Robust Estimation of a Location Parameter in the Presence of Asymmetry
Collins, John R.
Ann. Statist., Tome 4 (1976) no. 1, p. 68-85 / Harvested from Project Euclid
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.
Publié le : 1976-01-14
Classification:  Robust estimation,  asymmetry,  location parameter,  $M$-estimators,  minimax variance,  62G35,  62G05,  62G20
@article{1176343348,
     author = {Collins, John R.},
     title = {Robust Estimation of a Location Parameter in the Presence of Asymmetry},
     journal = {Ann. Statist.},
     volume = {4},
     number = {1},
     year = {1976},
     pages = { 68-85},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176343348}
}
Collins, John R. Robust Estimation of a Location Parameter in the Presence of Asymmetry. Ann. Statist., Tome 4 (1976) no. 1, pp.  68-85. http://gdmltest.u-ga.fr/item/1176343348/