Classical least squares regression consists of minimizing the sum of the squared residuals. Many authors have produced more robust versions of this estimator by replacing the square by something else, such as the absolute value. In this article a different approach is introduced in which the sum is replaced by the median of the squared residuals. The resulting estimator can resist the effect of nearly 50% of contamination in the data. In the special case of simple regression, it corresponds to finding the narrowest strip covering half of the observations. Generalizations are possible to multivariate location, orthogonal regression, and hypothesis testing in linear models.

Least Median of Squares Regression

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

National Institute of Standards and Technology における超伝導研究及び生活

We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$. General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176347265

The Jackknife and the Bootstrap for General Stationary Observations

A class of robust estimates for the linear model is introduced. These estimates, called MM-estimates, have simultaneously the following properties: (i) they are highly efficient when the errors have a normal distribution and (ii) their breakdown-point is 0.5. The MM-estimates are defined by a three-stage procedure. In the first stage an initial regression estimate is computed which is consistent robust and with high breakdown-point but not necessarily efficient. In the second stage an M-estimate of the errors scale is computed using residuals based on the initial estimate. Finally, in the third stage an M-estimate of the regression parameters based on a proper redescending psi-function is computed. Consistency and asymptotical normality of the MM-estimates assuming random carriers are proved. A convergent iterative numerical algorithm is given. Finally, the asymptotic biases under contamination of optimal bounded influence estimates and MM-estimates are compared.

/pdf/high-breakdown-point-and-high-efficiency-robust-estimates-2wef8i0mwr.pdf

High breakdown-point and high efficiency robust estimates for regression

Two new classes of robust estimates for ARMA models are introduced: estimates based on residual autocovariances (RA estimates), and estimates based on truncated residual autocovariances (TRA estimates). A heuristic derivation of the asymptotic normal distribution is given. We also perform a Monte Carlo study to compare the robustness properties of these estimates with the least squares, M, and GM estimates. In this study we consider observations that correspond to a Gaussian model with additive outliers. The Monte Carlo results show that RA and TRA estimates compare favorably with respect to least squares, M, and GM estimates.

Robust Estimates for ARMA Models

Bias- and efficiency-robustness of general M-estimators for regression with random carriers

Among the frameworks for Synthetic Aperture Radar (SAR) image modelling and analysis, the multiplicative model is very accurate and successful. It is based on the assumption that the observed random field is the result of the product of two independent and unobserved random fields: X and Y. The random field X models the terrain backscatter and, thus, depends only on the type of area to which each pixel belongs. The random field Y takes into account that SAR images are the result of a coherent imaging system that produces the well-known phenomenon called speckle noise, and that they are generated by performing an average of n statistically independent images (looks) in order to reduce the noise effect. There are various ways of modelling the random field X; recently the Γ−1/2(α, γ) distribution was proposed. This, with the usual Γ1/2(n, n) distribution for the amplitude speckle, resulted in a new distribution for the return: the (α, γ, n) law. The parameters α and γ depend only on the ground truth, and n i...

Classification of SAR images using a general and tractable multiplicative model

This work is concerned with simultaneous estimation of coefficients and a scale parameter of a p th-order autoregressive process (X

 t
 ). The observations are Y

 t
 =V

 t
 
Z

 t
 +(1-V

 t
 )X

 t
 where (Z
t) is a contaminating process and (V

 t
 ) represents the proportion of contamination. If (X

 t
 ) or (Z

 t
 ) have heavy tails both least squares estimates and ordinary M-estimates are seriously affected. Under general conditions we prove consistency and asymptotic normality of a general class of M-estimates which contains some M-estimates studied by Denby and Martin [6].

General M -estimates for contaminated p th-order autoregressive processes: Consistency and asymptotic normality

Among the frameworks for Synthetic Aperture Radar (SAR) image modelling and analysis, the multiplicative model is very accurate and successful. It is based on the assumption that the observed random field is the result of the product of two independent and unobserved random fields: X and Y . The random field X models the terrain backscatter and, thus, depends only on the type of area each pixel belongs to. The random field Y takes into account that SAR images are the result of a coherent imaging system that produces the well known phenomenon called speckle noise, and that they are generated by performing an average of n statistically independent images (looks) in order to reduce the noise eect. There are various = = ‡ ∞

Oscar H. Bustos

Papers

Robust Estimates for ARMA Models

Bias- and efficiency-robustness of general M-estimators for regression with random carriers

Classification of SAR images using a general and tractable multiplicative model

General M -estimates for contaminated p th-order autoregressive processes: Consistency and asymptotic normality

Classification of SAR Images using a General and Tractable