Mathematical Statistics I · Likelihood, score, and Fisher information
An iid Bernoulli sample of size n ≥ 1 contains K successes
Problem
An iid Bernoulli sample of size \(n\ge 1\) contains \(K\) successes. Find all maximum-likelihood estimates when the parameter space is \([0,1]\), handling \(K=0\) and \(K=n\) without taking logarithms of zero. Repeat the existence analysis when the model instead restricts \(p\) to \((0,1)\). Distinguish an attained maximum from a supremum, and explain why solving the interior score equation alone does not settle all cases.
Hint
Separate mixed samples from all-success and all-failure samples.
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