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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big STATISTICS SEMINAR}
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\centerline{4:15 p.m., Tuesday, January 28, 2003}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Jiashun Jin}
\centerline{\sl Stanford University}
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\centerline{\bf Asymptotic Minimaxity of False Discovery Rate Thresholding for
Sparse nonGaussian Data}
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{\it False Discovery Rate} (FDR) control is a recent innovation in
multiple hypothesis testing, in which one seeks to ensure that
at most a certain fraction of the rejected null hypotheses
correspond to false rejections (i.e. false discoveries).
The FDR principle also can be
used in highly multivariate estimation problems, where
it has recently been shown to provide an asymptotically minimax
solution to the problem of estimating
a sparse mean vector in the presence
of Gaussian white noise.
In effect, FDR provides an effective
method of setting a threshold for separating signal from noise
when the signal is sparse and the noise is Gaussian.

In this talk, we consider the application of FDR
thresholding to non-Gaussian settings -- exponential and Poisson,  in
hopes
of learning whether the good asymptotic properties of FDR
thresholding as an estimation tool hold more broadly
than just at the standard Gaussian model. 
We consider a vector $X_i$, $i=1,\dots,n$
whose coordinates are independent
exponential/poisson with individual means $\lambda_i$.
The vector $\lambda$
is thought to be sparse, with most coordinates $1$
and a small fraction significantly larger than $1$.
This models a situation where
most coordinates are simply `noise',
but a small fraction of
the coordinates contain `signal'.

For  exponential data, we develop an estimation theory
working with $\log(\lambda_i)$ as the estimand,
and use the per-coordinate mean-squared
error in recovering $\log(\lambda_i)$
to measure risk.  We consider minimax estimation over
parameter spaces defined by constraints on
the per-coordinate $\ell^p$ norm  of $\log(\lambda_i)$:
$Ave_i \log^p(\lambda_i) \leq \eta^p$.
Members of such spaces are vectors $(\lambda_i)$
which are sparsely heterogeneous.

For Poisson data, we use the per-coordinate weighted mean-squared
error ${(\hat{\lambda}_i - \lambda_i)^2}\over{\lambda_i}$ 
to measure risk.  We consider minimax estimation over
parameter spaces defined by constraints on
the per-coordinate $\ell^p$ norm  of $(\lambda_i-1)$:
$Ave_i (\lambda_i-1)^p \leq \eta^p$.
Members of such spaces are vectors $(\lambda_i)$
which are sparsely heterogeneous.

For Poisson data, we use the per-coordinate weighted mean-squared
error ${(\hat{\lambda}_i - \lambda_i)^2}\over{\lambda_i}$ 
to measure risk.  We consider minimax estimation over
parameter spaces defined by constraints on
the per-coordinate $\ell^p$ norm  of $(\lambda_i-1)$:
$Ave_i (\lambda_i-1)^p \leq \eta^p$.
Members of such spaces are vectors $(\lambda_i)$
which are sparsely heterogeneous.

We find that,  for large $n$ and small $\eta$,
the FDR thresholding is nearly minimax  for both the exponential and the
Poisson model,  increasingly so
as $\eta$ decreases.

We find that the FDR control parameter $0 < q <1$ plays an important
role, for both the exponential and the Poisson models:  when $q \leq
{{1}\over{2}}$, the FDR estimator is nearly minimax,
while choosing a fixed  $ q > {{1}\over{2}}$
prevents near minimaxity. We compare our results with work in the
Gaussian setting by  {\it Abramovich, Benjamini, Donoho, Johnstone}
(2000). 

Joint with David L. Donoho

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