LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The lower median of an odd empirical sample is its middle order statistic

Statement

Let $n = 2 r + 1 \geq 1$ and $x_{1}, \dots, x_{n} \in R$ . Let $P_{n} = n^{- 1} \sum_{i} δ_{x_{i}}$ be their empirical probability law and write $x_{(1)} \leq \dots \leq x_{(n)}$ for the sorted sample. Then the lower median $m (P_{n})$ is $x_{(r + 1)}$ .

Facts & Assumptions

Given: An odd sample size $n = 2 r + 1 \geq 1$ , real observations $x_{1}, \dots, x_{n}$ , and their empirical law $P_{n}$ .

[F1]

A finite empirical measure is a probability law: $P_{n} (A) = n^{- 1} \sum_{i} δ_{x_{i}} (A)$ is a probability law.

[F2]

The lower median of a real probability law: $m (P_{n}) = \inf {t : P_{n} ((- \infty, t]) \geq 1 / 2}$ .

Proof

technique · direct

1.1givenF1F2algebra

If $t < x_{(r + 1)}$ , at most $r$ sample coordinates are at most $t$ . Therefore $P_{n} ((- \infty, t]) \leq r / (2 r + 1) < 1 / 2$ , so no such $t$ belongs to the defining set in [F2].

1.2givenF1F2algebra

At $t = x_{(r + 1)}$ , at least $r + 1$ sample coordinates are at most $t$ . Hence $P_{n} ((- \infty, t]) \geq (r + 1) / (2 r + 1) > 1 / 2$ , so $x_{(r + 1)}$ belongs to that defining set.

2.1step 1.1step 1.2∎

Steps 1.1–1.2 show that the defining set has no point below $x_{(r + 1)}$ and contains $x_{(r + 1)}$ itself. Its infimum is therefore $x_{(r + 1)}$ , as claimed.

Depends on

Used by

Bounded influence can coexist with vanishing replacement breakdown Counterexample

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. B. Stark, Statistics 240 Lecture Notes, Part 9: Robustness and Related Topics, §1.2, PDF pp. 2–3 (standard reference, not scraped)
Elvezio Ronchetti, Introduction to Robust Statistics, slides 13 and 22 (standard reference, not scraped)