Approximate Procedures

Ordinary Poisson Binomial Distribution

Poisson Approximation

The Poisson Approximation (DC) approach is requested with method = "Poisson". It is based on a Poisson distribution, whose parameter is the sum of the probabilities of success.

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

dpbinom(NULL, pp, wt, "Poisson")
#>  [1] 2.263593e-16 8.154460e-15 1.468798e-13 1.763753e-12 1.588454e-11
#>  [6] 1.144462e-10 6.871428e-10 3.536273e-09 1.592402e-08 6.373926e-08
#> [11] 2.296169e-07 7.519830e-07 2.257479e-06 6.255718e-06 1.609704e-05
#> [16] 3.865908e-05 8.704191e-05 1.844490e-04 3.691482e-04 6.999128e-04
#> [21] 1.260697e-03 2.162661e-03 3.541299e-03 5.546660e-03 8.325631e-03
#> [26] 1.199704e-02 1.662255e-02 2.217842e-02 2.853445e-02 3.544609e-02
#> [31] 4.256414e-02 4.946284e-02 5.568342e-02 6.078674e-02 6.440607e-02
#> [36] 6.629115e-02 6.633610e-02 6.458699e-02 6.122916e-02 5.655755e-02
#> [41] 5.093630e-02 4.475488e-02 3.838734e-02 3.216003e-02 2.633059e-02
#> [46] 2.107875e-02 1.650760e-02 1.265269e-02 9.495953e-03 6.981348e-03
#> [51] 5.029979e-03 3.552981e-03 2.461424e-03 1.673044e-03 1.116119e-03
#> [56] 7.310458e-04 4.702766e-04 2.972182e-04 1.846053e-04 1.127169e-04
#> [61] 6.767601e-05 3.996702e-05
ppbinom(NULL, pp, wt, "Poisson")
#>  [1] 2.263593e-16 8.380820e-15 1.552606e-13 1.919013e-12 1.780355e-11
#>  [6] 1.322498e-10 8.193925e-10 4.355666e-09 2.027968e-08 8.401894e-08
#> [11] 3.136359e-07 1.065619e-06 3.323097e-06 9.578815e-06 2.567585e-05
#> [16] 6.433494e-05 1.513768e-04 3.358259e-04 7.049740e-04 1.404887e-03
#> [21] 2.665584e-03 4.828245e-03 8.369543e-03 1.391620e-02 2.224184e-02
#> [26] 3.423887e-02 5.086142e-02 7.303984e-02 1.015743e-01 1.370204e-01
#> [31] 1.795845e-01 2.290474e-01 2.847308e-01 3.455175e-01 4.099236e-01
#> [36] 4.762147e-01 5.425508e-01 6.071378e-01 6.683670e-01 7.249245e-01
#> [41] 7.758608e-01 8.206157e-01 8.590031e-01 8.911631e-01 9.174937e-01
#> [46] 9.385724e-01 9.550800e-01 9.677327e-01 9.772287e-01 9.842100e-01
#> [51] 9.892400e-01 9.927930e-01 9.952544e-01 9.969275e-01 9.980436e-01
#> [56] 9.987746e-01 9.992449e-01 9.995421e-01 9.997267e-01 9.998394e-01
#> [61] 9.999071e-01 1.000000e+00

A comparison with exact computation shows that the approximation quality of the PA procedure increases with smaller probabilities of success. The reason is that the Poisson Binomial distribution approaches a Poisson distribution when the probabilities are very small.

set.seed(1)

# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "Poisson")
#>  [1] 0.0000150619 0.0001672374 0.0009284471 0.0034362888 0.0095385726
#>  [6] 0.0211820073 0.0391985129 0.0621763578 0.0862956727 0.1064633767
#> [11] 0.1182099310 0.1193204840 0.1104046811 0.0942969970 0.0747865595
#> [16] 0.0553587178 0.0384166744 0.0250913815 0.0154776776 0.0090449448
#> [21] 0.0050214559
dpbinom(NULL, pp)
#>  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#>  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "Poisson") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -9.555e-02  1.506e-05  9.038e-03 -2.461e-04  2.407e-02  4.379e-02

# U(0, 0.01) random probabilities of success
pp <- runif(20, 0, 0.01)
dpbinom(NULL, pp, method = "Poisson")
#>  [1] 9.095763e-01 8.620639e-02 4.085167e-03 1.290592e-04 3.057942e-06
#>  [6] 5.796418e-08 9.156063e-10 1.239684e-11 1.468661e-13 1.546605e-15
#> [11] 1.465817e-17 1.262953e-19 9.974852e-22 7.272161e-24 4.923067e-26
#> [16] 3.110605e-28 1.842575e-30 1.027251e-32 5.408845e-35 2.698058e-37
#> [21] 1.278562e-39
dpbinom(NULL, pp)
#>  [1] 9.093051e-01 8.672423e-02 3.861917e-03 1.066765e-04 2.048094e-06
#>  [6] 2.902198e-08 3.145829e-10 2.667571e-12 1.794592e-14 9.656258e-17
#> [11] 4.170114e-19 1.444465e-21 3.994453e-24 8.738444e-27 1.490372e-29
#> [16] 1.938487e-32 1.859939e-35 1.249654e-38 5.381374e-42 1.245845e-45
#> [21] 9.511846e-50
summary(dpbinom(NULL, pp, method = "Poisson") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -5.178e-04  0.000e+00  0.000e+00  0.000e+00  6.000e-10  2.712e-04

Arithmetic Mean Binomial Approximation

The Arithmetic Mean Binomial Approximation (AMBA) approach is requested with method = "Mean". It is based on a Binomial distribution, whose parameter is the arithmetic mean of the probabilities of success.

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
mean(rep(pp, wt))
#> [1] 0.5905641

dpbinom(NULL, pp, wt, "Mean")
#>  [1] 2.204668e-24 1.939788e-22 8.393759e-21 2.381049e-19 4.979863e-18
#>  [6] 8.188480e-17 1.102354e-15 1.249300e-14 1.216331e-13 1.033156e-12
#> [11] 7.749086e-12 5.182139e-11 3.114432e-10 1.693217e-09 8.373498e-09
#> [16] 3.784379e-08 1.569327e-07 5.991812e-07 2.112610e-06 6.896287e-06
#> [21] 2.088890e-05 5.882491e-05 1.542694e-04 3.773093e-04 8.616897e-04
#> [26] 1.839474e-03 3.673702e-03 6.868933e-03 1.203071e-02 1.974641e-02
#> [31] 3.038072e-02 4.382068e-02 5.925587e-02 7.510979e-02 8.921887e-02
#> [36] 9.927353e-02 1.034154e-01 1.007871e-01 9.181496e-02 7.810121e-02
#> [41] 6.195859e-02 4.577391e-02 3.143980e-02 2.003761e-02 1.182352e-02
#> [46] 6.442647e-03 3.232269e-03 1.487928e-03 6.259647e-04 2.395401e-04
#> [51] 8.292214e-05 2.579729e-05 7.155695e-06 1.752667e-06 3.745215e-07
#> [56] 6.875325e-08 1.062521e-08 1.344354e-09 1.337294e-10 9.807924e-12
#> [61] 4.715599e-13 1.115034e-14
ppbinom(NULL, pp, wt, "Mean")
#>  [1] 2.204668e-24 1.961834e-22 8.589942e-21 2.466948e-19 5.226557e-18
#>  [6] 8.711136e-17 1.189465e-15 1.368247e-14 1.353155e-13 1.168472e-12
#> [11] 8.917558e-12 6.073895e-11 3.721822e-10 2.065399e-09 1.043890e-08
#> [16] 4.828268e-08 2.052154e-07 8.043966e-07 2.917007e-06 9.813294e-06
#> [21] 3.070220e-05 8.952711e-05 2.437965e-04 6.211058e-04 1.482796e-03
#> [26] 3.322270e-03 6.995972e-03 1.386490e-02 2.589561e-02 4.564203e-02
#> [31] 7.602274e-02 1.198434e-01 1.790993e-01 2.542091e-01 3.434279e-01
#> [36] 4.427015e-01 5.461169e-01 6.469040e-01 7.387189e-01 8.168201e-01
#> [41] 8.787787e-01 9.245526e-01 9.559924e-01 9.760300e-01 9.878536e-01
#> [46] 9.942962e-01 9.975285e-01 9.990164e-01 9.996424e-01 9.998819e-01
#> [51] 9.999648e-01 9.999906e-01 9.999978e-01 9.999995e-01 9.999999e-01
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00

