Exact Procedures

Ordinary Poisson Binomial Distribution

Direct Convolution

The Direct Convolution (DC) approach is requested with method = "Convolve".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

dpbinom(NULL, pp, wt, "Convolve")
#>  [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26
#>  [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19
#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13
#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10
#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17
#> [61] 9.411166e-19 6.727527e-21
ppbinom(NULL, pp, wt, "Convolve")
#>  [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26
#>  [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19
#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13
#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00

Divide & Conquer FFT Tree Convolution

The Divide & Conquer FFT Tree Convolution (DC-FFT) approach is requested with method = "DivideFFT".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

dpbinom(NULL, pp, wt, "DivideFFT")
#>  [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26
#>  [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19
#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13
#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10
#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17
#> [61] 9.411166e-19 6.727527e-21
ppbinom(NULL, pp, wt, "DivideFFT")
#>  [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26
#>  [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19
#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13
#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00

By design, as proposed by Biscarri, Zhao & Brunner (2018), its results are identical to the DC procedure, if \(n \leq 750\). Thus, differences can be observed for larger \(n > 750\):

set.seed(1)
pp1 <- runif(751)
pp2 <- pp1[1:750]

sum(abs(dpbinom(NULL, pp2, method = "DivideFFT") - dpbinom(NULL, pp2, method = "Convolve")))
#> [1] 0
sum(abs(dpbinom(NULL, pp1, method = "DivideFFT") - dpbinom(NULL, pp1, method = "Convolve")))
#> [1] 0

The reason is that the DC-FFT method splits the input probs vector into as equally sized parts as possible and computes their distributions separately with the DC approach. The results of the portions are then convoluted by means of the Fast Fourier Transformation. As proposed by Biscarri, Zhao & Brunner (2018), no splitting is done for \(n \leq 750\). In addition, the DC-FFT procedure does not produce probabilities \(\leq 5.55e\text{-}17\), i.e. smaller values are rounded off to 0, if \(n > 750\), whereas the smallest possible result of the DC algorithm is \(\sim 1e\text{-}323\). This is most likely caused by the used FFTW3 library.

set.seed(1)
pp1 <- runif(751)

d1 <- dpbinom(NULL, pp1, method = "DivideFFT")
d2 <- dpbinom(NULL, pp1, method = "Convolve")

min(d1[d1 > 0])
#> [1] 1.635357e-321
min(d2[d2 > 0])
#> [1] 1.635357e-321

Discrete Fourier Transformation of the Characteristic Function

The Discrete Fourier Transformation of the Characteristic Function (DFT-CF) approach is requested with method = "Characteristic".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

dpbinom(NULL, pp, wt, "Characteristic")
#>  [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [11] 0.000000e+00 2.238353e-16 3.549132e-15 4.829828e-14 5.804377e-13
#> [16] 6.158818e-12 5.784702e-11 4.822438e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110923e-10
#> [56] 2.392079e-11 1.468354e-12 6.994931e-14 2.513558e-15 0.000000e+00
#> [61] 0.000000e+00 0.000000e+00
ppbinom(NULL, pp, wt, "Characteristic")
#>  [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [11] 0.000000e+00 2.238353e-16 3.772968e-15 5.207125e-14 6.325089e-13
#> [16] 6.791327e-12 6.463834e-11 5.468822e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00

As can be seen, the DFT-CF procedure does not produce probabilities \(\leq 2.22e\text{-}16\), i.e. smaller values are rounded off to 0, most likely due to the used FFTW3 library.

