Is the following statement true? The MLE is unbiased

No. σ̂² = (1/n)Σ(Xᵢ − X̄)² for N(μ, σ²), or X₍ₙ₎ for Uniform(0,θ) Both underestimate systematically; E[X₍ₙ₎] = nθ/(n+1).

Counterexample

Is this true?

The MLE is unbiased

No — it is false.

The counterexample

$\sigma$ ̂ $^{2} = (1/n)\sum(X_{i} - {\bar{X}})^{2}$ for $N(\mu, \sigma^{2})$ , or $X_{(n)}$ for Uniform $(0,\theta)$

Why it fails

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

Keep practicing

Boundary and endpoint

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

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.