A comparison with exact computation shows that the approximation quality of the AMBA procedure increases when the probabilities of success are closer to each other. The reason is that, although the expectation remains unchanged, the distribution’s variance becomes smaller the less the probabilities differ. Since this variance is minimized by equal probabilities (but still underestimated), the AMBA method is best suited for situations with very similar probabilities of success.

set.seed(1)

# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "Mean")
#>  [1] 9.203176e-08 2.297178e-06 2.723611e-05 2.039497e-04 1.081780e-03
#>  [6] 4.320318e-03 1.347977e-02 3.364646e-02 6.823695e-02 1.135495e-01
#> [11] 1.558851e-01 1.768638e-01 1.655492e-01 1.271454e-01 7.934094e-02
#> [16] 3.960811e-02 1.544760e-02 4.536271e-03 9.435709e-04 1.239589e-04
#> [21] 7.735255e-06
dpbinom(NULL, pp)
#>  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#>  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "Mean") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -3.801e-02  2.290e-06  6.360e-04  0.000e+00  8.837e-03  1.662e-02

# U(0.3, 0.5) random probabilities of success
pp <- runif(20, 0.3, 0.5)
dpbinom(NULL, pp, method = "Mean")
#>  [1] 4.348271e-05 5.672598e-04 3.515127e-03 1.375712e-02 3.813748e-02
#>  [6] 7.960444e-02 1.298114e-01 1.693472e-01 1.795010e-01 1.561137e-01
#> [11] 1.120132e-01 6.642197e-02 3.249439e-02 1.304339e-02 4.253984e-03
#> [16] 1.109919e-03 2.262438e-04 3.472347e-05 3.774915e-06 2.591904e-07
#> [21] 8.453263e-09
dpbinom(NULL, pp)
#>  [1] 4.015121e-05 5.344728e-04 3.370391e-03 1.338738e-02 3.756479e-02
#>  [6] 7.915145e-02 1.299445e-01 1.702071e-01 1.806555e-01 1.569062e-01
#> [11] 1.121277e-01 6.604356e-02 3.200604e-02 1.269255e-02 4.078679e-03
#> [16] 1.045709e-03 2.088926e-04 3.133484e-05 3.320483e-06 2.216332e-07
#> [21] 7.008006e-09
summary(dpbinom(NULL, pp, method = "Mean") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -1.155e-03  1.400e-09  1.735e-05  0.000e+00  3.508e-04  5.727e-04

# U(0.39, 0.41) random probabilities of success
pp <- runif(20, 0.39, 0.41)
dpbinom(NULL, pp, method = "Mean")
#>  [1] 3.638616e-05 4.854405e-04 3.076305e-03 1.231262e-02 3.490673e-02
#>  [6] 7.451247e-02 1.242621e-01 1.657824e-01 1.797056e-01 1.598344e-01
#> [11] 1.172824e-01 7.112295e-02 3.558286e-02 1.460687e-02 4.871885e-03
#> [16] 1.299951e-03 2.709859e-04 4.253314e-05 4.728746e-06 3.320414e-07
#> [21] 1.107470e-08
dpbinom(NULL, pp)
#>  [1] 3.636149e-05 4.851935e-04 3.075192e-03 1.230970e-02 3.490204e-02
#>  [6] 7.450845e-02 1.242626e-01 1.657891e-01 1.797153e-01 1.598415e-01
#> [11] 1.172840e-01 7.112011e-02 3.557873e-02 1.460374e-02 4.870251e-03
#> [16] 1.299328e-03 2.708111e-04 4.249771e-05 4.723809e-06 3.316172e-07
#> [21] 1.105772e-08
summary(dpbinom(NULL, pp, method = "Mean") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -9.641e-06  1.700e-11  1.747e-07  0.000e+00  2.844e-06  4.689e-06

Geometric Mean Binomial Approximation - Variant A

The Geometric Mean Binomial Approximation (Variant A) (GMBA-A) approach is requested with method = "GeoMean". It is based on a Binomial distribution, whose parameter is the geometric mean of the probabilities of success: \[\hat{p} = \sqrt[n]{p_1 \cdot ... \cdot p_n}\]

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
prod(rep(pp, wt))^(1/sum(wt))
#> [1] 0.4669916

dpbinom(NULL, pp, wt, "GeoMean")
#>  [1] 2.141782e-17 1.144670e-15 3.008684e-14 5.184208e-13 6.586057e-12
#>  [6] 6.578175e-11 5.379195e-10 3.703028e-09 2.189958e-08 1.129911e-07
#> [11] 5.147813e-07 2.091103e-06 7.633772e-06 2.520966e-05 7.572779e-05
#> [16] 2.078916e-04 5.236606e-04 1.214475e-03 2.601021e-03 5.157435e-03
#> [21] 9.489168e-03 1.623184e-02 2.585712e-02 3.841422e-02 5.328923e-02
#> [26] 6.909972e-02 8.382634e-02 9.520502e-02 1.012875e-01 1.009827e-01
#> [31] 9.437363e-02 8.268481e-02 6.791600e-02 5.229152e-02 3.772988e-02
#> [36] 2.550094e-02 1.613623e-02 9.552467e-03 5.285892e-03 2.731219e-03
#> [41] 1.316117e-03 5.906156e-04 2.464113e-04 9.539397e-05 3.419132e-05
#> [46] 1.131690e-05 3.448772e-06 9.643463e-07 2.464308e-07 5.728188e-08
#> [51] 1.204491e-08 2.276152e-09 3.835067e-10 5.705775e-11 7.406038e-12
#> [56] 8.258409e-13 7.752374e-14 5.958061e-15 3.600079e-16 1.603823e-17
#> [61] 4.683928e-19 6.727527e-21
ppbinom(NULL, pp, wt, "GeoMean")
#>  [1] 2.141782e-17 1.166088e-15 3.125293e-14 5.496737e-13 7.135731e-12
#>  [6] 7.291748e-11 6.108370e-10 4.313865e-09 2.621345e-08 1.392046e-07
#> [11] 6.539859e-07 2.745088e-06 1.037886e-05 3.558852e-05 1.113163e-04
#> [16] 3.192079e-04 8.428685e-04 2.057343e-03 4.658364e-03 9.815799e-03
#> [21] 1.930497e-02 3.553681e-02 6.139393e-02 9.980815e-02 1.530974e-01
#> [26] 2.221971e-01 3.060234e-01 4.012285e-01 5.025160e-01 6.034986e-01
#> [31] 6.978723e-01 7.805571e-01 8.484731e-01 9.007646e-01 9.384945e-01
#> [36] 9.639954e-01 9.801316e-01 9.896841e-01 9.949700e-01 9.977012e-01
#> [41] 9.990173e-01 9.996080e-01 9.998544e-01 9.999498e-01 9.999840e-01
#> [46] 9.999953e-01 9.999987e-01 9.999997e-01 9.999999e-01 1.000000e+00
#> [51] 1.000000e+00 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

