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A smooth median has bounded influence and about one-half breakdown

Example

Suppose $F$ has a unique median $m$ , satisfies $F (m) = 1 / 2$ , and has a continuous density with $f (m) > 0$ . The median's gross-error sensitivity is $1 / (2 f (m)) < \infty$ . For odd sample sizes $n = 2 r + 1$ , the sample median's replacement breakdown fraction is $(r + 1) / (2 r + 1)$ , which tends to $1 / 2$ as $n$ increases through odd integers.

Facts & Assumptions

Given: A probability law $F$ and median $m$ satisfying the density and uniqueness assumptions in the Statement of Influence function and sensitivity of the median, and an odd sample size $n = 2 r + 1$ .

[F1]

Influence function and sensitivity of the median: the median has gross-error sensitivity $1 / (2 f (m))$ under these assumptions.

[F2]

Replacement breakdown of the sample mean and odd-sample median: the odd-sample median's replacement breakdown fraction is $(r + 1) / (2 r + 1)$ for every fixed sample.

Verification

1.1F1algebra

By [F1], the influence values have absolute value at most $1 / (2 f (m))$ , and their supremum is exactly that value, which is finite because $f (m) > 0$ .

1.2F2algebra

By [F2], the finite-sample replacement breakdown fraction is $(r + 1) / (2 r + 1) = 1 / 2 + 1 / [2 (2 r + 1)]$ .

2.1step 1.2algebra

Since $1 / [2 (2 r + 1)] \to 0$ as $r \to \infty$ , the odd-sample median breakdown fractions tend to $1 / 2$ .

3.1step 1.1step 2.1∎

The bounded pointwise sensitivity and the separate replacement count follow from different definitions and theorems; here the median has both properties under the stated density hypothesis.

Depends on

Used by

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Sources

P. B. Stark, Statistics 240 Lecture Notes, Part 9: Robustness and Related Topics, §§1.2–1.3, PDF pp. 2–3, 13–18 (standard reference, not scraped)
Elvezio Ronchetti, Introduction to Robust Statistics, slides 15–17 and 22 (standard reference, not scraped)