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Cited by 7,228Advanced Statistical Methods and ModelsStatistical Distribution Estimation and ApplicationsStatistical and numerical algorithms

Robust estimation of a location parameter

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Huber, Peter J.·1964·The Annals of Mathematical Statistics
DOI 10.1214/aoms/1177703732
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Abstract

This paper contains a new approach toward a theory of robust estimation; it treats in detail the asymptotic theory of estimating a location parameter for contaminated normal distributions, and exhibits estimators--intermediaries between sample mean and sample median--that are asymptotically most robust (in a sense to be specified) among all translation invariant estimators. For the general background, see Tukey (1960) (p. 448 ff.) Let $x_1, \cdots, x_n$ be independent random variables with common distribution function $F(t - \xi)$. The problem is to estimate the location parameter $\xi$, but with the complication that the prototype distribution $F(t)$ is only approximately known. I shall primarily be concerned with the model of indeterminacy $F = (1 - \epsilon)\Phi + \epsilon H$, where $0 \leqq \epsilon < 1$ is a known number, $\Phi(t) = (2\pi)^{-\frac{1}{2}} \int^t_{-\infty} \exp(-\frac{1}{2}s^2) ds$ is the standard normal cumulative and $H$ is an unknown contaminating distribution. This model arises for instance if the observations are assumed to be normal with variance 1, but a fraction $\epsilon$ of them is affected by gross errors. Later on, I shall also consider other models of indeterminacy, e.g., $\sup_t |F(t) - \Phi(t)| \leqq \epsilon$. Some inconvenience is caused by the fact that location and scale parameters are not uniquely determined: in general, for fixed $\epsilon$, there will be several values of $\xi$ and $\sigma$ such that $\sup_t|F(t) - \Phi((t - \xi)/\sigma)| \leqq \epsilon$, and similarly for the contaminated case. Although this inherent and unavoidable indeterminacy is small if $\epsilon$ is small and is rather irrelevant for practical purposes, it poses awkward problems for the theory, especially for optimality questions. To remove this difficulty, one may either (i) restrict attention to symmetric distributions, and estimate the location of the center of symmetry (this works for $\xi$ but not for $\sigma$); or (ii) one may define the parameter to be estimated in terms of the estimator itself, namely by its asymptotic value for sample size $n \rightarrow \infty$; or (iii) one may define the parameters by arbitrarily chosen functionals of the distribution (e.g., by the expectation, or the median of $F$). All three possibilities have unsatisfactory aspects, and I shall usually choose the variant which is mathematically most convenient. It is interesting to look back to the very origin of the theory of estimation, namely to Gauss and his theory of least squares. Gauss was fully aware that his main reason for assuming an underlying normal distribution and a quadratic loss function was mathematical, i.e., computational, convenience. In later times, this was often forgotten, partly because of the central limit theorem. However, if one wants to be honest, the central limit theorem can at most explain why many distributions occurring in practice are approximately normal. The stress is on the word "approximately." This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): What happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance: seemingly quite mild deviations may already explode its variance. Tukey and others proposed several more robust substitutes--trimmed means, Winsorized means, etc.--and explored their performance for a few typical violations of normality. A general theory of robust estimation is still lacking; it is hoped that the present paper will furnish the first few steps toward such a theory. At the core of the method of least squares lies the idea to minimize the sum of the squared "errors," that is, to adjust the unknown parameters such that the sum of the squares of the differences between observed and computed values is minimized. In the simplest case, with which we are concerned here, namely the estimation of a location parameter, one has to minimize the expression $\sum_i (x_i - T)^2$; this is of course achieved by the sample mean $T = \sum_i x_i/n$. I should like to emphasize that no loss function is involved here; I am only describing how the least squares estimator is defined, and neither the underlying family of distributions nor the true value of the parameter to be estimated enters so far. It is quite natural to ask whether one can obtain more robustness by minimizing another function of the errors than the sum of their squares. We shall therefore concentrate our attention to estimators that can be defined by a minimum principle of the form (for a location parameter): $T = T_n(x_1, \cdots, x_n) minimizes \sum_i \rho(x_i - T),$ \begin{equation*} \tag{M} where \rho is a non-constant function. \end{equation*} Of course, this definition generalizes at once to more general least squares type problems, where several parameters have to be determined. This class of estimato

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  • The Huber contamination model and the breakdown point are the classical framework for estimation under arbitrary corruption.

    “The Huber contamination model (Huber, 1964) and the breakdown point (Tukey, 1960) are the classical framework for estimation under arbitrary corruption.”

    From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Huber, 1964

  • Because each coordinate-wise estimator must commit per coordinate without seeing the joint pattern, its per-coordinate bias is Ω(α) on this class, and the per-coordinate errors compose in l2 with a √d factor.

    “Because each coordinate-wise estimator must commit per coordinate without seeing the joint pattern, its per-coordinate bias is Ω(α) on this class (a onedimensional Le Cam two-point argument under the symmetric corruption marginal, Huber, 1964, Thm.5.1), and the per-coordinate errors compose in ℓ 2 with a √ d factor;”

    From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Huber, 1964, Thm.5.1

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  1. CRAmÉR, H. (1946). Mathematical Methods of Statistics. Princeton Univ. Press.

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Authors
Huber, P. J.
Affiliations
University of California System (US)
Corresponding author
Huber, Peter J.
Venue
The Annals of Mathematical Statistics
Publisher
Institute of Mathematical Statistics
Published
1964-03
Volume
35
Issue
1
Pages
73-101
ISSN
0003-4851 (ISSN-L)
DOI
10.1214/aoms/1177703732
OpenAlex
W2046033161
Cited by
7,228
References
9
Citations by year
2026: 677 · 2025: 452 · 2024: 351 · 2023: 386 · 2022: 303 · 2021: 490 · 2020: 503 · 2019: 408 · 2018: 335 · 2017: 257 · 2016: 259 · 2015: 226 · 2014: 244 · 2013: 171 · 2012: 154
FWCI
9.39
Type
journal article
Language
en
Open access
Locations
  • The Annals of Mathematical Statistics (publishedVersion)
  • Project Euclid (Cornell University) (submittedVersion)
Topics
  • Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
  • Physical Sciences › Mathematics › Applied Mathematics › Statistical and numerical algorithms
Keywords
robust estimation, location parameter, contaminated normal distribution, asymptotic robustness, sample mean, sample median, gross errors, optimal estimators, asymptotic normality, asymptotic efficiency, scale parameter estimation

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