It is known that the geometric mean of the probabilities of success is always smaller than their arithmetic mean. Thus, we get a stochastically smaller binomial distribution. A comparison with exact computation shows that the approximation quality of the GMBA-A procedure increases when the probabilities of success are closer to each other:

set.seed(1)

# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "GeoMean")
#>  [1] 4.557123e-06 7.742984e-05 6.249130e-04 3.185359e-03 1.150098e-02
#>  [6] 3.126602e-02 6.640491e-02 1.128282e-01 1.557610e-01 1.764351e-01
#> [11] 1.648790e-01 1.273387e-01 8.113517e-02 4.241734e-02 1.801777e-02
#> [16] 6.122779e-03 1.625497e-03 3.249263e-04 4.600672e-05 4.114199e-06
#> [21] 1.747603e-07
dpbinom(NULL, pp)
#>  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#>  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "GeoMean") - dpbinom(NULL, pp))
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#> -0.11151 -0.01493  0.00000  0.00000  0.01140  0.10279

# U(0.4, 0.6) random probabilities of success
pp <- runif(20, 0.4, 0.6)
dpbinom(NULL, pp, method = "GeoMean")
#>  [1] 1.317886e-06 2.551200e-05 2.345875e-04 1.362363e-03 5.604265e-03
#>  [6] 1.735823e-02 4.200318e-02 8.131092e-02 1.278907e-01 1.650496e-01
#> [11] 1.757292e-01 1.546280e-01 1.122499e-01 6.686047e-02 3.235759e-02
#> [16] 1.252775e-02 3.789307e-03 8.629936e-04 1.392173e-04 1.418425e-05
#> [21] 6.864565e-07
dpbinom(NULL, pp)
#>  [1] 1.046635e-06 2.098187e-05 1.993006e-04 1.192678e-03 5.043114e-03
#>  [6] 1.601621e-02 3.964022e-02 7.829406e-02 1.253351e-01 1.642218e-01
#> [11] 1.770816e-01 1.574210e-01 1.151700e-01 6.896627e-02 3.347297e-02
#> [16] 1.296524e-02 3.913788e-03 8.873960e-04 1.421738e-04 1.435144e-05
#> [21] 6.864565e-07
summary(dpbinom(NULL, pp, method = "GeoMean") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -0.0029201 -0.0004375  0.0000000  0.0000000  0.0005612  0.0030169

# U(0.49, 0.51) random probabilities of success
pp <- runif(20, 0.49, 0.51)
dpbinom(NULL, pp, method = "GeoMean")
#>  [1] 9.491177e-07 1.899145e-05 1.805052e-04 1.083550e-03 4.607292e-03
#>  [6] 1.475040e-02 3.689366e-02 7.382266e-02 1.200193e-01 1.601024e-01
#> [11] 1.761970e-01 1.602558e-01 1.202494e-01 7.403508e-02 3.703527e-02
#> [16] 1.482120e-02 4.633845e-03 1.090839e-03 1.818935e-04 1.915586e-05
#> [21] 9.582517e-07
dpbinom(NULL, pp)
#>  [1] 9.472606e-07 1.895984e-05 1.802539e-04 1.082315e-03 4.603107e-03
#>  [6] 1.474011e-02 3.687497e-02 7.379784e-02 1.199969e-01 1.600932e-01
#> [11] 1.762060e-01 1.602781e-01 1.202742e-01 7.405383e-02 3.704562e-02
#> [16] 1.482542e-02 4.635093e-03 1.091093e-03 1.819256e-04 1.915775e-05
#> [21] 9.582517e-07
summary(dpbinom(NULL, pp, method = "GeoMean") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -2.485e-05 -4.219e-06  0.000e+00  0.000e+00  4.185e-06  2.482e-05

Geometric Mean Binomial Approximation - Variant B

The Geometric Mean Binomial Approximation (Variant B) (GMBA-B) approach is requested with method = "GeoMeanCounter". It is based on a Binomial distribution, whose parameter is 1 minus the geometric mean of the probabilities of failure: \[\hat{p} = 1 - \sqrt[n]{(1 - p_1) \cdot ... \cdot (1 - p_n)}\]

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
1 - prod(1 - rep(pp, wt))^(1/sum(wt))
#> [1] 0.7275426

dpbinom(NULL, pp, wt, "GeoMeanCounter")
#>  [1] 3.574462e-35 5.822379e-33 4.664248e-31 2.449471e-29 9.484189e-28
#>  [6] 2.887121e-26 7.195512e-25 1.509685e-23 2.721134e-22 4.279009e-21
#> [11] 5.941642e-20 7.356037e-19 8.184508e-18 8.237686e-17 7.541858e-16
#> [16] 6.310225e-15 4.844429e-14 3.424255e-13 2.235148e-12 1.350769e-11
#> [21] 7.574609e-11 3.948978e-10 1.917264e-09 8.681177e-09 3.670379e-08
#> [26] 1.450549e-07 5.363170e-07 1.856461e-06 6.019586e-06 1.829121e-05
#> [31] 5.209921e-05 1.391205e-04 3.482749e-04 8.172712e-04 1.797236e-03
#> [36] 3.702208e-03 7.139892e-03 1.288219e-02 2.172588e-02 3.421374e-02
#> [41] 5.024851e-02 6.872559e-02 8.738947e-02 1.031108e-01 1.126377e-01
#> [46] 1.136267e-01 1.055364e-01 8.994057e-02 7.004907e-02 4.962603e-02
#> [51] 3.180393e-02 1.831737e-02 9.406320e-03 4.265268e-03 1.687339e-03
#> [56] 5.734528e-04 1.640669e-04 3.843049e-05 7.077304e-06 9.609416e-07
#> [61] 8.553338e-08 3.744258e-09
ppbinom(NULL, pp, wt, "GeoMeanCounter")
#>  [1] 3.574462e-35 5.858123e-33 4.722829e-31 2.496699e-29 9.733859e-28
#>  [6] 2.984460e-26 7.493958e-25 1.584624e-23 2.879597e-22 4.566969e-21
#> [11] 6.398339e-20 7.995871e-19 8.984095e-18 9.136095e-17 8.455467e-16
#> [16] 7.155772e-15 5.560007e-14 3.980256e-13 2.633173e-12 1.614086e-11
#> [21] 9.188695e-11 4.867847e-10 2.404049e-09 1.108523e-08 4.778901e-08
#> [26] 1.928440e-07 7.291610e-07 2.585622e-06 8.605207e-06 2.689642e-05
#> [31] 7.899562e-05 2.181161e-04 5.663910e-04 1.383662e-03 3.180899e-03
#> [36] 6.883107e-03 1.402300e-02 2.690519e-02 4.863107e-02 8.284481e-02
#> [41] 1.330933e-01 2.018189e-01 2.892084e-01 3.923192e-01 5.049569e-01
#> [46] 6.185836e-01 7.241200e-01 8.140606e-01 8.841097e-01 9.337357e-01
#> [51] 9.655396e-01 9.838570e-01 9.932633e-01 9.975286e-01 9.992159e-01
#> [56] 9.997894e-01 9.999534e-01 9.999919e-01 9.999989e-01 9.999999e-01
#> [61] 1.000000e+00 1.000000e+00

