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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-34/2/h/solution

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Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 2 / h / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

Assuming independent digits under Benford law, put

N = \sum_{i} y_{i}

and

p_{i} = lo g_{10} (1 + 1/ i)

. The count vector has a multinomial distribution, with expected counts

E_{i} = N p_{i}

. Suitable predictive discrepancy statistics include the Pearson chi-squared statistic and multinomial deviance:

T_{P} = i = 1 \sum 9 \frac{( y _{i} - E _{i} ) ^{2}}{E _{i}}, T_{G} = 2 i : y_{i} > 0 \sum y_{i} lo g (y_{i} / E_{i}) .

(1)

The zero-count terms of

T_{G}

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:

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)
}

Each column returned by 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

p_{i}, N

, 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.

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