The p value line: a way to choose the tuning constant in tests based on the Huber $${\varvec{M}}$$ M -estimator

In this paper, we propose a method to choose a test result from several competing ones. The choice is based on the p value under the alternative, i.e., the probability of obtaining an extreme or a more extreme result than the one actually observed, assuming that the alternative hypothesis is true. The function obtained, by moving the alternative, is called the p value line. Because this combines the power of a test and its sensitivity to outliers, it is considered to be the basis for the selection between several classical and/or robust test results. When we change the trimming fraction in the trimmed mean, or the tuning constant when we consider a test based on the Huber $$M$$ M -estimator, we obtain competing robust test results. For this reason, we can use the p value line to choose the trimming fraction or the tuning constant in an objective way. In this article, we shall focus on the second class of estimators. Because computing the distribution of the test statistic under the alternative could be a very hard problem, a linear approximation of the tail probability functional based on the von Mises expansion can be used to compute the p value line. Applications of the proposal to the one-sample location problem, generalized linear models and generalized additive models are considered in the article. We conclude the article with the random p value line to decide if the observed differences between p value lines are significant or not. Copyright Sociedad de Estadística e Investigación Operativa 2014