It is known that the geometric mean of the probabilities of failure is always smaller than their arithmetic mean. As a result, 1 minus the geometric mean is larger than 1 minus the arithmetic mean. Thus, we get a stochastically larger binomial distribution. A comparison with exact computation shows that the approximation quality of the GMBA-B procedure again increases when the probabilities of success are closer to each other:

set.seed(1)

# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "GeoMeanCounter")
#>  [1] 4.401037e-11 2.019854e-09 4.403304e-08 6.062685e-07 5.912743e-06
#>  [6] 4.341843e-05 2.490859e-04 1.143179e-03 4.262876e-03 1.304297e-02
#> [11] 3.292337e-02 6.868258e-02 1.182069e-01 1.669263e-01 1.915269e-01
#> [16] 1.758024e-01 1.260695e-01 6.807004e-02 2.603394e-02 6.288561e-03
#> [21] 7.215333e-04
dpbinom(NULL, pp)
#>  [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#>  [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "GeoMeanCounter") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -1.469e-01 -1.724e-02 -3.200e-07  0.000e+00  2.592e-02  1.528e-01

# U(0.4, 0.6) random probabilities of success
pp <- runif(20, 0.4, 0.6)
dpbinom(NULL, pp, method = "GeoMeanCounter")
#>  [1] 1.046635e-06 2.073844e-05 1.951870e-04 1.160254e-03 4.885321e-03
#>  [6] 1.548796e-02 3.836059e-02 7.600922e-02 1.223688e-01 1.616443e-01
#> [11] 1.761588e-01 1.586582e-01 1.178895e-01 7.187414e-02 3.560358e-02
#> [16] 1.410928e-02 4.368234e-03 1.018282e-03 1.681387e-04 1.753458e-05
#> [21] 8.685930e-07
dpbinom(NULL, pp)
#>  [1] 1.046635e-06 2.098187e-05 1.993006e-04 1.192678e-03 5.043114e-03
#>  [6] 1.601621e-02 3.964022e-02 7.829406e-02 1.253351e-01 1.642218e-01
#> [11] 1.770816e-01 1.574210e-01 1.151700e-01 6.896627e-02 3.347297e-02
#> [16] 1.296524e-02 3.913788e-03 8.873960e-04 1.421738e-04 1.435144e-05
#> [21] 6.864565e-07
summary(dpbinom(NULL, pp, method = "GeoMeanCounter") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -0.0029663 -0.0005283  0.0000000  0.0000000  0.0004544  0.0029079

# U(0.49, 0.51) random probabilities of success
pp <- runif(20, 0.49, 0.51)
dpbinom(NULL, pp, method = "GeoMeanCounter")
#>  [1] 9.472606e-07 1.895800e-05 1.802225e-04 1.082065e-03 4.601880e-03
#>  [6] 1.473596e-02 3.686475e-02 7.377926e-02 1.199722e-01 1.600709e-01
#> [11] 1.761969e-01 1.602871e-01 1.202964e-01 7.407854e-02 3.706427e-02
#> [16] 1.483571e-02 4.639289e-03 1.092334e-03 1.821786e-04 1.918963e-05
#> [21] 9.601293e-07
dpbinom(NULL, pp)
#>  [1] 9.472606e-07 1.895984e-05 1.802539e-04 1.082315e-03 4.603107e-03
#>  [6] 1.474011e-02 3.687497e-02 7.379784e-02 1.199969e-01 1.600932e-01
#> [11] 1.762060e-01 1.602781e-01 1.202742e-01 7.405383e-02 3.704562e-02
#> [16] 1.482542e-02 4.635093e-03 1.091093e-03 1.819256e-04 1.915775e-05
#> [21] 9.582517e-07
summary(dpbinom(NULL, pp, method = "GeoMeanCounter") - dpbinom(NULL, pp))
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -2.467e-05 -4.159e-06  0.000e+00  0.000e+00  4.196e-06  2.470e-05

Normal Approximation

The Normal Approximation (NA) approach is requested with method = "Normal". It is based on a Normal distribution, whose parameters are derived from the theoretical mean and variance of the input probabilities of success.

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

dpbinom(NULL, pp, wt, "Normal")
#>  [1] 2.552770e-32 1.207834e-30 5.219650e-29 2.022022e-27 7.021785e-26
#>  [6] 2.185917e-24 6.100302e-23 1.526188e-21 3.423032e-20 6.882841e-19
#> [11] 1.240755e-17 2.005270e-16 2.905604e-15 3.774712e-14 4.396661e-13
#> [16] 4.591569e-12 4.299381e-11 3.609645e-10 2.717342e-09 1.834224e-08
#> [21] 1.110185e-07 6.025326e-07 2.932337e-06 1.279682e-05 5.007841e-05
#> [26] 1.757379e-04 5.530339e-04 1.560683e-03 3.949650e-03 8.963710e-03
#> [31] 1.824341e-02 3.329786e-02 5.450317e-02 8.000636e-02 1.053238e-01
#> [36] 1.243451e-01 1.316535e-01 1.250080e-01 1.064497e-01 8.129267e-02
#> [41] 5.567468e-02 3.419491e-02 1.883477e-02 9.303614e-03 4.121280e-03
#> [46] 1.637186e-03 5.832371e-04 1.863241e-04 5.337829e-05 1.371282e-05
#> [51] 3.159002e-06 6.525712e-07 1.208800e-07 2.007813e-08 2.990389e-09
#> [56] 3.993563e-10 4.782059e-11 5.134327e-12 4.942641e-13 4.266130e-14
#> [61] 3.301422e-15 2.441468e-16
ppbinom(NULL, pp, wt, "Normal")
#>  [1] 2.552770e-32 1.233362e-30 5.342987e-29 2.075452e-27 7.229330e-26
#>  [6] 2.258210e-24 6.326123e-23 1.589449e-21 3.581977e-20 7.241039e-19
#> [11] 1.313165e-17 2.136587e-16 3.119262e-15 4.086639e-14 4.805325e-13
#> [16] 5.072102e-12 4.806591e-11 4.090305e-10 3.126373e-09 2.146861e-08
#> [21] 1.324871e-07 7.350197e-07 3.667357e-06 1.646417e-05 6.654258e-05
#> [26] 2.422805e-04 7.953144e-04 2.355997e-03 6.305647e-03 1.526936e-02
#> [31] 3.351276e-02 6.681062e-02 1.213138e-01 2.013201e-01 3.066439e-01
#> [36] 4.309891e-01 5.626426e-01 6.876506e-01 7.941003e-01 8.753930e-01
#> [41] 9.310676e-01 9.652625e-01 9.840973e-01 9.934009e-01 9.975222e-01
#> [46] 9.991594e-01 9.997426e-01 9.999290e-01 9.999823e-01 9.999960e-01
#> [51] 9.999992e-01 9.999999e-01 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

