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Is this true?

The MLE is unbiased

No — it is false.

The counterexample

$σ$ ̂ $^{2} = (1/ n) \sum (X_{i} - \overset{ˉ}{X})^{2}$ for $N (μ, σ^{2})$ , or $X_{(n)}$ for Uniform $(0, θ)$

Both underestimate systematically; $E [X_{(n)}] = n θ / (n + 1)$ .

The kind of mistake this is

Boundary and endpoint

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

Drill statements like this

Others that fail the same way

From Estimation › MLE and method of moments

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