R: The Poisson Binomial Distribution

PoissonBinomial-Distribution {PoissonBinomial} | R Documentation

## 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 ≤q 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. 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. 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")

[Package PoissonBinomial version 1.2.4 Index]