ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The endpoint MLE for a uniform scale

Example

For $X_{1}, \dots, X_{n}$ iid with density $p_{θ} (x) = θ^{- 1} 1_{[0, θ]} (x)$ , $θ > 0$ , the likelihood is maximized almost surely at the sample endpoint $X_{(n)} = \max_{i} X_{i}$ . The error has the nonregular limit $n (θ - X_{(n)}) \Rightarrow Exp (1 / θ)$ , where the parameter denotes the exponential rate.

Verification

Given: The Uniform model, $n \geq 1$ , and the IID sample in the Example.

[F1] IID observations have one common law and are mutually independent (Identical distribution and IID families); a probability measure has total mass one (Probability measures and probability spaces).

[F2] For this model, $n (θ - X_{(n)})$ converges to the exponential law with rate $1 / θ$ , with CDF $1 - e^{- t / θ}$ for $t \geq 0$ (The uniform endpoint MLE has rate n, not square-root n).

1.1F1givenalgebra

For a sample in the support, independence makes the joint likelihood $L_{n} (ϑ; x) = \prod_{i = 1}^{n} p_{ϑ} (x_{i}) = ϑ^{- n} 1 {x_{(n)} \leq ϑ}$ for $ϑ > 0$ . If $x_{(n)} > 0$ , this likelihood is zero below $x_{(n)}$ and strictly decreases as $ϑ^{- n}$ above it; its maximum is at ${\hat{θ}}_{n} = x_{(n)}$ .

2.1F1F2givenstep 1.1algebra∎

Since each uniform observation has no atom at zero, $P_{θ} (X_{(n)} = 0) \leq P_{θ} (X_{1} = 0) = 0$ . Thus the endpoint formula is the MLE almost surely. On the null event $X_{(n)} = 0$ , the likelihood is $ϑ^{- n}$ and is unbounded as $ϑ ↓ 0$ , so no maximizer exists there; a fixed positive fallback is only a measurable extension, not an MLE on that event. Applying [F2] identifies the exact rate and limit law.

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 15 (standard reference, not scraped)
Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 22 (standard reference, not scraped)