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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 1689-1716

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2010.255

\vspace{3pc} \normalsize \centerline{\large\bf ESTIMATING THE
PROPORTION OF TRUE NULL} \vspace{2pt} \centerline{\large\bf
HYPOTHESES  UNDER DEPENDENCE} \vspace{.25cm}
\author{Irina Ostrovnaya and  Dan L. Nicolae}
\vspace{.4cm} \centerline{\it Memorial Sloan-Kettering Cancer Center
and The University of Chicago} \vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:}
 Multiple testing procedures, such as the
False Discovery Rate control, often rely on estimating the
proportion of true null hypotheses. This proportion is directly
related to the minimum of the density of the p-value distribution.
We propose a new estimator for the minimum of a density that is
based on constrained multinomial likelihood functions. The proposed
method involves partitioning the support of the density into several
intervals, and estimating multinomial probabilities that are a
function of the density. The motivation for this new  approach comes
from multiple testing settings where the test statistics are
dependent, since this framework can be extended to the case of the
dependent observations by using weighted univariate likelihoods. The
optimal weights are obtained using the theory of estimating
equations, and depend on the discretized pairwise joint
distributions of the observations. We discuss how  optimal weights
can be estimated when the test statistics have multivariate normal
distribution, and their correlation matrix is available or
estimated. We evaluate the performance of the proposed estimator in
simulations that mimic the testing for differential expression using
microarray data.\par\vspace{9pt} \noindent {\it Key words and
phrases:} Constrained multinomial likelihood, correlated tests, FDR,
minimum of a density, multiple testing under dependence, proportion
of null hypotheses.
\par
\end{quotation}
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