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Register a new userSequential Multiple Testing
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Added by
Shiyun Chen
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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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.
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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.
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.
This paper considers Bayesian multiple testing under sparsity for polynomial-tailed distributions satisfying a monotone likelihood ratio property. Included in this class of distributions are the Students t, the Pareto, and many other distributions. We prove some general asymptotic optimality results under fixed and random thresholding. As examples of these general results, we establish the Bayesian asymptotic optimality of several multiple testing procedures in the literature for appropriately chosen false discovery rate levels. We also show by simulation that the Benjamini-Hochberg procedure with a false discovery rate level different from the asymptotically optimal one can lead to high Bayes risk.
This paper studies the problem of high-dimensional multiple testing and sparse recovery from the perspective of sequential analysis. In this setting, the probability of error is a function of the dimension of the problem. A simple sequential testing procedure is proposed. We derive necessary conditions for reliable recovery in the non-sequential setting and contrast them with sufficient conditions for reliable recovery using the proposed sequential testing procedure. Applications of the main results to several commonly encountered models show that sequential testing can be exponentially more sensitive to the difference between the null and alternative distributions (in terms of the dependence on dimension), implying that subtle cases can be much more reliably determined using sequential methods.
The sequential multiple testing problem is considered under two generalized error metrics. Under the first one, the probability of at least $k$ mistakes, of any kind, is controlled. Under the second, the probabilities of at least $k_1$ false positives and at least $k_2$ false negatives are simultaneously controlled. For each formulation, the optimal expected sample size is characterized, to a first-order asymptotic approximation as the error probabilities go to 0, and a novel multiple testing procedure is proposed and shown to be asymptotically efficient under every signal configuration. These results are established when the data streams for the various hypotheses are independent and each local log-likelihood ratio statistic satisfies a certain Strong Law of Large Numbers. In the special case of i.i.d. observations in each stream, the gains of the proposed sequential procedures over fixed-sample size schemes are quantified.
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