx, size = N, prob = s1 / (s1+s2))),
beside = TRUE, col = c("blue","green"), las = 1,
main = paste("Beta-binomial (size=",N,", shape1=", s1,
", shape2=", s2, ") (blue) vs\n",
" Binomial(size=", N, ", prob=", s1/(s1+s2), ") (green)",
sep = ""),
names.arg = as.character(xx), cex.main = 0.8)
#> Error: object 'dy' not found
sum(dy * xx) # Check expected values are equal
#> Error: object 'dy' not found
sum(dbinom(xx, size = N, prob = s1 / (s1+s2)) * xx)
#> Error: object 'xx' not found
# Should be all 0:
cumsum(dy) - pbetabinom.ab(xx, N, shape1 = s1, shape2 = s2)
#> Error: object 'dy' not found
y <- rbetabinom.ab(n = 1e4, size = N, shape1 = s1, shape2 = s2)
#> Error: object 'N' not found
ty <- table(y)
#> Error: object 'y' not found
barplot(rbind(dy, ty / sum(ty)),
beside = TRUE, col = c("blue", "orange"), las = 1,
main = paste("Beta-binomial (size=", N, ", shape1=", s1,
", shape2=", s2, ") (blue) vs\n",
" Random generated beta-binomial(size=", N, ", prob=",
s1/(s1+s2), ") (orange)", sep = ""), cex.main = 0.8,
names.arg = as.character(xx))
#> Error: object 'dy' not found
N <- 1e5; size <- 20; pstr0 <- 0.2; pstrsize <- 0.2
kk <- rzoibetabinom.ab(N, size, s1, s2, pstr0, pstrsize)
#> Error: object 's1' not found
hist(kk, probability = TRUE, border = "blue", ylim = c(0, 0.25),
main = "Blue/green = inflated; orange = ordinary beta-binomial",
breaks = -0.5 : (size + 0.5))
#> Error: object 'kk' not found
sum(kk == 0) / N # Proportion of 0
#> Error: object 'kk' not found
sum(kk == size) / N # Proportion of size
#> Error: object 'kk' not found
lines(0 : size,
dbetabinom.ab(0 : size, size, s1, s2), col = "orange")
#> Error: object 's1' not found
lines(0 : size, col = "green", type = "b",
dzoibetabinom.ab(0 : size, size, s1, s2, pstr0, pstrsize))
#> Error: object 's1' not found
# \dontrun{}