A comparison with exact computation shows that the approximation quality of the NA procedure increases with larger numbers of probabilities of success:

set.seed(1)

# 10 random probabilities of success
pp <- runif(10)
dpn <- dpbinom(NULL, pp, method = "Normal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -0.0053305 -0.0010422  0.0005271  0.0000000  0.0016579  0.0026553

# 1000 random probabilities of success
pp <- runif(1000)
dpn <- dpbinom(NULL, pp, method = "Normal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -8.412e-06  0.000e+00  0.000e+00  0.000e+00  0.000e+00  3.815e-06

# 100000 random probabilities of success
pp <- runif(100000)
dpn <- dpbinom(NULL, pp, method = "Normal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -4.484e-09  0.000e+00  8.720e-13  0.000e+00  4.914e-10  2.734e-09

Refined Normal Approximation

The Refined Normal Approximation (RNA) approach is requested with method = "RefinedNormal". It is based on a Normal distribution, whose parameters are derived from the theoretical mean, variance and skewness of the input probabilities of success.

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

dpbinom(NULL, pp, wt, "RefinedNormal")
#>  [1] 2.579548e-31 1.128297e-29 4.507210e-28 1.611452e-26 5.156486e-25
#>  [6] 1.476806e-23 3.785627e-22 8.685911e-21 1.783953e-19 3.280039e-18
#> [11] 5.399492e-17 7.959230e-16 1.050796e-14 1.242802e-13 1.317210e-12
#> [16] 1.251531e-11 1.066498e-10 8.155390e-10 5.599786e-09 3.455053e-08
#> [21] 1.917106e-07 9.574753e-07 4.308224e-06 1.748069e-05 6.401569e-05
#> [26] 2.117447e-04 6.329842e-04 1.710740e-03 4.180480e-03 9.234968e-03
#> [31] 1.843341e-02 3.322175e-02 5.401115e-02 7.912655e-02 1.043358e-01
#> [36] 1.236782e-01 1.316360e-01 1.256489e-01 1.074322e-01 8.218619e-02
#> [41] 5.618825e-02 3.428872e-02 1.865323e-02 9.032795e-03 3.886960e-03
#> [46] 1.483178e-03 5.004545e-04 1.487517e-04 3.873113e-05 8.757189e-06
#> [51] 1.693868e-06 2.722346e-07 3.388544e-08 2.218356e-09 0.000000e+00
#> [56] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [61] 0.000000e+00 0.000000e+00
ppbinom(NULL, pp, wt, "RefinedNormal")
#>  [1] 2.579548e-31 1.154092e-29 4.622620e-28 1.657678e-26 5.322254e-25
#>  [6] 1.530028e-23 3.938629e-22 9.079774e-21 1.874750e-19 3.467514e-18
#> [11] 5.746244e-17 8.533855e-16 1.136134e-14 1.356415e-13 1.452852e-12
#> [16] 1.396817e-11 1.206179e-10 9.361569e-10 6.535943e-09 4.108647e-08
#> [21] 2.327971e-07 1.190272e-06 5.498496e-06 2.297918e-05 8.699487e-05
#> [26] 2.987396e-04 9.317238e-04 2.642463e-03 6.822944e-03 1.605791e-02
#> [31] 3.449132e-02 6.771307e-02 1.217242e-01 2.008508e-01 3.051866e-01
#> [36] 4.288648e-01 5.605008e-01 6.861497e-01 7.935820e-01 8.757682e-01
#> [41] 9.319564e-01 9.662451e-01 9.848984e-01 9.939312e-01 9.978181e-01
#> [46] 9.993013e-01 9.998018e-01 9.999505e-01 9.999892e-01 9.999980e-01
#> [51] 9.999997e-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

A comparison with exact computation shows that the approximation quality of the RNA procedure increases with larger numbers of probabilities of success:

set.seed(1)

# 10 random probabilities of success
pp <- runif(10)
dpn <- dpbinom(NULL, pp, method = "RefinedNormal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -0.0039538 -0.0006920  0.0003543  0.0000000  0.0017167  0.0023597

# 1000 random probabilities of success
pp <- runif(1000)
dpn <- dpbinom(NULL, pp, method = "RefinedNormal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -2.974e-06  0.000e+00  0.000e+00  0.000e+00  0.000e+00  2.270e-06

# 100000 random probabilities of success
pp <- runif(100000)
dpn <- dpbinom(NULL, pp, method = "RefinedNormal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -3.126e-09  0.000e+00  6.200e-13  0.000e+00  4.616e-10  2.293e-09

Processing Speed Comparisons

To assess the performance of the approximation 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(4000), method = "Normal")
f2 <- function() dpbinom(NULL, runif(4000), method = "RefinedNormal")
f3 <- function() dpbinom(NULL, runif(4000), method = "Poisson")
f4 <- function() dpbinom(NULL, runif(4000), method = "Mean")
f5 <- function() dpbinom(NULL, runif(4000), method = "GeoMean")
f6 <- function() dpbinom(NULL, runif(4000), method = "GeoMeanCounter")
f7 <- function() dpbinom(NULL, runif(4000), method = "DivideFFT")

microbenchmark(f1(), f2(), f3(), f4(), f5(), f6(), f7())
#> Unit: microseconds
#>  expr    min      lq     mean  median      uq     max neval    cld
#>  f1()  599.2  637.80  697.765  649.50  669.00  2716.5   100 a     
#>  f2()  754.1  806.20  867.631  820.05  841.30  2949.5   100  b    
#>  f3()  909.9  984.95 1035.070  993.30 1008.35  3635.5   100   c   
#>  f4() 1313.2 1413.40 1481.377 1428.50 1447.95  3635.5   100    d  
#>  f5() 1464.7 1587.60 1616.613 1600.80 1618.45  2884.4   100    de 
#>  f6() 1453.8 1572.10 1645.180 1586.45 1605.20  3901.6   100     e 
#>  f7() 7513.9 7729.15 7988.867 7805.40 7869.70 12405.8   100      f

Clearly, the NA procedure is the fastest, followed by the RNA and PA methods, which need approximately 25% (RNA) and 50% (PA) more time, and the AMBA and GMBA approaches that need 2.5 times as long as the NA algorithm. AMBA, GMBA-A and GMBA-B procedures exhibit almost equal mean execution speed, with the AMBA algorithm being slightly faster. All of the approximation procedures outperform the fastest exact approach, DC-FFT, by far. Even the slowest approximate algorithm is roughly five times as fast as DC-FFT.

