dkbinom: Probability functions for the sum of k independent binomials
In Smisc: Sego Miscellaneous

Description Usage Arguments Details Value Note Author(s) References See Also Examples

The mass and distribution functions of the sum of k independent binomial random variables, with possibly different probabilities.

dkbinom(x, size, prob, log = FALSE, verbose = FALSE,
        method = c("butler", "fft"))
pkbinom(q, size, prob, log.p = FALSE, verbose = FALSE,
        method = c("butler", "naive", "fft"))

`x`	Vector of values at which to evaluate the mass function of the sum of the k binomial variates
`size`	Vector of the number of trials
`prob`	Vector of the probabilities of success
`log, log.p`	logical; if TRUE, probabilities p are given as log(p) (see Note).
`verbose`	`= TRUE` produces output that shows the iterations of the convolutions and 3 arrays, A, B, and C that are used to convolve and reconvolve the distributions. Array C is the final result. See the source code in `dkbinom.c` for more details.
`method`	A character string that uniquely indicates the method. `butler` is the preferred (and default) method, which uses the algorithm given by Butler, et al. The `naive` method is an alternative approach that can be much slower that can handle no more the sum of five binomials, but is useful for validating the other methods. The `naive` method only works for a single value of `q`. The `fft` method uses the fast Fourier transform to compute the convolution of k binomial random variates, and is also useful for checking the other methods.
`q`	Vector of quantiles (value at which to evaluate the distribution function) of the sum of the k binomial variates

size[1] and prob[1] are the size and probability of the first binomial variate, size[2] and prob[2] are the size and probability of the second binomial variate, etc.

If the elements of prob are all the same, then pbinom or dbinom is used. Otherwise, repeating convolutions of the k binomials are used to calculate the mass or the distribution functions.

dkbinom gives the mass function, pkbinom gives the distribution function.

When log.p or log is TRUE, these functions do not have the same precision as dbinom or pbinom when the probabilities are very small, i.e, the values tend to go to -Inf more quickly.

Landon Sego and Alex Venzin

The Butler method is based on the exact algorithm discussed by: Butler, Ken and Stephens, Michael. (1993) The Distribution of a Sum of Binomial Random Variables. Technical Report No. 467, Department of Statistics, Stanford University. http://www.dtic.mil/dtic/tr/fulltext/u2/a266969.pdf

dbinom, pbinom

# A sum of 3 binomials...
dkbinom(c(0, 4, 7), c(3, 4, 2), c(0.3, 0.5, 0.8))
dkbinom(c(0, 4, 7), c(3, 4, 2), c(0.3, 0.5, 0.8), method = "b")
pkbinom(c(0, 4, 7), c(3, 4, 2), c(0.3, 0.5, 0.8))
pkbinom(c(0, 4, 7), c(3, 4, 2), c(0.3, 0.5, 0.8), method = "b")

 
# Compare the output of the 3 methods
pkbinom(4, c(3, 4, 2), c(0.3, 0.5, 0.8), method = "fft")
pkbinom(4, c(3, 4, 2), c(0.3, 0.5, 0.8), method = "butler")
pkbinom(4, c(3, 4, 2), c(0.3, 0.5, 0.8), method = "naive")

# Some inputs
n <- c(30000, 40000, 20000)
p <- c(0.02, 0.01, 0.005)

# Compare timings
x1 <- timeIt(pkbinom(1100, n, p, method = "butler"))
x2 <- timeIt(pkbinom(1100, n, p, method = "naive"))
x3 <- timeIt(pkbinom(1100, n, p, method = "fft"))
pvar(x1, x1 - x2, x2 - x3, x1 - x3, digits = 12)