Recursive Formula

The Recursive Formula (RF) approach is requested with method = "Recursive".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

dpbinom(NULL, pp, wt, "Recursive")
#>  [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26
#>  [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19
#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13
#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10
#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17
#> [61] 9.411166e-19 6.727527e-21
ppbinom(NULL, pp, wt, "Recursive")
#>  [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26
#>  [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19
#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13
#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00

Obviously, the RF procedure does produce probabilities \(\leq 5.55e\text{-}17\), because it does not rely on the FFTW3 library. Furthermore, it yields the same results as the DC method.

set.seed(1)
pp <- runif(1000)
wt <- sample(1:10, 1000, TRUE)

sum(abs(dpbinom(NULL, pp, wt, "Convolve") - dpbinom(NULL, pp, wt, "Recursive")))
#> [1] 0

Processing Speed Comparisons

To assess the performance of the exact procedures, we use the microbenchmark package. Each algorithm has to calculate the PMF repeatedly based on random probability vectors. The run times are then summarized in a table that presents, among other statistics, their minima, maxima and means. The following results were recorded on an AMD Ryzen 7 1800X with 32 GiB of RAM and Windows 10 Education (20H2).

library(microbenchmark)
set.seed(1)

f1 <- function() dpbinom(NULL, runif(5000), method = "DivideFFT")
f2 <- function() dpbinom(NULL, runif(5000), method = "Convolve")
f3 <- function() dpbinom(NULL, runif(5000), method = "Recursive")
f4 <- function() dpbinom(NULL, runif(5000), method = "Characteristic")

microbenchmark(f1(), f2(), f3(), f4())
#> Unit: milliseconds
#>  expr     min       lq     mean   median       uq     max neval  cld
#>  f1() 13.4042 13.65425 13.96337 13.77990 13.91975 17.1088   100 a   
#>  f2() 20.6392 20.93765 21.15733 21.10240 21.32325 22.6958   100  b  
#>  f3() 27.5966 27.97090 28.28354 28.17785 28.43690 31.1628   100   c 
#>  f4() 65.7154 66.30750 66.72891 66.55035 66.91000 70.2184   100    d

Clearly, the DC-FFT procedure is the fastest, followed by the DC method, which needs roughly 1.5 times as much time here. RF and DFT-CF are almost two and five times, respectively, slower than DC-FFT, respectively (although RF has some advantages in precision, as stated before).

Generalized Poisson Binomial Distribution

Generalized Direct Convolution

The Generalized Direct Convolution (G-DC) approach is requested with method = "Convolve".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)

dgpbinom(NULL, pp, va, vb, wt, "Convolve")
#>   [1] 1.140600e-31 5.349930e-30 1.164698e-28 1.572037e-27 1.491024e-26
#>   [6] 1.077204e-25 6.336147e-25 3.215011e-24 1.466295e-23 6.127671e-23
#>  [11] 2.363402e-22 8.484857e-22 2.866109e-21 9.171228e-21 2.788507e-20
#>  [16] 8.091940e-20 2.254155e-19 6.051395e-19 1.570129e-18 3.953458e-18
#>  [21] 9.696098e-18 2.321913e-17 5.442392e-17 1.251302e-16 2.824507e-16
#>  [26] 6.264454e-16 1.366745e-15 2.934598e-15 6.203639e-15 1.292697e-14
#>  [31] 2.657759e-14 5.394727e-14 1.081983e-13 2.144873e-13 4.201625e-13
#>  [36] 8.135609e-13 1.557745e-12 2.949821e-12 5.527695e-12 1.025815e-11
#>  [41] 1.885777e-11 3.434641e-11 6.196981e-11 1.106787e-10 1.956340e-10
#>  [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753751e-09 2.972596e-09
#>  [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08
#>  [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07
#>  [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06
#>  [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05
#>  [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05
#>  [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04
#>  [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03
#>  [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03
#>  [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03
#>  [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02
#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02
#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02
#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02
#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02
#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02
#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02
#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02
#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03
#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03
#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03
#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04
#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04
#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05
#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05
#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06
#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07
#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08
#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09
#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11
#> [196] 1.676154e-11 7.585978e-12 3.326429e-12 1.407527e-12 5.717370e-13
#> [201] 2.216349e-13 8.149241e-14 2.824954e-14 9.179165e-15 2.780017e-15
#> [206] 7.803525e-16 2.018046e-16 4.775552e-17 1.025798e-17 1.979767e-18
#> [211] 3.386554e-19 5.038594e-20 6.336865e-21 6.424747e-22 4.821385e-23
#> [216] 2.108301e-24
pgpbinom(NULL, pp, va, vb, wt, "Convolve")
#>   [1] 1.140600e-31 5.463990e-30 1.219337e-28 1.693971e-27 1.660421e-26
#>   [6] 1.243246e-25 7.579393e-25 3.972950e-24 1.863590e-23 7.991261e-23
#>  [11] 3.162528e-22 1.164739e-21 4.030847e-21 1.320208e-20 4.108715e-20
#>  [16] 1.220065e-19 3.474220e-19 9.525615e-19 2.522691e-18 6.476149e-18
#>  [21] 1.617225e-17 3.939138e-17 9.381530e-17 2.189455e-16 5.013962e-16
#>  [26] 1.127842e-15 2.494586e-15 5.429184e-15 1.163282e-14 2.455979e-14
#>  [31] 5.113739e-14 1.050847e-13 2.132829e-13 4.277703e-13 8.479327e-13
#>  [36] 1.661494e-12 3.219239e-12 6.169059e-12 1.169675e-11 2.195491e-11
#>  [41] 4.081268e-11 7.515909e-11 1.371289e-10 2.478076e-10 4.434415e-10
#>  [46] 7.859810e-10 1.380789e-09 2.406013e-09 4.159763e-09 7.132360e-09
#>  [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08
#>  [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07
#>  [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06
#>  [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05
#>  [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04
#>  [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03
#>  [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03
#>  [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02
#>  [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02
#>  [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01
#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01
#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01
#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01
#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01
#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01
#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01
#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01
#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01
#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01
#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01
#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01
#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01
#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01
#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01
#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01
#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01
#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00
#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [216] 1.000000e+00

