PoissonBinomial-Distribution: The Poisson Binomial Distribution

In PoissonBinomial: Efficient Computation of Ordinary and Generalized Poisson Binomial Distributions

The Poisson Binomial Distribution

Description

Density, distribution function, quantile function and random generation for the Poisson binomial distribution with probability vector probs.

Usage

dpbinom(x, probs, wts = NULL, method = "DivideFFT", log = FALSE)

ppbinom(
  x,
  probs,
  wts = NULL,
  method = "DivideFFT",
  lower.tail = TRUE,
  log.p = FALSE
)

qpbinom(
  p,
  probs,
  wts = NULL,
  method = "DivideFFT",
  lower.tail = TRUE,
  log.p = FALSE
)

rpbinom(n, probs, wts = NULL, method = "DivideFFT", generator = "Sample")

Arguments

x	Either a vector of observed numbers of successes or NULL. If NULL, probabilities of all possible observations are returned.
probs	Vector of probabilities of success of each Bernoulli trial.
wts	Vector of non-negative integer weights for the input probabilities.
method	Character string that specifies the method of computation and must be one of "DivideFFT", "Convolve", "Characteristic", "Recursive", "Mean", "GeoMean", "GeoMeanCounter", "Poisson", "Normal" or "RefinedNormal" (abbreviations are allowed).
log, log.p	Logical value indicating if results are given as logarithms.
lower.tail	Logical value indicating if results are P[X \leq x] (if TRUE; default) or P[X > x] (if FALSE).
p	Vector of probabilities for computation of quantiles.
n	Number of observations. If length(n) > 1, the length is taken to be the number required.
generator	Character string that specifies the random number generator and must either be "Sample" (default) or "Bernoulli" (abbreviations are allowed). See Details for more information.

Details

See the references for computational details. The Divide and Conquer ("DivideFFT") and Direct Convolution ("Convolve") algorithms are derived and described in Biscarri, Zhao & Brunner (2018). The Discrete Fourier Transformation of the Characteristic Function ("Characteristic"), the Recursive Formula ("Recursive"), the Poisson Approximation ("Poisson"), the Normal Approach ("Normal") and the Refined Normal Approach ("RefinedNormal") are described in Hong (2013). The calculation of the Recursive Formula was modified to overcome the excessive memory requirements of Hong's implementation.

The "Mean" method is a naive binomial approach using the arithmetic mean of the probabilities of success. Similarly, the "GeoMean" and "GeoMeanCounter" procedures are binomial approximations, too, but they form the geometric mean of the probabilities of success ("GeoMean") and their counter probabilities ("GeoMeanCounter"), respectively.

In some special cases regarding the values of probs, the method parameter is ignored (see Introduction vignette).

Random numbers can be generated in two ways. The "Sample" method uses R's sample function to draw random values according to their probabilities that are calculated by dgpbinom. The "Bernoulli" procedure ignores the method parameter and simulates Bernoulli-distributed random numbers according to the probabilities in probs and sums them up. It is a bit slower than the "Sample" generator, but may yield better results, as it allows to obtain observations that cannot be generated by the "Sample" procedure, because dgpbinom may compute 0-probabilities, due to rounding, if the length of probs is large and/or its values contain a lot of very small values.

Value

dpbinom gives the density, ppbinom computes the distribution function, qpbinom gives the quantile function and rpbinom generates random deviates.

For rpbinom, the length of the result is determined by n, and is the lengths of the numerical arguments for the other functions.

References

Hong, Y. (2013). On computing the distribution function for the Poisson binomial distribution. Computational Statistics & Data Analysis, 59, pp. 41-51. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.csda.2012.10.006")}

Biscarri, W., Zhao, S. D. and Brunner, R. J. (2018) A simple and fast method for computing the Poisson binomial distribution. Computational Statistics and Data Analysis, 31, pp. 216–222. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.csda.2018.01.007")}

Examples

set.seed(1)
pp <- c(0, 0, runif(995), 1, 1, 1)
qq <- seq(0, 1, 0.01)

dpbinom(NULL, pp, method = "DivideFFT")
ppbinom(450:550, pp, method = "DivideFFT")
qpbinom(qq, pp, method = "DivideFFT")
rpbinom(100, pp, method = "DivideFFT")

dpbinom(NULL, pp, method = "Convolve")
ppbinom(450:550, pp, method = "Convolve")
qpbinom(qq, pp, method = "Convolve")
rpbinom(100, pp, method = "Convolve")

dpbinom(NULL, pp, method = "Characteristic")
ppbinom(450:550, pp, method = "Characteristic")
qpbinom(qq, pp, method = "Characteristic")
rpbinom(100, pp, method = "Characteristic")

dpbinom(NULL, pp, method = "Recursive")
ppbinom(450:550, pp, method = "Recursive")
qpbinom(qq, pp, method = "Recursive")
rpbinom(100, pp, method = "Recursive")

dpbinom(NULL, pp, method = "Mean")
ppbinom(450:550, pp, method = "Mean")
qpbinom(qq, pp, method = "Mean")
rpbinom(100, pp, method = "Mean")

dpbinom(NULL, pp, method = "GeoMean")
ppbinom(450:550, pp, method = "GeoMean")
qpbinom(qq, pp, method = "GeoMean")
rpbinom(100, pp, method = "GeoMean")

dpbinom(NULL, pp, method = "GeoMeanCounter")
ppbinom(450:550, pp, method = "GeoMeanCounter")
qpbinom(qq, pp, method = "GeoMeanCounter")
rpbinom(100, pp, method = "GeoMeanCounter")

dpbinom(NULL, pp, method = "Poisson")
ppbinom(450:550, pp, method = "Poisson")
qpbinom(qq, pp, method = "Poisson")
rpbinom(100, pp, method = "Poisson")

dpbinom(NULL, pp, method = "Normal")
ppbinom(450:550, pp, method = "Normal")
qpbinom(qq, pp, method = "Normal")
rpbinom(100, pp, method = "Normal")

dpbinom(NULL, pp, method = "RefinedNormal")
ppbinom(450:550, pp, method = "RefinedNormal")
qpbinom(qq, pp, method = "RefinedNormal")
rpbinom(100, pp, method = "RefinedNormal")

PoissonBinomial documentation built on Sept. 30, 2024, 9:43 a.m.