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Register a new userA Scan Procedure for Multiple Testing
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Andrew Ying
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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In a multiple testing framework, we propose a method that identifies the interval with the highest estimated false discovery rate of P-values and rejects the corresponding null hypotheses. Unlike the Benjamini-Hochberg method, which does the same but over intervals with an endpoint at the origin, the new procedure `scans all intervals. In parallel with citep*{storey2004strong}, we show that this scan procedure provides strong control of asymptotic false discovery rate. In addition, we investigate its asymptotic false non-discovery rate, deriving conditions under which it outperforms the Benjamini-Hochberg procedure. For example, the scan procedure is superior in power-law location models.
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We study an online multiple testing problem where the hypotheses arrive sequentially in a stream. The test statistics are independent and assumed to have the same distribution under their respective null hypotheses. We investigate two procedures LORD and LOND, proposed by (Javanmard and Montanari, 2015), which are proved to control the FDR in an online manner. In some (static) model, we show that LORD is optimal in some asymptotic sense, in particular as powerful as the (static) Benjamini-Hochberg procedure to first asymptotic order. We also quantify the performance of LOND. Some numerical experiments complement our theory.
We study a stylized multiple testing problem where the test statistics are independent and assumed to have the same distribution under their respective null hypotheses. We first show that, in the normal means model where the test statistics are normal Z-scores, the well-known method of (Benjamini and Hochberg, 1995) is optimal in some asymptotic sense. We then show that this is also the case of a recent distribution-free method proposed by Foygel-Barber and Cand`es (2015). The method is distribution-free in the sense that it is agnostic to the null distribution - it only requires that the null distribution be symmetric. We extend these optimality results to other location models with a base distribution having fast-decaying tails.
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Classification rules can be severely affected by the presence of disturbing observations in the training sample. Looking for an optimal classifier with such data may lead to unnecessarily complex rules. So, simpler effective classification rules could be achieved if we relax the goal of fitting a good rule for the whole training sample but only consider a fraction of the data. In this paper we introduce a new method based on trimming to produce classification rules with guaranteed performance on a significant fraction of the data. In particular, we provide an automatic way of determining the right trimming proportion and obtain in this setting oracle bounds for the classification error on the new data set.
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