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 $1/(1 + e^\theta)$ , where $\theta \in \mathbb{R}$ 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 $\theta$ is equal to 0.
B.For M = 10, the maximum likelihood estimate of $\theta$ does not exist.✓ Correct
C.The maximum likelihood estimator of $\theta$ exists for M ∈ {1, 2, …, 9} and is equal to ln(10/M − 1).✓ Correct
D.The method of moments estimator of $\theta$ 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, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: The MLE is unbiased

The parameter space is open so the extremes have no mle

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