Generalized Divide & Conquer FFT Tree Convolution

The Generalized Divide & Conquer FFT Tree Convolution (G-DC-FFT) approach is requested with method = "DivideFFT".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)

dgpbinom(NULL, pp, va, vb, wt, "DivideFFT")
#>   [1] 1.140600e-31 5.349930e-30 1.164698e-28 1.572037e-27 1.491024e-26
#>   [6] 1.077204e-25 6.336147e-25 3.215011e-24 1.466295e-23 6.127671e-23
#>  [11] 2.363402e-22 8.484857e-22 2.866109e-21 9.171228e-21 2.788507e-20
#>  [16] 8.091940e-20 2.254155e-19 6.051395e-19 1.570129e-18 3.953458e-18
#>  [21] 9.696098e-18 2.321913e-17 5.442392e-17 1.251302e-16 2.824507e-16
#>  [26] 6.264454e-16 1.366745e-15 2.934598e-15 6.203639e-15 1.292697e-14
#>  [31] 2.657759e-14 5.394727e-14 1.081983e-13 2.144873e-13 4.201625e-13
#>  [36] 8.135609e-13 1.557745e-12 2.949821e-12 5.527695e-12 1.025815e-11
#>  [41] 1.885777e-11 3.434641e-11 6.196981e-11 1.106787e-10 1.956340e-10
#>  [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753751e-09 2.972596e-09
#>  [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08
#>  [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07
#>  [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06
#>  [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05
#>  [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05
#>  [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04
#>  [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03
#>  [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03
#>  [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03
#>  [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02
#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02
#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02
#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02
#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02
#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02
#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02
#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02
#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03
#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03
#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03
#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04
#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04
#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05
#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05
#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06
#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07
#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08
#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09
#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11
#> [196] 1.676154e-11 7.585978e-12 3.326429e-12 1.407527e-12 5.717370e-13
#> [201] 2.216349e-13 8.149241e-14 2.824954e-14 9.179165e-15 2.780017e-15
#> [206] 7.803525e-16 2.018046e-16 4.775552e-17 1.025798e-17 1.979767e-18
#> [211] 3.386554e-19 5.038594e-20 6.336865e-21 6.424747e-22 4.821385e-23
#> [216] 2.108301e-24
pgpbinom(NULL, pp, va, vb, wt, "DivideFFT")
#>   [1] 1.140600e-31 5.463990e-30 1.219337e-28 1.693971e-27 1.660421e-26
#>   [6] 1.243246e-25 7.579393e-25 3.972950e-24 1.863590e-23 7.991261e-23
#>  [11] 3.162528e-22 1.164739e-21 4.030847e-21 1.320208e-20 4.108715e-20
#>  [16] 1.220065e-19 3.474220e-19 9.525615e-19 2.522691e-18 6.476149e-18
#>  [21] 1.617225e-17 3.939138e-17 9.381530e-17 2.189455e-16 5.013962e-16
#>  [26] 1.127842e-15 2.494586e-15 5.429184e-15 1.163282e-14 2.455979e-14
#>  [31] 5.113739e-14 1.050847e-13 2.132829e-13 4.277703e-13 8.479327e-13
#>  [36] 1.661494e-12 3.219239e-12 6.169059e-12 1.169675e-11 2.195491e-11
#>  [41] 4.081268e-11 7.515909e-11 1.371289e-10 2.478076e-10 4.434415e-10
#>  [46] 7.859810e-10 1.380789e-09 2.406013e-09 4.159763e-09 7.132360e-09
#>  [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08
#>  [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07
#>  [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06
#>  [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05
#>  [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04
#>  [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03
#>  [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03
#>  [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02
#>  [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02
#>  [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01
#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01
#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01
#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01
#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01
#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01
#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01
#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01
#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01
#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01
#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01
#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01
#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01
#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01
#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01
#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01
#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01
#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00
#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [216] 1.000000e+00

