Robust Estimation in Linear Model and its Computational Aspects

Heuristics of robust estimation is briefly explained and a survey of the most typical robust methods of the point estimation both of the location and scale parameters and in the linear model is presented. Basic notions of robust statistics as well as of some particular topics of robust diagnostics are discussed, too. Testing of submodels and the possibilities of adaptive approach in the estimation of parameters in the linear model are also commented.

[1]  F. Hampel The Influence Curve and Its Role in Robust Estimation , 1974 .

[2]  S. Stigler Do Robust Estimators Work with Real Data , 1977 .

[3]  Hendrik P. Lopuhaä Estimation of location and covariance with high breakdown point , 1990 .

[4]  T. P. Hettmansperger,et al.  Statistical Inference Based on Ranks. , 1985 .

[5]  Jan Ámos Vísek Adaptive estimation in linear regression model. II. Asymptotic normality , 1992, Kybernetika.

[6]  Edward L. Frome,et al.  A revised simplex algorithm for the absolute deviation curve fitting problem , 1979 .

[7]  R. Serfling Approximation Theorems of Mathematical Statistics , 1980 .

[8]  Ricardo Fraiman,et al.  Qualitative Robustness for Stochastic Processes , 1987 .

[9]  Subhash C. NarulaI,et al.  The Minimum Sum of Absolute Errors Regression: A State of the Art Survey , 1982 .

[10]  Roger Koenker,et al.  Tests of Linear Hypotheses and l[subscript]1 Estimation , 1982 .

[11]  Virginia Ann Johnson,et al.  State-of-the-art Survey , 2022 .

[12]  G WERNER,et al.  The measurement of uncertainty , 1961, Clinical pharmacology and therapeutics.

[13]  M. Johns,et al.  Robust Pitman-like Estimators , 1979 .

[14]  Muni S. Srivastava,et al.  Regression Analysis: Theory, Methods, and Applications , 1991 .

[15]  R. Koenker,et al.  Asymptotic Theory of Least Absolute Error Regression , 1978 .

[16]  D. Ruppert,et al.  Transformation and Weighting in Regression , 1988 .

[17]  P. Bickel,et al.  On Some Analogues to Linear Combinations of Order Statistics in the Linear Model , 1973 .

[18]  P. Rousseeuw Least Median of Squares Regression , 1984 .

[19]  Jana Jurečková Tail-Behavior of Location Estimators , 1981 .

[20]  C. J. Lawrence Robust estimates of location : survey and advances , 1975 .

[21]  Frederick Mosteller,et al.  Understanding robust and exploratory data analysis , 1983 .

[22]  P. J. Huber Robust Estimation of a Location Parameter , 1964 .

[23]  S. Weisberg Plots, transformations, and regression , 1985 .

[24]  J. Antoch Behaviour of L-Estimators of Location from the Point of View of Large Deviations , 1984 .

[25]  I. Barrodale,et al.  An Improved Algorithm for Discrete $l_1 $ Linear Approximation , 1973 .

[26]  W. Steiger,et al.  Least Absolute Deviations: Theory, Applications and Algorithms , 1984 .

[27]  P. Révész,et al.  Strong approximations in probability and statistics , 1981 .

[28]  R. Koenker,et al.  Computing regression quantiles , 1987 .

[29]  Moti Lal Tiku,et al.  Robust Inference , 1986 .

[30]  Minimizing the sum of absolute deviations , 1978 .

[31]  D. F. Andrews,et al.  Finding the Outliers that Matter , 1978 .

[32]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[33]  Computational aspects of adaptive combination of least squares and least absolute deviations estimators , 1991 .

[34]  Vic Barnett,et al.  Outliers in Statistical Data , 1980 .

[35]  R. Maronna Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[36]  H. Ekblom A new algorithm for the huber estimator in linear models , 1988 .

[37]  C. Stein Efficient Nonparametric Testing and Estimation , 1956 .

[38]  I. Barrodale,et al.  Algorithms for restricted least absolute value estimation , 1977 .

[39]  R. Wolke,et al.  Iteratively Reweighted Least Squares: Algorithms, Convergence Analysis, and Numerical Comparisons , 1988 .

[40]  P. J. Huber The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[41]  Thomas P. Hettmansperger,et al.  Statistical inference based on ranks , 1985 .

[42]  Jana Jurečková,et al.  On adaptive scale-equivariant m-estimators in linear models , 1982 .

[43]  W. R. van Zwet,et al.  VAN DE HULST ON ROBUST STATISTICS: A HISTORICAL NOTE , 1985 .

[44]  S. Weisberg,et al.  Residuals and Influence in Regression , 1982 .

[45]  U. Peters,et al.  Up- and down-dating procedures for linearL1 regression , 1983 .

