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Replacement breakdown of the sample mean and odd-sample median

Statement

For any fixed sample $x \in R^{n}$ , the sample mean $\overset{ˉ}{x} = n^{- 1} \sum_{i = 1}^{n} x_{i}$ has finite-sample replacement breakdown fraction $1 / n$ . If $n = 2 r + 1$ is odd and the conventional sample median is the $(r + 1)$ -st order statistic, its finite-sample replacement breakdown fraction is $(r + 1) / n = ⌊ (n + 1) / 2 ⌋ / n$ .

Facts & Assumptions

Given: A sample $x = (x_{1}, \dots, x_{n}) \in R^{n}$ , the mean statistic $\overset{ˉ}{x}$ , and, when $n = 2 r + 1$ , the sample median $x_{(r + 1)}$ after sorting the observations in nondecreasing order.

[F1]

Finite-sample replacement breakdown: breakdown is the smallest replacement count $m$ for which the supremum displacement over changes to at most $m$ coordinates is infinite, divided by $n$ .

Proof

technique · direct

1.1givenF1algebra

With no replacements the mean displacement is zero. Replacing coordinate $i$ by $M$ changes the sample mean to $\overset{ˉ}{x} + (M - x_{i}) / n$ , whose displacement is unbounded as $M \to + \infty$ . Thus the smallest breaking count for the mean is one, and [F1] gives $1 / n$ .

1.2givenF1algebra

Let $n = 2 r + 1$ . If at most $r$ coordinates are replaced, at least $r + 1$ original observations remain. Writing $x_{\min} = \min_{i} x_{i}$ and $x_{\max} = \max_{i} x_{i}$ , at least $r + 1$ resulting observations are at most $x_{\max}$ and at least $r + 1$ are at least $x_{\min}$ . Therefore the $(r + 1)$ -st order statistic lies in $[x_{\min}, x_{\max}]$ , so its displacement from the original median is bounded uniformly over all such replacements.

1.3givenalgebra

Replace $r + 1$ coordinates by the same value $M > x_{\max}$ . The $r$ unchanged observations are the first $r$ order statistics and the $r + 1$ copies of $M$ occupy positions $r + 1$ through $2 r + 1$ . The median is therefore $M$ , which is unbounded as $M \to + \infty$ .

2.1step 1.2step 1.3F1algebra∎

Steps 1.2–1.3 show that the smallest breaking count for the odd-sample median is $r + 1$ . By [F1] its breakdown fraction is $(r + 1) / (2 r + 1) = ⌊ (n + 1) / 2 ⌋ / n$ .

Depends on

Finite-sample replacement breakdown

Used by

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Sources

P. B. Stark, Statistics 240 Lecture Notes, Part 9: Robustness and Related Topics, §1.2, PDF pp. 2–3, 11–12 (standard reference, not scraped)
Elvezio Ronchetti, Introduction to Robust Statistics, slides 22–23 (standard reference, not scraped)