26,723 research outputs found
Estimation and confidence sets for sparse normal mixtures
For high dimensional statistical models, researchers have begun to focus on
situations which can be described as having relatively few moderately large
coefficients. Such situations lead to some very subtle statistical problems. In
particular, Ingster and Donoho and Jin have considered a sparse normal means
testing problem, in which they described the precise demarcation or detection
boundary. Meinshausen and Rice have shown that it is even possible to estimate
consistently the fraction of nonzero coordinates on a subset of the detectable
region, but leave unanswered the question of exactly in which parts of the
detectable region consistent estimation is possible. In the present paper we
develop a new approach for estimating the fraction of nonzero means for
problems where the nonzero means are moderately large. We show that the
detection region described by Ingster and Donoho and Jin turns out to be the
region where it is possible to consistently estimate the expected fraction of
nonzero coordinates. This theory is developed further and minimax rates of
convergence are derived. A procedure is constructed which attains the optimal
rate of convergence in this setting. Furthermore, the procedure also provides
an honest lower bound for confidence intervals while minimizing the expected
length of such an interval. Simulations are used to enable comparison with the
work of Meinshausen and Rice, where a procedure is given but where rates of
convergence have not been discussed. Extensions to more general Gaussian
mixture models are also given.Comment: Published in at http://dx.doi.org/10.1214/009053607000000334 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Global testing against sparse alternatives in time-frequency analysis
In this paper, an over-sampled periodogram higher criticism (OPHC) test is
proposed for the global detection of sparse periodic effects in a
complex-valued time series. An explicit minimax detection boundary is
established between the rareness and weakness of the complex sinusoids hidden
in the series. The OPHC test is shown to be asymptotically powerful in the
detectable region. Numerical simulations illustrate and verify the
effectiveness of the proposed test. Furthermore, the periodogram over-sampled
by is proven universally optimal in global testing for
periodicities under a mild minimum separation condition.Comment: Published at http://dx.doi.org/10.1214/15-AOS1412 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Detection of Sparse Positive Dependence
In a bivariate setting, we consider the problem of detecting a sparse
contamination or mixture component, where the effect manifests itself as a
positive dependence between the variables, which are otherwise independent in
the main component. We first look at this problem in the context of a normal
mixture model. In essence, the situation reduces to a univariate setting where
the effect is a decrease in variance. In particular, a higher criticism test
based on the pairwise differences is shown to achieve the detection boundary
defined by the (oracle) likelihood ratio test. We then turn to a Gaussian
copula model where the marginal distributions are unknown. Standard invariance
considerations lead us to consider rank tests. In fact, a higher criticism test
based on the pairwise rank differences achieves the detection boundary in the
normal mixture model, although not in the very sparse regime. We do not know of
any rank test that has any power in that regime
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