Jana Jurečková

Robust statistical procedures : asymptotics and interrelations

ASYMPTOTICS AND INTERRELATIONS Preliminaries Robust Estimation of Location and Regression Asymptotic Representations for L-Estimators Asymptotic Representations for M-Estimators Asymptotic Representations for R-Estimators Asymptotic Interrelations of Estimators ROBUST STATISTICAL INFERENCE Robust Sequential and Recursive Point Estimation Robust Confidence Sets and Intervals Robust Statistical Tests Appendix References Indexes.

Tests of linear hypotheses based on regression rank scores

We propose a general class of asymptotically distribution-free tests of a linear hypothesis in the linear regression model. The tests are based on regression rank scores, recently introduced by Gutenbrunner and Jureckova (1992) as dual variables to the regression quantiles of Koenker and Bassett (1978). Their properties are analogous to those of the corresponding rank tests in location model. Unlike the other regression tests based on aligned rank statistics, however, our tests do not require preliminary estimation of nuisance parameters, indeed they are invariant with respect to a regression shift of the nuisance parameters.

Asymptotics for one-step m-estimators in regression with application to combining efficiency and high breakdown point

In the linear regression model, a one-step version of the M-estimator M n starting with an initial estimator T n is proposed which inherits the efficiency properties of M n and the breakdown-point of T n respectively; this even in the case that the rate of consistency of T n is lower than Such results suggest a potentially valuable method for combining high efficiency with high breakdown point. They follow from a general asymptotic linearity result which holds for M -estimators with kernels which are not everywhere differentiable.

Sequential procedures based on M-estimators with discontinuous score functions

Abstract Bounded-width sequential confidence intervals and sequential tests for regression parameter based on M-estimators are extended to the case where the score-functions generating the M-estimators have jump-discontinuities. In the context of the asymptotic normality of the stopping variable, for the confidence interval problem, it is observed that the jump-discontinuities induce a slower rate of convergence. The proofs of the main theorems rest on the weak convergence of some related processes and this is also studied.

Moderate and Cramér-type large deviation theorems for M-estimators

The known central limit result for broad classes of M-estimators is refined to moderate and large deviation behaviour. The results are applied in relating the local inaccuracy rate and the asymptotic variance of M-estimators in the location and scale problem.

Nonparametric multivariate rank tests and their unbiasedness

Although unbiasedness is a basic property of a good test, many tests on vector parameters or scalar parameters against two-sided alternatives are not finite-sample unbiased. This was already noticed by Sugiura [Ann. Inst. Statist. Math. 17 (1965) 261--263]; he found an alternative against which the Wilcoxon test is not unbiased. The problem is even more serious in multivariate models. When testing the hypothesis against an alternative which fits well with the experiment, it should be verified whether the power of the test under this alternative cannot be smaller than the significance level. Surprisingly, this serious problem is not frequently considered in the literature. The present paper considers the two-sample multivariate testing problem. We construct several rank tests which are finite-sample unbiased against a broad class of location/scale alternatives and are finite-sample distribution-free under the hypothesis and alternatives. Each of them is locally most powerful against a specific alternative of the Lehmann type. Their powers against some alternatives are numerically compared with each other and with other rank and classical tests. The question of affine invariance of two-sample multivariate tests is also discussed.

Robust Estimators of Location anD Regression Parameters and their Second Order Asymptotic Relations

Second order asymptotic relations-namely the orders of asymptotic equivalence — of some pairs of robust estimators are studied. More specifically, we shall study the relations of α-trimmed mean to Huber’s estimator of location and of α-trimmed least-squares estimator to Huber’s estimator of regression, respectively, and the order of the approxiaation of an M-estimator by its one-step version.

Rank tests and regression rank score tests in measurement error models

The rank and regression rank score tests of linear hypothesis in the linear regression model are modified for measurement error models. The modified tests are still distribution free. Some tests of linear subhypotheses are invariant to the nuisance parameter, others are based on the aligned ranks using the R-estimators. The asymptotic relative efficiencies of tests with respect to tests in models without measurement errors are evaluated. The simulation study illustrates the powers of the tests.

Regression Rank Scores Scale Statistics and Studentization in Linear Models

In linear models, regression-invariant and scale-equivariant regression rank scores scale statistics are proposed. Among their potential applications in statistical inference in linear models, studentization and testing for homoscedasticity are considered.

A Class of Tests on the Tail Index

AbstractIn the family of distribution functions with nondegenerate right tail, we test the hypothesis