The Annals of Statistics

Robust Estimation of a Location Parameter in the Presence of Asymmetry

John R. Collins

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Huber's theory of robust estimation of a location parameter is adapted to obtain estimators that are robust against a class of asymmetric departures from normality. Let $F$ be a distribution function that is governed by the standard normal density on the set $\lbrack - d, d\rbrack$ and is otherwise arbitrary. Let $X_1,\cdots, X_n$ be a random sample from $F(x - \theta)$, where $\theta$ is the unknown location parameter. If $\psi$ is in a class of continuous skew-symmetric functions $\Psi_c$ which vanish outside a certain set $\lbrack -c, c\rbrack$, then the estimator $T_n$, obtained by solving $\sum\psi (X_i - T_n) = 0$ by Newton's method with the sample median as starting value, is a consistent estimator of $\theta$. Also $n^{\frac{1}{2}}(T_n - \theta)$ is asymptotically normal. For a model of symmetric contamination of the normal center of $F$, an asymptotic minimax variance problem is solved for the optimal $\psi$. The solution has the form $\psi(x) = x$ for $|x| \leqq x_0, \psi(x) = x_1 \tanh \lbrack\frac{1}{2}x_1(c - |x|)\rbrack\operatorname{sgn} (x)$ for $x_0 \leqq |x| \leqq c$, and $\psi(x) = 0$ for $|x| \geqq c$. The results are extended to include an unknown scale parameter in the model.

Article information

Ann. Statist., Volume 4, Number 1 (1976), 68-85.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G35: Robustness
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Robust estimation asymmetry location parameter $M$-estimators minimax variance


Collins, John R. Robust Estimation of a Location Parameter in the Presence of Asymmetry. Ann. Statist. 4 (1976), no. 1, 68--85. doi:10.1214/aos/1176343348.

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