Generalized Poisson Binomial Distribution

Generalized Normal Approximation

The Generalized Normal Approximation (G-NA) approach is requested with method = "Normal". It is based on a Normal distribution, whose parameters are derived from the theoretical mean and variance of the input probabilities of success (see Introduction.

set.seed(2)
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, "Normal")
#>   [1] 5.607923e-34 8.868899e-34 2.266907e-33 5.759009e-33 1.454159e-32
#>   [6] 3.649437e-32 9.103112e-32 2.256856e-31 5.561194e-31 1.362016e-30
#>  [11] 3.315478e-30 8.021587e-30 1.928965e-29 4.610400e-29 1.095224e-28
#>  [16] 2.585931e-28 6.068497e-28 1.415453e-27 3.281403e-27 7.560907e-27
#>  [21] 1.731562e-26 3.941418e-26 8.916960e-26 2.005077e-25 4.481212e-25
#>  [26] 9.954281e-25 2.197730e-24 4.822684e-24 1.051849e-23 2.280173e-23
#>  [31] 4.912836e-23 1.052075e-22 2.239296e-22 4.737247e-22 9.960718e-22
#>  [36] 2.081639e-21 4.323844e-21 8.926573e-21 1.831680e-20 3.735634e-20
#>  [41] 7.572323e-20 1.525612e-19 3.054984e-19 6.080284e-19 1.202787e-18
#>  [46] 2.364851e-18 4.621350e-18 8.976023e-18 1.732802e-17 3.324790e-17
#>  [51] 6.340586e-17 1.201834e-16 2.264174e-16 4.239603e-16 7.890246e-16
#>  [56] 1.459506e-15 2.683313e-15 4.903282e-15 8.905378e-15 1.607563e-14
#>  [61] 2.884254e-14 5.143387e-14 9.116221e-14 1.605945e-13 2.811877e-13
#>  [66] 4.893417e-13 8.464047e-13 1.455104e-12 2.486337e-12 4.222561e-12
#>  [71] 7.127579e-12 1.195799e-11 1.993996e-11 3.304764e-11 5.443857e-11
#>  [76] 8.912982e-11 1.450405e-10 2.345880e-10 3.771137e-10 6.025440e-10
#>  [81] 9.568753e-10 1.510330e-09 2.369401e-09 3.694497e-09 5.725614e-09
#>  [86] 8.819398e-09 1.350224e-08 2.054578e-08 3.107347e-08 4.670967e-08
#>  [91] 6.978689e-08 1.036313e-07 1.529531e-07 2.243755e-07 3.271469e-07
#>  [96] 4.740893e-07 6.828536e-07 9.775638e-07 1.390954e-06 1.967117e-06
#> [101] 2.765018e-06 3.862920e-06 5.363935e-06 7.402890e-06 1.015475e-05
#> [106] 1.384482e-05 1.876097e-05 2.526814e-05 3.382528e-05 4.500488e-05
#> [111] 5.951520e-05 7.822512e-05 1.021915e-04 1.326884e-04 1.712386e-04
#> [116] 2.196444e-04 2.800198e-04 3.548195e-04 4.468649e-04 5.593647e-04
#> [121] 6.959275e-04 8.605635e-04 1.057674e-03 1.292025e-03 1.568701e-03
#> [126] 1.893038e-03 2.270537e-03 2.706749e-03 3.207136e-03 3.776912e-03
#> [131] 4.420856e-03 5.143112e-03 5.946968e-03 6.834635e-03 7.807017e-03
#> [136] 8.863494e-03 1.000172e-02 1.121747e-02 1.250446e-02 1.385431e-02
#> [141] 1.525651e-02 1.669842e-02 1.816543e-02 1.964112e-02 2.110749e-02
#> [146] 2.254536e-02 2.393468e-02 2.525505e-02 2.648616e-02 2.760831e-02
#> [151] 2.860294e-02 2.945314e-02 3.014411e-02 3.066363e-02 3.100235e-02
#> [156] 3.115414e-02 3.111624e-02 3.088932e-02 3.047753e-02 2.988830e-02
#> [161] 2.913216e-02 2.822242e-02 2.717477e-02 2.600684e-02 2.473770e-02
#> [166] 2.338736e-02 2.197622e-02 2.052462e-02 1.905228e-02 1.757799e-02
#> [171] 1.611912e-02 1.469141e-02 1.330871e-02 1.198280e-02 1.072335e-02
#> [176] 9.537908e-03 8.431904e-03 7.408807e-03 6.470249e-03 5.616215e-03
#> [181] 4.845254e-03 4.154698e-03 3.540890e-03 2.999407e-03 2.525274e-03
#> [186] 2.113156e-03 1.757538e-03 1.452874e-03 1.193717e-03 9.748208e-04
#> [191] 7.912218e-04 6.382955e-04 5.117942e-04 4.078674e-04 3.230671e-04
#> [196] 2.543411e-04 1.990171e-04 1.547798e-04 1.196432e-04 9.192046e-05
#> [201] 7.019178e-05 5.327340e-05 4.018691e-05 3.013068e-05 2.245346e-05
#> [206] 1.663059e-05 1.224284e-05 8.957907e-06 6.514501e-06 1.614725e-05
pgpbinom(NULL, pp, va, vb, wt, "Normal")
#>   [1] 5.607923e-34 1.447682e-33 3.714589e-33 9.473598e-33 2.401518e-32
#>   [6] 6.050955e-32 1.515407e-31 3.772263e-31 9.333457e-31 2.295361e-30
#>  [11] 5.610840e-30 1.363243e-29 3.292208e-29 7.902608e-29 1.885484e-28
#>  [16] 4.471416e-28 1.053991e-27 2.469444e-27 5.750847e-27 1.331175e-26
#>  [21] 3.062738e-26 7.004156e-26 1.592112e-25 3.597189e-25 8.078401e-25
#>  [26] 1.803268e-24 4.000998e-24 8.823682e-24 1.934217e-23 4.214390e-23
#>  [31] 9.127226e-23 1.964798e-22 4.204093e-22 8.941340e-22 1.890206e-21
#>  [36] 3.971844e-21 8.295689e-21 1.722226e-20 3.553906e-20 7.289540e-20
#>  [41] 1.486186e-19 3.011798e-19 6.066782e-19 1.214707e-18 2.417494e-18
#>  [46] 4.782345e-18 9.403695e-18 1.837972e-17 3.570774e-17 6.895564e-17
#>  [51] 1.323615e-16 2.525449e-16 4.789624e-16 9.029227e-16 1.691947e-15
#>  [56] 3.151453e-15 5.834767e-15 1.073805e-14 1.964343e-14 3.571905e-14
#>  [61] 6.456159e-14 1.159955e-13 2.071577e-13 3.677521e-13 6.489399e-13
#>  [66] 1.138282e-12 1.984686e-12 3.439790e-12 5.926127e-12 1.014869e-11
#>  [71] 1.727627e-11 2.923425e-11 4.917421e-11 8.222186e-11 1.366604e-10
#>  [76] 2.257903e-10 3.708308e-10 6.054188e-10 9.825325e-10 1.585076e-09
#>  [81] 2.541952e-09 4.052282e-09 6.421683e-09 1.011618e-08 1.584179e-08
#>  [86] 2.466119e-08 3.816343e-08 5.870922e-08 8.978268e-08 1.364924e-07
#>  [91] 2.062792e-07 3.099106e-07 4.628636e-07 6.872392e-07 1.014386e-06
#>  [96] 1.488475e-06 2.171329e-06 3.148893e-06 4.539847e-06 6.506964e-06
#> [101] 9.271982e-06 1.313490e-05 1.849884e-05 2.590173e-05 3.605648e-05
#> [106] 4.990129e-05 6.866226e-05 9.393040e-05 1.277557e-04 1.727606e-04
#> [111] 2.322758e-04 3.105009e-04 4.126924e-04 5.453808e-04 7.166194e-04
#> [116] 9.362638e-04 1.216284e-03 1.571103e-03 2.017968e-03 2.577333e-03
#> [121] 3.273260e-03 4.133824e-03 5.191498e-03 6.483523e-03 8.052224e-03
#> [126] 9.945263e-03 1.221580e-02 1.492255e-02 1.812968e-02 2.190660e-02
#> [131] 2.632745e-02 3.147056e-02 3.741753e-02 4.425217e-02 5.205918e-02
#> [136] 6.092268e-02 7.092440e-02 8.214187e-02 9.464633e-02 1.085006e-01
#> [141] 1.237572e-01 1.404556e-01 1.586210e-01 1.782621e-01 1.993696e-01
#> [146] 2.219150e-01 2.458497e-01 2.711047e-01 2.975909e-01 3.251992e-01
#> [151] 3.538021e-01 3.832553e-01 4.133994e-01 4.440630e-01 4.750653e-01
#> [156] 5.062195e-01 5.373357e-01 5.682250e-01 5.987026e-01 6.285909e-01
#> [161] 6.577230e-01 6.859454e-01 7.131202e-01 7.391271e-01 7.638648e-01
#> [166] 7.872521e-01 8.092283e-01 8.297529e-01 8.488052e-01 8.663832e-01
#> [171] 8.825023e-01 8.971938e-01 9.105025e-01 9.224853e-01 9.332086e-01
#> [176] 9.427465e-01 9.511784e-01 9.585872e-01 9.650575e-01 9.706737e-01
#> [181] 9.755189e-01 9.796736e-01 9.832145e-01 9.862139e-01 9.887392e-01
#> [186] 9.908524e-01 9.926099e-01 9.940628e-01 9.952565e-01 9.962313e-01
#> [191] 9.970225e-01 9.976608e-01 9.981726e-01 9.985805e-01 9.989036e-01
#> [196] 9.991579e-01 9.993569e-01 9.995117e-01 9.996314e-01 9.997233e-01
#> [201] 9.997935e-01 9.998467e-01 9.998869e-01 9.999171e-01 9.999395e-01
#> [206] 9.999561e-01 9.999684e-01 9.999773e-01 9.999839e-01 1.000000e+00

