We consider the following problem arising in robust estimation theory: Find the maximum asymptotic variance of a trimmed mean used to estimate an unknown location parameter when the error distribution is subject to asymmetric contamination. The model for the error distribution is $F = (1 - \varepsilon)F_0 + \varepsilon G$, where $F_0$ is a known distribution symmetric about $0, \varepsilon$ is fixed proportion of contamination, and $G$ is an unknown and possibly asymmetric…
In the framework of Huber's theory of robust estimation of a location parameter, minimax variance M-estimators are studied for error distributions with densities of the form $f(x) = (1 - \varepsilon)h(x) + \varepsilon g(x)$, where $g$ is unknown. A well-known result of Huber (1964) is that when $h$ is strongly unimodal, the least informative density $f_0 = (1 - \varepsilon)h + \varepsilon g_0$ has exponential tails. We study the minimax variance solutions when the known…
Consider the problem of estimating the unknown location parameter $\theta$ based on a random sample from $F(x - \theta)$, where $F$ is an unknown member of the class of distribution functions $\mathscr{F} = \{F: F$ is symmetric about 0 and $\sup_x |F(x) - \Phi(x)| \leq \varepsilon\}$, where $\Phi$ denotes the standard normal distribution function. Huber (1964) showed that $M$-estimation has a minimax property for this model, whereas Sacks and Ylvisaker (1972) showed that $L$-…
The problem considered is that of optimizing a function of a finite number of linear functionals over an infinite dimensional convex set $S$. It is shown that under some reasonably general conditions the method of moment spaces can be used to reduce the problem to one of optimizing over a simple finite dimensional set (generally a set of convex combinations of extreme points of $S$). The results are applied to finding the maximum asymptotic variance of M-estimators over…
If $X_1, \cdots, X_n$ is a random sample from $F(x - \theta)$, where $F$ is an unknown member of a specified class $\mathscr{F}$ of approximately normal symmetric distributions, then an $M$-estimator of the unknown location parameter $\theta$ is obtained by solving the equation $\sum^n_{i=1} \psi(X_i - \hat{\theta}_n) = 0$ for $\hat{\theta}_n$. A suitable measure of the robustness of the $M$-estimator is $\sup \{V(\psi, F): F \in \mathscr{F}\}$, where $V(\psi, F) = \int \psi^2 dF/(\int \psi' dF)^2$…
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…
Methods for statistical analysis of reliability and life data , Methods for statistical analysis of reliability and life data , کتابخانههای دانشگاه کردستان
We consider the following problem arising in robust estimation theory: Find the maximum asymptotic variance of a trimmed mean used to estimate an unknown location parameter when the error distribution is subject to asymmetric contamination. The model for the error distribution is F=(1−ε)F0+εG, where F0 is a known distribution symmetric about 0,ε is fixed proportion of contamination, and G is an unknown and possibly asymmetric…
Minimax Variance M-Estimators in ε-Contamination Models
In the framework of Huber's theory of robust estimation of a location parameter, minimax variance M-estimators are studied for error distributions with densities of the form f(x)=(1−ε)h(x)+εg(x), where g is unknown. A well-known result of Huber (1964) is that when h is strongly unimodal, the least informative density f0=(1−ε)h+εg0 has exponential tails. We study the minimax variance solutions when the known…
On the Minimax Property for R-Estimators of Location
Consider the problem of estimating the unknown location parameter θ based on a random sample from F(x−θ), where F is an unknown member of the class of distribution functions F={F:F is symmetric about 0 and supx∣F(x)−Φ(x)∣≤ε}, where Φ denotes the standard normal distribution function. Huber (1964) showed that M-estimation has a minimax property for this model, whereas Sacks and Ylvisaker (1972) showed that L-…
Maximizing the Variance of M-Estimators Using the Generalized Method of Moment Spaces
The problem considered is that of optimizing a function of a finite number of linear functionals over an infinite dimensional convex set S. It is shown that under some reasonably general conditions the method of moment spaces can be used to reduce the problem to one of optimizing over a simple finite dimensional set (generally a set of convex combinations of extreme points of S). The results are applied to finding the maximum asymptotic variance of M-estimators over…
Upper Bounds on Asymptotic Variances of M-Estimators of Location
If X1,⋯,Xn is a random sample from F(x−θ), where F is an unknown member of a specified class F of approximately normal symmetric distributions, then an M-estimator of the unknown location parameter θ is obtained by solving the equation ∑i=1nψ(Xi−θ^n)=0 for θ^n. A suitable measure of the robustness of the M-estimator is sup{V(ψ,F):F∈F}, where V(ψ,F)=∫ψ2dF/(∫ψ′dF)2…
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 [−d,d] and is otherwise arbitrary. Let X1,⋯,Xn be a random sample from F(x−θ), where θ is the unknown location parameter. If ψ is in a class of continuous skew-symmetric…