[46]  Jana Jurecková,et al.  Regression quantiles and trimmed least squares estimator under a general design , 1984, Kybernetika.

[47]  D. F. Andrews,et al.  Robust Estimates of Location , 1972 .

[48]  J. Antoch,et al.  Algoritmic Development in Variable Selection Procedures , 1986 .

[49]  Michael R. Greenberg,et al.  Chapter 1 – Theory, Methods, and Applications , 1978 .

[50]  P. J. Huber Minimax Aspects of Bounded-Influence Regression , 1983 .

[51]  V. Yohai HIGH BREAKDOWN-POINT AND HIGH EFFICIENCY ROBUST ESTIMATES FOR REGRESSION , 1987 .

[52]  Roger Koenker,et al.  An Empirical Quantile Function for Linear Models with | operatornameiid Errors , 1982 .

[53]  Ralf Wolke Iteratively reweighted least squares: A comparison of several single step algorithms for linear models , 1992 .

[54]  P. Bickel One-Step Huber Estimates in the Linear Model , 1975 .

[55]  Jan Ámos Vísek Efficiency rate and local deficiency of the most powerful tests in the model of contaminacy with general neighbourhoods , 1987, Kybernetika.

[56]  P. J. Huber Robust Statistical Procedures , 1977 .

[57]  R. Beran An Efficient and Robust Adaptive Estimator of Location , 1978 .

[58]  Nabih N. Abdelmalek,et al.  On the discrete linear L1 approximation and L1 solutions of overdetermined linear equations , 1974 .

[59]  Y. Dodge on Statistical data analysis based on the L1-norm and related methods , 1987 .

[60]  P. Hennequin,et al.  Quelques Aspects De La Statistique Robuste , 1981 .

[61]  P. J. Huber,et al.  Minimax Tests and the Neyman-Pearson Lemma for Capacities , 1973 .

[62]  Harold Davenport,et al.  A Historical Note , 1947 .

[63]  Kaj Madsen,et al.  Algorithms for non-linear huber estimation , 1989 .

[64]  Jan Ámos Vísek Estimation of contamination level in model of contaminacy with general neighbourhoods , 1989, Kybernetika.

[65]  Alfio Marazzi,et al.  Probabilistic algorithms for least median of squares regression , 1989 .

[66]  Flexible L-estimation in the linear model , 1991 .

[67]  W. Rey Introduction to Robust and Quasi-Robust Statistical Methods , 1983 .

[68]  D. Ruppert,et al.  Trimmed Least Squares Estimation in the Linear Model , 1980 .

[69]  S. Chatterjee Sensitivity analysis in linear regression , 1988 .

[70]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .

[71]  V. Yohai,et al.  Influence Functionals for Time Series , 1986 .

[72]  Roy E. Welsch,et al.  Efficient Computing of Regression Diagnostics , 1981 .

[73]  I. Vajda Theory of statistical inference and information , 1989 .

[74]  S. Stigler Simon Newcomb, Percy Daniell, and the History of Robust Estimation 1885–1920 , 1972 .

[75]  V. Yohai,et al.  Asymptotic behavior of general M-estimates for regression and scale with random carriers , 1981 .

[76]  Werner A. Stahel,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[77]  F. Hampel A General Qualitative Definition of Robustness , 1971 .

[78]  N. Draper,et al.  Applied Regression Analysis , 1966 .

[79]  P. J. Huber,et al.  Numerical methods for the nonlinear robust regression problem , 1981 .

[80]  J. Collins Robust Estimation of a Location Parameter in the Presence of Asymmetry , 1976 .

[81]  Subhash C. Narula,et al.  The Minimum Sum of Absolute Errors Regression , 1987 .

[82]  Z. Šidák Rectangular Confidence Regions for the Means of Multivariate Normal Distributions , 1967 .

[83]  Xuming He TAIL BEHAVIOR OF REGRESSION ESTIMATORS AND THEIR BREAKDOWN POINTS , 1990 .

[84]  S. Stigler Gauss and the Invention of Least Squares , 1981 .

[85]  Elvezio Ronchetti,et al.  Small Sample Asymptotics , 1990 .

[86]  Roy E. Welsch,et al.  Computational Procedures for Bounded-Influence Regression , 1982 .

[87]  A. Siegel Robust regression using repeated medians , 1982 .

[88]  W. W. Muir,et al.  Regression Diagnostics: Identifying Influential Data and Sources of Collinearity , 1980 .

[89]  École d'été de probabilités de Saint-Flour,et al.  Ecole d'été de probabilités de Saint-Flour IX-1979 , 1981 .

[90]  P. J. Huber Minimax Aspects of Bounded-Influence Regression: Rejoinder , 1983 .

[91]  Jana Jurečková,et al.  Asymptotics for one-step m-estimators in regression with application to combining efficiency and high breakdown point , 1987 .