A comparison with exact computation shows that the approximation quality of the NA procedure increases with larger numbers of probabilities of success:

set.seed(2)

# 10 random probabilities of success
pp <- runif(10)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "Normal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -0.0346309 -0.0042919  0.0001378  0.0000000  0.0038447  0.0317044

# 100 random probabilities of success
pp <- runif(100)
va <- sample(0:100, 100, TRUE)
vb <- sample(0:100, 100, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "Normal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -3.006e-05 -1.126e-09  0.000e+00  0.000e+00  1.854e-09  2.967e-05

# 1000 random probabilities of success
pp <- runif(1000)
va <- sample(0:1000, 1000, TRUE)
vb <- sample(0:1000, 1000, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "Normal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -3.152e-08  0.000e+00  3.060e-12  0.000e+00  8.991e-10  3.707e-08

Generalized Refined Normal Approximation

The Generalized Refined Normal Approximation (G-RNA) approach is requested with method = "RefinedNormal". It is based on a Normal distribution, whose parameters are derived from the theoretical mean, variance and skewness of the input probabilities of success.

set.seed(2)
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, "RefinedNormal")
#>   [1] 5.100768e-32 7.816039e-32 1.959106e-31 4.880045e-31 1.208047e-30
#>   [6] 2.971921e-30 7.265798e-30 1.765311e-29 4.262362e-29 1.022751e-28
#>  [11] 2.438814e-28 5.779315e-28 1.361012e-27 3.185186e-27 7.407878e-27
#>  [16] 1.712136e-26 3.932484e-26 8.975930e-26 2.035985e-25 4.589352e-25
#>  [21] 1.028037e-24 2.288476e-24 5.062470e-24 1.112900e-23 2.431235e-23
#>  [26] 5.278047e-23 1.138660e-22 2.441116e-22 5.200621e-22 1.101015e-21
#>  [31] 2.316333e-21 4.842591e-21 1.006056e-20 2.076983e-20 4.260973e-20
#>  [36] 8.686571e-20 1.759748e-19 3.542530e-19 7.086575e-19 1.408697e-18
#>  [41] 2.782630e-18 5.461965e-18 1.065359e-17 2.064884e-17 3.976912e-17
#>  [46] 7.611065e-17 1.447413e-16 2.735176e-16 5.135966e-16 9.582999e-16
#>  [51] 1.776730e-15 3.273256e-15 5.992053e-15 1.089949e-14 1.970017e-14
#>  [56] 3.538058e-14 6.313772e-14 1.119541e-13 1.972495e-13 3.453144e-13
#>  [61] 6.006676e-13 1.038179e-12 1.782897e-12 3.042246e-12 5.157913e-12
#>  [66] 8.688860e-12 1.454315e-11 2.418568e-11 3.996319e-11 6.560867e-11
#>  [71] 1.070186e-10 1.734408e-10 2.792769e-10 4.467944e-10 7.101774e-10
#>  [76] 1.121527e-09 1.759679e-09 2.743061e-09 4.248282e-09 6.536785e-09
#>  [81] 9.992759e-09 1.517660e-08 2.289965e-08 3.432780e-08 5.112383e-08
#>  [86] 7.564129e-08 1.111860e-07 1.623661e-07 2.355550e-07 3.394997e-07
#>  [91] 4.861107e-07 6.914779e-07 9.771650e-07 1.371840e-06 1.913307e-06
#>  [96] 2.651012e-06 3.649099e-06 4.990081e-06 6.779222e-06 9.149662e-06
#> [101] 1.226837e-05 1.634294e-05 2.162919e-05 2.843967e-05 3.715276e-05
#> [106] 4.822249e-05 6.218875e-05 7.968764e-05 1.014618e-04 1.283702e-04
#> [111] 1.613972e-04 2.016606e-04 2.504176e-04 3.090698e-04 3.791651e-04
#> [116] 4.623982e-04 5.606082e-04 6.757744e-04 8.100102e-04 9.655553e-04
#> [121] 1.144767e-03 1.350110e-03 1.584150e-03 1.849543e-03 2.149024e-03
#> [126] 2.485405e-03 2.861561e-03 3.280420e-03 3.744950e-03 4.258135e-03
#> [131] 4.822941e-03 5.442277e-03 6.118927e-03 6.855467e-03 7.654163e-03
#> [136] 8.516833e-03 9.444692e-03 1.043817e-02 1.149671e-02 1.261856e-02
#> [141] 1.380053e-02 1.503782e-02 1.632377e-02 1.764978e-02 1.900514e-02
#> [146] 2.037702e-02 2.175055e-02 2.310888e-02 2.443348e-02 2.570445e-02
#> [151] 2.690096e-02 2.800177e-02 2.898579e-02 2.983278e-02 3.052397e-02
#> [156] 3.104271e-02 3.137515e-02 3.151071e-02 3.144261e-02 3.116818e-02
#> [161] 3.068902e-02 3.001109e-02 2.914456e-02 2.810352e-02 2.690563e-02
#> [166] 2.557147e-02 2.412399e-02 2.258773e-02 2.098813e-02 1.935073e-02
#> [171] 1.770044e-02 1.606093e-02 1.445398e-02 1.289904e-02 1.141287e-02
#> [176] 1.000927e-02 8.699011e-03 7.489773e-03 6.386301e-03 5.390581e-03
#> [181] 4.502114e-03 3.718233e-03 3.034469e-03 2.444914e-03 1.942594e-03
#> [186] 1.519822e-03 1.168521e-03 8.805066e-04 6.477360e-04 4.625001e-04
#> [191] 2.621189e-04 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [196] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [201] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [206] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
pgpbinom(NULL, pp, va, vb, wt, "RefinedNormal")
#>   [1] 5.100768e-32 1.291681e-31 3.250786e-31 8.130831e-31 2.021130e-30
