This paper addresses the following simple question about sparsity. For the estimation of an $n$-dimensional mean vector $\boldsymbolθ$ in the Gaussian sequence model, is it possible to find an adaptive optimal threshold estimator in a full range of sparsity levels where nonadaptive optimality can be achieved by threshold estimators? We provide an explicit affirmative answer as follows. Under the squared loss, adaptive minimaxity in strong and weak $\ell_p$ balls with $0\le p<2$ is achieved by a class of smooth threshold estimators with the threshold level of the Benjamini-Hochberg FDR rule or its a certain approximation, provided that the minimax risk is between $n^{-δ_n}$ and $δ_n n$ for some $δ_n\to 0$. For $p=0$, this means adaptive minimaxity in $\ell_0$ balls when $1\le \|\boldsymbolθ\|_0\ll n$. The class of smooth threshold estimators includes the soft and firm threshold estimators but not the hard threshold estimator. The adaptive minimaxity in such a wide range is a delicate problem since the same is not true for the FDR hard threshold estimator at certain threshold and nominal FDR levels. The above adaptive minimaxity of the FDR smooth-threshold estimator is established by proving a stronger notion of adaptive ratio optimality for the soft threshold estimator in the sense that the risk for the FDR threshold level is uniformly within an infinitesimal fraction of the risk for the optimal threshold level for each unknown vector, when the minimum risk of nonadaptive soft threshold estimator is between $n^{-δ_n}$ and $δ_n n$. It is an interesting consequence of this adaptive ratio optimality that the FDR smooth-threshold estimator outperforms the sample mean in the common mean model $θ_i=μ$ when $|μ|<n^{-1/2}$.
For multiple testing, we introduce Storey-type FDR procedures and the concept of "regular estimator of the proportion of true nulls". We show that the rejection threshold of a Storey-type FDR procedure is a stopping time with respect to the backward filtration generated by the p-values and that a Storey-type FDR estimator at this rejection threshold equals the pre-specified FDR level, when the estimator of the proportion of true nulls is regular. These results hold regardless of the dependence among or the types of distributions of the p-values. They directly imply that a Storey-type FDR proce…
The present paper introduces new adaptive multiple tests which rely on the estimation of the number of true null hypotheses and which control the false discovery rate (FDR) at level alpha for finite sample size. We derive exact formulas for the FDR for a large class of adaptive multiple tests which apply to a new class of testing procedures. In the following, generalized Storey estimators and weighted versions are introduced and it turns out that the corresponding adaptive step up and step down tests control the FDR. The present results also include particular dynamic adaptive step wise tests …
In the context of multiple hypotheses testing, the proportion $π_0$ of true null hypotheses in the pool of hypotheses to test often plays a crucial role, although it is generally unknown a priori. A testing procedure using an implicit or explicit estimate of this quantity in order to improve its efficency is called adaptive. In this paper, we focus on the issue of False Discovery Rate (FDR) control and we present new adaptive multiple testing procedures with control of the FDR. First, in the context of assuming independent $p$-values, we present two new procedures and give a unified review of …
In this work we study an adaptive step-down procedure for testing $m$ hypotheses. It stems from the repeated use of the false discovery rate controlling the linear step-up procedure (sometimes called BH), and makes use of the critical constants $iq/[(m+1-i(1-q)]$, $i=1,...,m$. Motivated by its success as a model selection procedure, as well as by its asymptotic optimality, we are interested in its false discovery rate (FDR) controlling properties for a finite number of hypotheses. We prove this step-down procedure controls the FDR at level $q$ for independent test statistics. We then numerical…
Some crucial issues about a recently proposed estimator for the proportion of true null hypotheses ($π_0$) under discrete setup are discussed. An estimator for $π_0$ is introduced under the same setup. The estimator may be seen as a modification of a very popular estimator for $π_0$, originally proposed under the assumption of continuous test statistics. It is shown that adaptive Benjamini-Hochberg procedure remains conservative with the new estimator for $π_0$ being plugged in.