By design, similar to the ordinary DC-FFT algorithm by Biscarri, Zhao & Brunner (2018), its results are identical to the G-DC procedure, if \(n\) and the number of possible observed values is small. Thus, differences can be observed for larger numbers:

set.seed(1)
pp1 <- runif(250)
va1 <- sample(0:50, 250, TRUE)
vb1 <- sample(0:50, 250, TRUE)
pp2 <- pp1[1:248]
va2 <- va1[1:248]
vb2 <- vb1[1:248]

sum(abs(dgpbinom(NULL, pp1, va1, vb1, method = "DivideFFT")
        - dgpbinom(NULL, pp1, va1, vb1, method = "Convolve")))
#> [1] 0

sum(abs(dgpbinom(NULL, pp2, va2, vb2, method = "DivideFFT")
        - dgpbinom(NULL, pp2, va2, vb2, method = "Convolve")))
#> [1] 0

The reason is that the G-DC-FFT method splits the input probs, val_p and val_q vectors into parts such that the numbers of possible observations of all parts are as equally sized as possible. Their distributions are then computed separately with the G-DC approach. The results of the portions are then convoluted by means of the Fast Fourier Transformation. For small \(n\) and small distribution sizes, no splitting is needed. In addition, the G-DC-FFT procedure, just like the DC-FFT method, does not produce probabilities \(\leq 5.55e\text{-}17\), i.e. smaller values are rounded off to \(0\), if the total number of possible observations is smaller than \(750\), whereas the smallest possible result of the DC algorithm is \(\sim 1e\text{-}323\). This is most likely caused by the used FFTW3 library.

d1 <- dgpbinom(NULL, pp1, va1, vb1, method = "DivideFFT")
d2 <- dgpbinom(NULL, pp1, va1, vb1, method = "Convolve")

min(d1[d1 > 0])
#> [1] 2.839368e-99
min(d2[d2 > 0])
#> [1] 2.839368e-99

Generalized Discrete Fourier Transformation of the Characteristic Function

The Generalized Discrete Fourier Transformation of the Characteristic Function (G-DFT-CF) approach is requested with method = "Characteristic".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)