#>   [6] 4.993051e-30 1.225885e-29 2.991196e-29 7.253558e-29 1.748106e-28
#>  [11] 4.186920e-28 9.966236e-28 2.357636e-27 5.542822e-27 1.295070e-26
#>  [16] 3.007206e-26 6.939690e-26 1.591562e-25 3.627547e-25 8.216899e-25
#>  [21] 1.849727e-24 4.138203e-24 9.200673e-24 2.032968e-23 4.464203e-23
#>  [26] 9.742250e-23 2.112885e-22 4.554002e-22 9.754623e-22 2.076477e-21
#>  [31] 4.392810e-21 9.235402e-21 1.929596e-20 4.006579e-20 8.267552e-20
#>  [36] 1.695412e-19 3.455160e-19 6.997690e-19 1.408427e-18 2.817123e-18
#>  [41] 5.599754e-18 1.106172e-17 2.171531e-17 4.236415e-17 8.213328e-17
#>  [46] 1.582439e-16 3.029852e-16 5.765028e-16 1.090099e-15 2.048399e-15
#>  [51] 3.825129e-15 7.098385e-15 1.309044e-14 2.398993e-14 4.369010e-14
#>  [56] 7.907068e-14 1.422084e-13 2.541625e-13 4.514120e-13 7.967264e-13
#>  [61] 1.397394e-12 2.435573e-12 4.218470e-12 7.260717e-12 1.241863e-11
#>  [66] 2.110749e-11 3.565064e-11 5.983632e-11 9.979950e-11 1.654082e-10
#>  [71] 2.724267e-10 4.458675e-10 7.251445e-10 1.171939e-09 1.882116e-09
#>  [76] 3.003643e-09 4.763322e-09 7.506383e-09 1.175466e-08 1.829145e-08
#>  [81] 2.828421e-08 4.346081e-08 6.636046e-08 1.006883e-07 1.518121e-07
#>  [86] 2.274534e-07 3.386394e-07 5.010055e-07 7.365605e-07 1.076060e-06
#>  [91] 1.562171e-06 2.253649e-06 3.230814e-06 4.602653e-06 6.515960e-06
#>  [96] 9.166972e-06 1.281607e-05 1.780615e-05 2.458537e-05 3.373504e-05
#> [101] 4.600341e-05 6.234634e-05 8.397554e-05 1.124152e-04 1.495680e-04
#> [106] 1.977905e-04 2.599792e-04 3.396668e-04 4.411286e-04 5.694988e-04
#> [111] 7.308960e-04 9.325566e-04 1.182974e-03 1.492044e-03 1.871209e-03
#> [116] 2.333607e-03 2.894215e-03 3.569990e-03 4.380000e-03 5.345555e-03
#> [121] 6.490322e-03 7.840432e-03 9.424583e-03 1.127413e-02 1.342315e-02
#> [126] 1.590855e-02 1.877011e-02 2.205053e-02 2.579549e-02 3.005362e-02
#> [131] 3.487656e-02 4.031884e-02 4.643777e-02 5.329323e-02 6.094740e-02
#> [136] 6.946423e-02 7.890892e-02 8.934709e-02 1.008438e-01 1.134624e-01
#> [141] 1.272629e-01 1.423007e-01 1.586245e-01 1.762743e-01 1.952794e-01
#> [146] 2.156564e-01 2.374070e-01 2.605159e-01 2.849493e-01 3.106538e-01
#> [151] 3.375548e-01 3.655565e-01 3.945423e-01 4.243751e-01 4.548991e-01
#> [156] 4.859418e-01 5.173169e-01 5.488276e-01 5.802702e-01 6.114384e-01
#> [161] 6.421274e-01 6.721385e-01 7.012831e-01 7.293866e-01 7.562922e-01
#> [166] 7.818637e-01 8.059877e-01 8.285754e-01 8.495636e-01 8.689143e-01
#> [171] 8.866147e-01 9.026757e-01 9.171296e-01 9.300287e-01 9.414415e-01
#> [176] 9.514508e-01 9.601498e-01 9.676396e-01 9.740259e-01 9.794165e-01
#> [181] 9.839186e-01 9.876368e-01 9.906713e-01 9.931162e-01 9.950588e-01
#> [186] 9.965786e-01 9.977471e-01 9.986276e-01 9.992754e-01 9.997379e-01
#> [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

A comparison with exact computation shows that the approximation quality of the RNA procedure increases with larger numbers of probabilities of success:

set.seed(2)

# 10 random probabilities of success
pp <- runif(10)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "RefinedNormal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -3.045e-02 -4.084e-03  1.727e-04  1.179e-05  4.324e-03  3.161e-02

# 100 random probabilities of success
pp <- runif(100)
va <- sample(0:100, 100, TRUE)
vb <- sample(0:100, 100, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "RefinedNormal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -8.831e-06  0.000e+00  1.000e-12  9.000e-12  3.642e-07  1.333e-05

# 1000 random probabilities of success
pp <- runif(1000)
va <- sample(0:1000, 1000, TRUE)
vb <- sample(0:1000, 1000, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "RefinedNormal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#>       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#> -1.980e-08  0.000e+00  4.960e-12  0.000e+00  1.561e-09  3.197e-08

Processing Speed Comparisons

To assess the performance of the approximation 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)
n <- 1500
set.seed(2)
va <- sample(1:50, n, TRUE)
vb <- sample(1:50, n, TRUE)

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

microbenchmark(f1(), f2(), f3())
#> Unit: milliseconds
#>  expr     min       lq      mean   median       uq      max neval cld
#>  f1()  7.6607  8.07170  9.705939  8.14100  8.64070 106.7701   100  a 
#>  f2()  9.1814  9.61555 10.250806  9.74215 10.01675  13.7779   100  a 
#>  f3() 42.7156 43.67715 44.416052 43.92380 44.33145  55.8752   100   b

Clearly, the G-NA procedure is the fastest, followed by the G-RNA method. Both are roughly 4-5 times as fast as G-DC-FFT.