Part CCSIR NET December 2024the-parameter-space-is-open-so-the-extremes-have-no-mle

The parameter space is open so the extremes have no mle

Let $X_{1}, X_{2}, \dots, X_{10}$ be a random sample of size 10 from a Bernoulli distribution with success probability $1/ (1 + e^{θ})$ , where $θ \in R$ is an unknown parameter. If $M = \sum (i = 1..10) X_{i}$ , then which of the following statements are true?

A.For M = 0, the maximum likelihood estimate of $θ$ is equal to 0.
B.For M = 10, the maximum likelihood estimate of $θ$ does not exist.✓
C.The maximum likelihood estimator of $θ$ exists for M ∈ {1, 2, …, 9} and is equal to ln(10/M − 1).✓
D.The method of moments estimator of $θ$ exists and is equal to M/10.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: The MLE is unbiased

The parameter space is open so the extremes have no mle

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