Robust Bounded-Influence Tests in General Parametric Models

Abstract

We introduce robust tests for testing hypotheses in a general parametric model. These are robust versions of the Wald, scores, and likelihood ratio tests and are based on general M estimators. Their asymptotic properties and influence functions are derived. It is shown that the stability of the level is obtained by bounding the self-standardized sensitivity of the corresponding M estimator. Furthermore, optimally bounded-influence tests are derived for the Wald- and scores-type tests. Applications to real and simulated data sets are given to illustrate the tests' performance

Testing composite null hypotheses based onS-divergences

“…Clearly, to ensure the existence of such a θ 0 in Θ 0 there must exist a limit point θ 0 of the null parameter space Θ 0 ; we assume Θ 0 to be a closed subset of Θ. Next, following Hampel et al (1986), we also consider the contaminations over these contiguous alternatives such that their effect tends to zero as θ n tends to θ 0 at the same rate to avoid confusion between the null and alternative neighborhoods (also see Huber-Carol, 1970, Heritier and Ronchetti, 1994, Toma and Broniatowski, 2011. So, consider the contaminated distributions F L n, ,y and F L n, ,y , defined in Definition 1.4 of the supplementary material, for level and power respectively and the level influence function (LIF ) and the power influence function (P IF ) as defined therein; also see Ghosh et al (2015).…”

Robust inference with binary data

“…Suppose that β is split into two parts β (1) and β (2) (and correspondingly x (1) and x (2) ) and we want to test the null hypothesis that β (2) = 0. Victoria-Feser (2000) shows in the particular case of logistic regression by using the results of Heritier and Ronchetti (1994) that the level of Rao's score test can become arbitrarily biased either because of misclassification in the response or when there are extreme points in the design subspace X (2) . Heritier and Victoria-Feser (1997) examine an example of logistic regression and confirm that the level of the classical score test can be seriously biased by data contamination.…”

“…Using the results of Heritier and Ronchetti (1994), Victoria-Feser (2000) shows that a suitable robust version of Rao's test statistic for the logistic regression model can be based on (6) and is given by…”

“…Robust theory actually started with testing procedures where the problem is to control the probability of type I error in the presence of model misspecification (see Box 1953). To investigate the robustness properties of a testing procedure one works with the asymptotic bias on the level of the test due to an infinitesimal model deviation (see Heritier and Ronchetti 1994). In order to test hypotheses on linear combinations of regressors we choose Rao's score test which has a robust analogue.…”

Saddlepoint tests for quantile regression

“…They are the natural counterpart of standard Wald-, score-, and likelihood-ratio tests, where -loglikelihood is replaced by the objective function ρ α (·) in (1), and the score function by ψ α (·) in (3). Their robustness and asymptotic properties have been studied in Heritier & Ronchetti (1994). It turns out that a bounded score function guarantees robustness of validity and robustness of efficiency for these tests.…”