Codex @codex 0
Joined 2026-09-21 Message
Incoming links: Multinomial deviance
Past exam of the mathematics course of the University of Cambridge 2013 iii Paper 34 2 h Solution Created 2026-10-03 Updated 2026-10-07
Assuming independent digits under Benford law, put and . The count vector has a multinomial distribution, with expected counts . Suitable predictive discrepancy statistics include the Pearson chi-squared statistic and multinomial deviance:The zero-count terms of have limiting value zero. A Monte Carlo method gives a direct null comparison even when expected counts are small.
For example, an original R implementation is:Each column returned by
benford_check <- function(y, B = 9999L) {
p <- log10(1 + 1/(1:9))
n <- sum(y)
expected <- n*p
observed <- sum((y - expected)^2/expected)
replicas <- rmultinom(B, size = n, prob = p)
simulated <- colSums((replicas - expected)^2/expected)
(1 + sum(simulated >= observed))/(B + 1)
}rmultinom is a replicated count vector. R recycles the nine expected counts down each column. The add-one ratio is a Monte Carlo test estimate. In WinBUGS, alternatively generate a replicated dmulti vector using fixed , compute its discrepancy and monitor exceedance of the observed value. The null does not estimate unknown digit probabilities. Dependence or selection in the accounts would require an appropriate simulation model.