3,286 research outputs found
Minimax Structured Normal Means Inference
We provide a unified treatment of a broad class of noisy structure recovery
problems, known as structured normal means problems. In this setting, the goal
is to identify, from a finite collection of Gaussian distributions with
different means, the distribution that produced some observed data. Recent work
has studied several special cases including sparse vectors, biclusters, and
graph-based structures. We establish nearly matching upper and lower bounds on
the minimax probability of error for any structured normal means problem, and
we derive an optimality certificate for the maximum likelihood estimator, which
can be applied to many instantiations. We also consider an experimental design
setting, where we generalize our minimax bounds and derive an algorithm for
computing a design strategy with a certain optimality property. We show that
our results give tight minimax bounds for many structure recovery problems and
consider some consequences for interactive sampling
Minimax Estimation of Nonregular Parameters and Discontinuity in Minimax Risk
When a parameter of interest is nondifferentiable in the probability, the
existing theory of semiparametric efficient estimation is not applicable, as it
does not have an influence function. Song (2014) recently developed a local
asymptotic minimax estimation theory for a parameter that is a
nondifferentiable transform of a regular parameter, where the nondifferentiable
transform is a composite map of a continuous piecewise linear map with a single
kink point and a translation-scale equivariant map. The contribution of this
paper is two fold. First, this paper extends the local asymptotic minimax
theory to nondifferentiable transforms that are a composite map of a Lipschitz
continuous map having a finite set of nondifferentiability points and a
translation-scale equivariant map. Second, this paper investigates the
discontinuity of the local asymptotic minimax risk in the true probability and
shows that the proposed estimator remains to be optimal even when the risk is
locally robustified not only over the scores at the true probability, but also
over the true probability itself. However, the local robustification does not
resolve the issue of discontinuity in the local asymptotic minimax risk
Sensitivity Analysis for Multiple Comparisons in Matched Observational Studies through Quadratically Constrained Linear Programming
A sensitivity analysis in an observational study assesses the robustness of
significant findings to unmeasured confounding. While sensitivity analyses in
matched observational studies have been well addressed when there is a single
outcome variable, accounting for multiple comparisons through the existing
methods yields overly conservative results when there are multiple outcome
variables of interest. This stems from the fact that unmeasured confounding
cannot affect the probability of assignment to treatment differently depending
on the outcome being analyzed. Existing methods allow this to occur by
combining the results of individual sensitivity analyses to assess whether at
least one hypothesis is significant, which in turn results in an overly
pessimistic assessment of a study's sensitivity to unobserved biases. By
solving a quadratically constrained linear program, we are able to perform a
sensitivity analysis while enforcing that unmeasured confounding must have the
same impact on the treatment assignment probabilities across outcomes for each
individual in the study. We show that this allows for uniform improvements in
the power of a sensitivity analysis not only for testing the overall null of no
effect, but also for null hypotheses on \textit{specific} outcome variables
while strongly controlling the familywise error rate. We illustrate our method
through an observational study on the effect of smoking on naphthalene
exposure
Shrinkage Confidence Procedures
The possibility of improving on the usual multivariate normal confidence was
first discussed in Stein (1962). Using the ideas of shrinkage, through Bayesian
and empirical Bayesian arguments, domination results, both analytic and
numerical, have been obtained. Here we trace some of the developments in
confidence set estimation.Comment: Published in at http://dx.doi.org/10.1214/10-STS319 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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