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Investigation of finite-sample properties of robust location and scale estimators

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Pith's one-line read Finite-sample breakdown points of the median, Hodges-Lehmann, MAD and Shamos estimators are derived in closed form, and Monte Carlo unbiasing factors and relative efficiencies are tabulated for sample sizes up to 100.

arxiv 1908.00462 v2 pith:436YLQNP submitted 2019-08-01 stat.ME

Chanseok Park , Haewon Kim , Min Wang This is my paper · ORCID
classification stat.ME
keywords estimatorssamplefinite-samplesizelocationrobustscaleunbiasing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

Robust statistics are tools that keep working when a few data points are bad. The median and the Hodges-Lehmann estimator are robust location tools, while the MAD and the Shamos estimator are robust scale tools. Their behavior for large samples is well known, but for small samples it is less studied.

This paper does two things. First, it derives exact formulas for the finite-sample breakdown point, the largest fraction of observations that can be corrupted before the estimator produces an arbitrarily large answer. For example, the median of 10 observations breaks down at 0.4, not the asymptotic 0.5. The paper gives similar closed forms for the three variants of the Hodges-Lehmann estimator and for the MAD and Shamos scale estimators.

Second, it uses 10 million simulations of normal data for each sample size from 2 to 100 to measure the bias and variance of these estimators. The MAD systematically underestimates the standard deviation at small n, while the Shamos estimator overestimates it. The paper provides correction factors, called unbiasing factors, and shows the corrections reduce bias and mean square error. It also fits simple curves to the simulated biases so the corrections can be extended to sample sizes above 100. A quality control chart example shows the bias-corrected Shamos estimator is nearly as efficient as the usual standard deviation when data are clean, and much more stable when one observation is corrupted.

Extended reading notes

Core claim

The finite-sample breakdown point of the HL1 and Shamos estimators is given in closed form by epsilon_n = floor(n - 1/2 - sqrt((n - 1/2)^2 - 2 floor((n^2 - n - 2)/4)))/n, with similar explicit formulas for HL2, HL3, median and MAD, and these values differ substantially from the asymptotic limits at small n.

Load-bearing premise

The finite-sample breakdown values are computed under the replacement-breakdown definition in Equation (1), which counts the largest m such that the bias remains finite. Under the alternative definition cited in the same section, Equation (1.6.1) of Hettmansperger and McKean (2010), the same estimators have different breakdown values (for example, the median at n = 3 would be 2/3 instead of 1/3), so the reported numbers are convention-dependent.

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Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central breakdown-point result rests on the standard median breakdown formula and the counting assumption for corrupted Walsh averages. The numerical unbiasing factors and variances are simulation outputs under normality; the least-squares coefficients listed as free parameters are fitted to those outputs to support extrapolation beyond n = 100. No new entities are introduced.

free parameters (5)
  • Least-squares coefficients for A_n (MAD bias), Hayes model = a1 = -0.76213, a2 = -0.86413
    Fitted by least squares to simulated MAD biases for finite n; used to extrapolate unbiasing factors for n > 100.
  • Least-squares coefficients for A_n (MAD bias), Williams model = a3 = -0.804168866, a4 = -1.008922
    Alternative fitted curve for MAD bias extrapolation.
  • Least-squares coefficients for B_n (Shamos bias), Hayes model = b1 = 0.414253297, b2 = 0.442396799
    Fitted to simulated Shamos biases.
  • Least-squares coefficients for B_n (Shamos bias), Williams model = b3 = 0.435760656, b4 = -1.0084443
    Fitted to simulated Shamos biases.
  • Least-squares coefficients for variance extrapolation formulas = e.g., median odd: -0.6589, -0.943; median even: -2.1950, 1.929; MAD odd: 0.2996, -149.357; MAD even: -2.417, -153.010…
    Fitted to simulated variances for n > 100 extrapolation.
assumptions (4)
  • standard math The replacement breakdown point of the median of N values is floor((N-1)/2)/N.
    Used in Section 2.1 to bound the breakdown of HL estimators, which are medians of Walsh averages.
  • domain assumption The supremum bias in the breakdown definition is achieved by sending corrupted observations to infinity, so the number of corrupted Walsh averages that become arbitrarily large determines breakdown.
    Standard counting argument for replacement breakdown; stated implicitly in Section 2.1.
  • domain assumption Simulation draws from N(0,1) and scale equivariance extend biases and variances to N(mu, sigma^2).
    Sections 3 and 4 use standard normal only; extension relies on equivariance.
  • domain assumption Asymptotic relative efficiencies ARE(MAD|S_n) = 0.37 and ARE(Shamos|S_n) = 0.863 are correct values from the literature.
    Used in Section 4 to anchor the finite-sample variance ratios.

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Pith. "Pith review of Investigation of finite-sample properties of robust location and scale estimators." pith.science (2026). https://pith.science/paper/436YLQNP

@misc{pith2026190800462,
  author       = {Pith},
  title        = {Pith review of: Investigation of finite-sample properties of robust location and scale estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/436YLQNP}},
  note         = {Machine review of arXiv:1908.00462}
}
read the original abstract

When the experimental data set is contaminated, we usually employ robust alternatives to common location and scale estimators such as the sample median and Hodges-Lehmann estimators for location and the sample median absolute deviation and Shamos estimators for scale. It is well known that these estimators have high positive asymptotic breakdown points and are Fisher-consistent as the sample size tends to infinity. To the best of our knowledge, the finite-sample properties of these estimators, depending on the sample size, have not well been studied in the literature. In this paper, we fill this gap by providing their closed-form finite-sample breakdown points and calculating the unbiasing factors and relative efficiencies of the robust estimators through the extensive Monte Carlo simulations up to the sample size 100. The numerical study shows that the unbiasing factor improves the finite-sample performance significantly. In addition, we provide the predicted values for the unbiasing factors obtained by using the least squares method which can be used for the case of sample size more than 100.

Figures

Figures reproduced from arXiv: 1908.00462 by the authors.

Figure 1
Figure 1. The finite-sample breakdown points under consideration. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Empirical biases of the MAD and Shamos estimators. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. The empirical biases, variances and mean square errors of the estimates under [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The values of n Var(median), n Var(HL), Var(MAD)/(1 − c4(n) 2 ), and Var(Shamos)/(1 − c4(n) 2 ). 33 [PITH_FULL_IMAGE:figures/full_fig_p033_4.png]
Figure 5
Figure 5. Figure 5: The relative efficiencies under consideration. [PITH_FULL_IMAGE:figures/full_fig_p034_5.png]

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Works this paper leans on

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