Major internet companies routinely perform tens of thousands of A/B tests each year. Such large-scale sequential experimentation has resulted in a recent spurt of new algorithms that can provably control the false discovery rate (FDR) in a fully online fashion. However, current state-of-the-art adaptive algorithms can suffer from a significant loss in power if null p-values are conservative (stochastically larger than the uniform distribution), a situation that occurs frequently in practice. In this work, we introduce a new adaptive discarding method called ADDIS that provably controls the FDR…
We apply FDR thresholding to a non-Gaussian vector whose coordinates X_i, i=1,..., n, are independent exponential with individual means $μ_i$. The vector $μ=(μ_i)$ is thought to be sparse, with most coordinates 1 but a small fraction significantly larger than 1; roughly, most coordinates are simply `noise,' but a small fraction contain `signal.' We measure risk by per-coordinate mean-squared error in recovering $\log(μ_i)$, and study minimax estimation over parameter spaces defined by constraints on the per-coordinate p-norm of $\log(μ_i)$: $\frac{1}{n}\sum_{i=1}^n\log^p(μ_i)\leq η^p$. We show…
Two recently introduced model based bias corrected estimators for proportion of true null hypotheses ($π_0$) under multiple hypotheses testing scenario have been restructured for exponentially distributed random observations available for each of the common hypotheses. Based on stochastic ordering, a new motivation behind formulation of some related estimators for $π_0$ is given. The reduction of bias for the model based estimators are theoretically justified and algorithms for computing the estimators are also presented. The estimators are also used to formulate a popular adaptive multiple te…
In the multiple testing problem with independent tests, the classical linear step-up procedure controls the false discovery rate (FDR) at level $π_0α$, where $π_0$ is the proportion of true null hypotheses and $α$ is the target FDR level. Adaptive procedures can improve power by incorporating estimates of $π_0$, which typically rely on a tuning parameter. Fixed adaptive procedures set their tuning parameters before seeing the data and can be shown to control the FDR in finite samples. We develop theoretical results for dynamic adaptive procedures whose tuning parameters are determined by the d…
This research deals with massive multiple hypothesis testing. First regarding multiple tests as an estimation problem under a proper population model, an error measurement called Erroneous Rejection Ratio (ERR) is introduced and related to the False Discovery Rate (FDR). ERR is an error measurement similar in spirit to FDR, and it greatly simplifies the analytical study of error properties of multiple test procedures. Next an improved estimator of the proportion of true null hypotheses and a data adaptive significance threshold criterion are developed. Some asymptotic error properties of the s…
Much effort has been done to control the "false discovery rate" (FDR) when $m$ hypotheses are tested simultaneously. The FDR is the expectation of the "false discovery proportion" $\text{FDP}=V/R$ given by the ratio of the number of false rejections $V$ and all rejections $R$. In this paper, we have a closer look at the FDP for adaptive linear step-up multiple tests. These tests extend the well known Benjamini and Hochberg test by estimating the unknown amount $m_0$ of the true null hypotheses. We give exact finite sample formulas for higher moments of the FDP and, in particular, for its varia…
We introduce a new class of methods for finite-sample false discovery rate (FDR) control in multiple testing problems with dependent test statistics where the dependence is fully or partially known. Our approach separately calibrates a data-dependent p-value rejection threshold for each hypothesis, relaxing or tightening the threshold as appropriate to target exact FDR control. In addition to our general framework we propose a concrete algorithm, the dependence-adjusted Benjamini-Hochberg (dBH) procedure, which adaptively thresholds the q-value for each hypothesis. Under positive regression de…
The False Discovery Rate (FDR) is a commonly used type I error rate in multiple testing problems. It is defined as the expected False Discovery Proportion (FDP), that is, the expected fraction of false positives among rejected hypotheses. When the hypotheses are independent, the Benjamini-Hochberg procedure achieves FDR control at any pre-specified level. By construction, FDR control offers no guarantee in terms of power, or type II error. A number of alternative procedures have been developed, including plug-in procedures that aim at gaining power by incorporating an estimate of the proportio…
In the online false discovery rate (FDR) problem, one observes a possibly infinite sequence of $p$-values $P_1,P_2,\dots$, each testing a different null hypothesis, and an algorithm must pick a sequence of rejection thresholds $α_1,α_2,\dots$ in an online fashion, effectively rejecting the $k$-th null hypothesis whenever $P_k \leq α_k$. Importantly, $α_k$ must be a function of the past, and cannot depend on $P_k$ or any of the later unseen $p$-values, and must be chosen to guarantee that for any time $t$, the FDR up to time $t$ is less than some pre-determined quantity $α\in (0,1)$. In this wo…
False Discovery Controlled Regions for Excursion Sets
University of California, San Diego（加州大学圣地亚哥分校）；Big Data Institute, Nuffield Department of Population Health, University of Oxford（牛津大学纳菲尔德人口健康学院大数据研究所）；School of Mathematics, University of Bristol（布里斯托尔大学数学学院）；Halıcıoğlu Data Science Institute, University of California, San Diego（加州大学圣地亚哥分校哈奇奥格鲁数据科学研究所）
This work is concerned with the study of the adaptivity properties of nonparametric regression estimators over the $d$-dimensional sphere within the global thresholding framework. The estimators are constructed by means of a form of spherical wavelets, the so-called needlets, which enjoy strong concentration properties in both harmonic and real domains. The author establishes the convergence rates of the $L^p$-risks of these estimators, focussing on their minimax properties and proving their optimality over a scale of nonparametric regularity function spaces, namely, the Besov spaces.
In the random coefficients binary choice model, a binary variable equals 1 iff an index $X^\topβ$ is positive.The vectors $X$ and $β$ are independent and belong to the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$.We prove lower bounds on the minimax risk for estimation of the density $f\_β$ over Besov bodies where the loss is a power of the $L^p(\mathbb{S}^{d-1})$ norm for $1\le p\le \infty$. We show that a hard thresholding estimator based on a needlet expansion with data-driven thresholds achieves these lower bounds up to logarithmic factors.
Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this paper, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert--Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects o…
