# GenPoissonBinomial-Distribution: The Generalized Poisson Binomial Distribution In PoissonBinomial: Efficient Computation of Ordinary and Generalized Poisson Binomial Distributions

## Description

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33``` ```dgpbinom(x, probs, val_p, val_q, wts = NULL, method = "DivideFFT", log = FALSE) pgpbinom( x, probs, val_p, val_q, wts = NULL, method = "DivideFFT", lower.tail = TRUE, log.p = FALSE ) qgpbinom( p, probs, val_p, val_q, wts = NULL, method = "DivideFFT", lower.tail = TRUE, log.p = FALSE ) rgpbinom( n, probs, val_p, val_q, wts = NULL, method = "DivideFFT", generator = "Sample" ) ```

## Arguments

 `x` Either a vector of observed sums or NULL. If NULL, probabilities of all possible observations are returned. `probs` Vector of probabilities of success of each Bernoulli trial. `val_p` Vector of values that each trial produces with probability in `probs`. `val_q` Vector of values that each trial produces with probability in `1 - probs`. `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"`, `"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"` or `"Bernoulli"` (abbreviations are allowed).

## 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). They have been modified for use with the generalized Poisson binomial distribution. The Discrete Fourier Transformation of the Characteristic Function (`"Characteristic"`) is derived in Zhang, Hong & Balakrishnan (2018), the Normal Approach (`"Normal"`) and the Refined Normal Approach (`"RefinedNormal"`) are described in Hong (2013). They were slightly adapted for the generalized Poisson binomial distribution.

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

`dgpbinom` gives the density, `pgpbinom` computes the distribution function, `qgpbinom` gives the quantile function and `rgpbinom` generates random deviates.

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

## References

Hong, Y. (2018). 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

Zhang, M., Hong, Y. and Balakrishnan, N. (2018). The generalized Poisson-binomial distribution and the computation of its distribution function. Journal of Statistical Computational and Simulation, 88(8), pp. 1515-1527. doi: 10.1080/00949655.2018.1440294

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30``` ```set.seed(1) pp <- c(1, 0, runif(10), 1, 0, 1) qq <- seq(0, 1, 0.01) va <- rep(5, length(pp)) vb <- 1:length(pp) dgpbinom(NULL, pp, va, vb, method = "DivideFFT") pgpbinom(75:100, pp, va, vb, method = "DivideFFT") qgpbinom(qq, pp, va, vb, method = "DivideFFT") rgpbinom(100, pp, va, vb, method = "DivideFFT") dgpbinom(NULL, pp, va, vb, method = "Convolve") pgpbinom(75:100, pp, va, vb, method = "Convolve") qgpbinom(qq, pp, va, vb, method = "Convolve") rgpbinom(100, pp, va, vb, method = "Convolve") dgpbinom(NULL, pp, va, vb, method = "Characteristic") pgpbinom(75:100, pp, va, vb, method = "Characteristic") qgpbinom(qq, pp, va, vb, method = "Characteristic") rgpbinom(100, pp, va, vb, method = "Characteristic") dgpbinom(NULL, pp, va, vb, method = "Normal") pgpbinom(75:100, pp, va, vb, method = "Normal") qgpbinom(qq, pp, va, vb, method = "Normal") rgpbinom(100, pp, va, vb, method = "Normal") dgpbinom(NULL, pp, va, vb, method = "RefinedNormal") pgpbinom(75:100, pp, va, vb, method = "RefinedNormal") qgpbinom(qq, pp, va, vb, method = "RefinedNormal") rgpbinom(100, pp, va, vb, method = "RefinedNormal") ```

PoissonBinomial documentation built on July 27, 2021, 9:08 a.m.