Solution
Binomial with Both Parameters Hidden
Show the problem again
Six independent observations 2, 4, 5, 5, 3, 7 are drawn from a Binomial(N, p) distribution in which both the integer N and the probability p are unknown. What is the maximum likelihood estimate of N? (You may check the answer by evaluating the profile likelihood, in which p is replaced by its maximizer for each N, namely p = x̄/N.)
Worked solution
For fixed N the best p is x̄/N = (13/3)/N, since the total number of successes is 26 across 6N trials. This gives a profile likelihood L(N) defined for N ≥ 7 (the largest count). Exact evaluation shows that L(N) rises from N = 7 up to N = 10, where L(10)/L(9) ≈ 1.020, and falls afterwards: L(11)/L(10) ≈ 0.997. The profile also decays toward the Poisson limit as N → ∞ (it stays below its value at N = 10 throughout), so the global maximizer is N̂ = 10, with p̂ = 13/30. Such estimates are notoriously unstable: when the sample variance is close to the sample mean, N̂ can explode.
Source: Classic joint estimation of the binomial parameters (Olkin, Petkau and Zidek discuss its instability); the data are original. Statement written for AxiomIQ.