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约翰·W. 图基对稳健统计的贡献AI 解读

We summarize John W. Tukey's contributions to robust statistics, separating them into four categories: conceptual; tools; techniques; procedures.

M 与 P 估计量的有限样本崩溃性质AI 解读

The finite sample breakdown properties of $M$-estimators, defined by $\sum\rho(x_i - T) = \min!$, and of the associated Pitman-type or $P$-estimators, defined by $T = \frac{\int \exp\{-\Sigma \rho(x_i - \theta)\}\theta d\theta}{\int \exp\{-\Sigma\rho(x_i - \theta)\} d\theta},$ are investigated. If $\rho$ is symmetric, and $\psi = \rho'$ is monotone and bounded, then the breakdown point of either estimator is $\varepsilon^\ast = 1/2$. If $\psi$ decreases to 0 for large $x$ ("r…

有界影响回归的极小极大性质AI 解读

Bounded-influence regression (in the sense of Hampel-Mallows-Krasker-Welsch) is examined critically from the point of view of finite-sample minimax robust estimation theory. The main conclusion is that the influence of position of very high leverage points—that is, of points where the diagonal element of the hat matrix H = X(XTX)−1 XT exceeds a certain bound—should be cut down selectively. This bound should be chosen roughly between .2 and .5; for the upper of these values,…

摘要暂缺

稳健统计程序AI 解读

Background - Why Robust Procedures? Qualitative and Quantitative Robustness, Qualitative Robustness, Quantitative Robustness, Breakdown, Infinitesimal Robustness, Influence Function M-L and R-Estimates, L-Estimates, R-Estimates, Asymptotic Properties of M-Estimates, Asymptotically Efficient M-L, R-Estimates, Scaling Question Asymptotic Minimax Theory, Minimax Asymptotic Bias, Minimax Asymptotic Variance Multiparameter Problems, Generalities, Regression, Robust Covariances -…

It is shown that among all cumulative distribution functions passing through $k \geqq 2$ given points there is a unique one with minimal Fisher information; it is obtained by a curious type of spline interpolation. This answers some questions raisd by D. G. Kendall and J. W. Tukey.

稳健回归：渐近理论、猜想与 Monte Carlo 模拟AI 解读

Maximum likelihood type robust estimates of regression are defined and their asymptotic properties are investigated both theoretically and empirically. Perhaps the most important new feature is that the number $p$ of parameters is allowed to increase with the number $n$ of observations. The initial terms of a formal power series expansion (essentially in powers of $p/n$) show an excellent agreement with Monte Carlo results, in most cases down to 4 observations per parameter.

容量下的极小极大检验与 Neyman–Pearson 引理AI 解读

Robust test problems between two approximately known simple hypotheses can be formalized as minimax test problems between two composite hypotheses. We show that if the composite hypotheses can be described in terms of alternating capacities of order 2 (in the sense of Choquet), then the minimax tests are ordinary Neyman-Pearson tests between a fixed representative pair of simple hypotheses; moreover, the condition is in a certain sense also necessary. All the neighborhoods…