dgpbinom(NULL, pp, va, vb, wt, "Characteristic")
#>   [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>   [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2.837237e-16
#>  [26] 6.270062e-16 1.364746e-15 2.935666e-15 6.201829e-15 1.292176e-14
#>  [31] 2.657237e-14 5.394193e-14 1.081902e-13 2.144802e-13 4.201557e-13
#>  [36] 8.135509e-13 1.557735e-12 2.949809e-12 5.527683e-12 1.025814e-11
#>  [41] 1.885776e-11 3.434640e-11 6.196980e-11 1.106787e-10 1.956340e-10
#>  [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753750e-09 2.972596e-09
#>  [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08
#>  [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07
#>  [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06
#>  [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05
#>  [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05
#>  [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04
#>  [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03
#>  [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03
#>  [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03
#>  [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02
#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02
#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02
#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02
#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02
#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02
#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02
#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02
#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03
#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03
#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03
#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04
#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04
#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05
#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05
#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06
#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07
#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08
#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09
#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11
#> [196] 1.676154e-11 7.585978e-12 3.326430e-12 1.407529e-12 5.717381e-13
#> [201] 2.216360e-13 8.149551e-14 2.825209e-14 9.182470e-15 2.781725e-15
#> [206] 7.813323e-16 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [211] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [216] 0.000000e+00
pgpbinom(NULL, pp, va, vb, wt, "Characteristic")
#>   [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>   [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#>  [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2.837237e-16
#>  [26] 9.107298e-16 2.275475e-15 5.211141e-15 1.141297e-14 2.433473e-14
#>  [31] 5.090710e-14 1.048490e-13 2.130392e-13 4.275194e-13 8.476751e-13
#>  [36] 1.661226e-12 3.218961e-12 6.168770e-12 1.169645e-11 2.195459e-11
#>  [41] 4.081235e-11 7.515875e-11 1.371285e-10 2.478072e-10 4.434412e-10
#>  [46] 7.859806e-10 1.380788e-09 2.406013e-09 4.159763e-09 7.132359e-09
#>  [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08
#>  [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07
#>  [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06
#>  [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05
#>  [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04
#>  [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03
#>  [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03
#>  [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02
#>  [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02
#>  [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01
#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01
#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01
#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01
#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01
#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01
#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01
#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01
#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01
#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01
#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01
#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01
#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01
#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01
#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01
#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01
#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01
#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00
#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [216] 1.000000e+00

As can be seen, the G-DFT-CF procedure does not produce probabilities \(\leq 2.2e\text{-}16\), i.e. smaller values are rounded off to 0, most likely due to the used FFTW3 library.

Processing Speed Comparisons

To assess the performance of the exact procedures, we use the microbenchmark package. Each algorithm has to calculate the PMF repeatedly based on random probability and value vectors. The run times are then summarized in a table that presents, among other statistics, their minima, maxima and means. The following results were recorded on an AMD Ryzen 7 1800X with 32 GiB of RAM and Windows 10 Education (20H2).

library(microbenchmark)
n <- 2500
set.seed(1)
va <- sample(1:50, n, TRUE)
vb <- sample(1:50, n, TRUE)

f1 <- function() dgpbinom(NULL, runif(n), va, vb, method = "DivideFFT")
f2 <- function() dgpbinom(NULL, runif(n), va, vb, method = "Convolve")
f3 <- function() dgpbinom(NULL, runif(n), va, vb, method = "Characteristic")

microbenchmark(f1(), f2(), f3())
#> Unit: milliseconds
#>  expr      min        lq      mean    median        uq      max neval cld
#>  f1()  56.8388  57.76210  63.44748  58.63540  60.26615 338.7246   100 a  
#>  f2()  91.1376  91.98105  93.34477  93.04655  94.50995  97.0263   100  b 
#>  f3() 509.4249 517.44770 521.62463 519.51470 522.53045 614.7971   100   c

Clearly, the G-DC-FFT procedure is the fastest one. If \(n\) and \(m\) are large enough, it outperforms both the G-DC and G-DFT-CF approaches. The latter one needs almost six times as much time as G-DC. Generally, the computational speed advantage of the G-DC-FFT procedure increases with larger \(n\) (and \(m\)).