Exercise 6.2: The Olkin-Petkau-Zidek example of MLE fragility

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In 1981, Olkin, Petkau, and Zidek demonstrated an example in which MLE estimates can very wildly with only small changes in the data. We will work through their example in this problem.

a) Say you are measuring the outcomes of \(N\) Bernoulli trials, but you can only measure a positive result; negative results are not detected in your experiment. You do know, however that \(N\), while unknown, is the same for all experiments. The number of positive results you get from a set of measurements (sorted for convenience) are n = 16, 18, 22, 25, 27. Modeling the generative process with Binomial distribution, \(n_i \sim \text{Binom}(\theta, N)\;\;\forall i\), obtain maximum likelihood estimates for \(\theta\) and \(N\). Hint: You can work out an analytical expression for the MLE of \(\theta\) in terms of \(N\), and then you can find \(N\) by enumerating \(N\).

b) Now, let’s say that the final measurement has 28 positive results instead of 27. Repeat your MLE calculation. How do the results vary?