pmultinom {pmultinom}

Calculate the probability that a multnomial random vector is between, elementwise, two other vectors.

Description

Calculate the probability that a multnomial random vector is between, elementwise, two other vectors.

Usage

pmultinom(lower = -Inf, upper = Inf, size, probs, method)

Arguments

lower

Vector of lower bounds. Lower bounds are excluded

upper

Vector of upper bounds. Upper bounds are included

size

Number of draws

probs

Cell probabilities

method

Method used for computation. Only method currently implemented is "exact"

Details

The calculation follows the scheme suggested in Levin (1981): begin with the equivalent probability for a Poisson random vector, and update it by conditioning on the sum of the vector being equal to the size parameter, using Bayes' theorem. This requires computation of the distribution of a sum of truncated Poisson random variables, which is accomplished using convolution, as per Levin's suggestion for an exact calculation. Levin's suggestion for an approximate calculation, using Edgeworth expansions, may be added to a later version. Fast convolution is achieved using the fastest Fourier transform in the west (Frigo, Johnson 1998).

Value

The probability that a multinomial random vector is greater than all the lower bounds, and less than or equal all the upper bounds:

P(l1 < N1 <= u1, ..., lk < Nk <= uk)

If only the upper bounds are given, then this is the multinomial CDF:

P(N1<=u1, ..., Nk<=uk)

If only the upper bounds are given, then this is the multinomial tail probability:

P(N1>l1, ..., Nk>lk)

References

Fougere, P. F. (1988). Maximum entropy calculations on a discrete probability space. In Maximum-Entropy and Bayesian Methods in Science and Engineering (pp. 205-234). Springer, Dordrecht. doi:10.1007/978-94-009-3049-0_10

Frigo, Matteo, and Steven G. Johnson. (2005). The design and implementation of FFTW3. Proceedings of the IEEE, 93(2), 216-231. doi:10.1109/JPROC.2004.840301

Levin, Bruce. (1981). "A Representation for Multinomial Cumulative Distribution Functions". Annals of Statistics 9 (5): 1123–6. doi:10.1214/aos/1176345593

See Also

invert.pmultinom

Examples

# To determine the bias of a die, Rudolph Wolf rolled it
# 20,000 times. Side 2 was the most frequently observed, and
# was observed 3631 times. What is the probability that a
# fair die would have a side observed this many times or
# more?

# Input: 
1 - pmultinom(upper=rep.int(3630, 6), size=20000,
              probs=rep.int(1/6, 6), method="exact")
# Output:
# [1] 7.379909e-08

# Therefore we conclude that the die is biased. Fougere
# (1988) attempted to account for these biases by assuming
# certain manufacturing errors. Repeating the calculation
# with the distribution Fougere derived:

# Input:
theoretical.dist <- c(.17649, .17542, .15276, .15184, .17227, .17122)
1 - pmultinom(upper=rep.int(3630, 6), size=20000,
              probs=theoretical.dist, method="exact")
# Output:
# [1] 0.043362

# Therefore we conclude that the die still seems more biased
# than Fougere's model can explain.