In the context of large-scale multiple testing, hypotheses are often
accompanied with certain prior information. In this paper, we present a
single-index modulated (SIM) multiple testing procedure, which maintains
control of the false discovery rate while incorporating prior information, by
assuming the availability of a bivariate $p$-value, $(p_1,p_2)$, for each
hypothesis, where $p_1$ is a preliminary $p$-value from prior information and
$p_2$ is the primary $p$-value for the ultimate analysis. To find the optimal
rejection region for the bivariate $p$-value, we propose a criteria based on
the ratio of probability density functions of $(p_1,p_2)$ under the true null
and nonnull. This criteria in the bivariate normal setting further motivates us
to project the bivariate $p$-value to a single-index, $p(\theta)$, for a wide
range of directions $\theta$. The true null distribution of $p(\theta)$ is
estimated via parametric and nonparametric approaches, leading to two
procedures for estimating and controlling the false discovery rate. To derive
the optimal projection direction $\theta$, we propose a new approach based on
power comparison, which is further shown to be consistent under some mild
conditions. Simulation evaluations indicate that the SIM multiple testing
procedure improves the detection power significantly while controlling the
false discovery rate. Analysis of a real dataset will be illustrated.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1222 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
This paper develops an estimation approach for nonparametric regression analysis with measurement error in covariates, assuming the availability of independent validation data on them, in addition to primary data on the response variable and surrogate covariates. Without specifying any error model structure between the surrogate and true covariates, we propose an estimator that integrates local linear regression and Fourier transformation methods. Under mild conditions, the consistency of the proposed estimator is established and the convergence rate is also obtained. Numerical examples show that it performs well in applications.
In this article, we propose a factor-adjusted multiple testing (FAT) procedure based on factor-adjusted p-values in a linear factor model involving some observable and unobservable factors, for the purpose of selecting skilled funds in empirical finance (Barras et al., 2010). The factor-adjusted p-values were obtained after extracting the latent common factors by the principal component method (Wang, 2012). Under some mild conditions, the false discovery proportion can be consistently estimated even if the idiosyncratic errors are allowed to be weakly correlated across units. Furthermore, by appropriately setting a sequence of threshold values approaching zero, the proposed FAT procedure enjoys model selection consistency. Extensive simulation studies and a real data analysis for selecting skilled funds in the U.S. financial market are presented to illustrate the practical utility of the proposed method.
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