11,914 research outputs found
Rank-based optimal tests of the adequacy of an elliptic VARMA model
We are deriving optimal rank-based tests for the adequacy of a vector
autoregressive-moving average (VARMA) model with elliptically contoured
innovation density. These tests are based on the ranks of pseudo-Mahalanobis
distances and on normed residuals computed from Tyler's [Ann. Statist. 15
(1987) 234-251] scatter matrix; they generalize the univariate signed rank
procedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1-29].
Two types of optimality properties are considered, both in the local and
asymptotic sense, a la Le Cam: (a) (fixed-score procedures) local asymptotic
minimaxity at selected radial densities, and (b) (estimated-score procedures)
local asymptotic minimaxity uniform over a class F of radial densities.
Contrary to their classical counterparts, based on cross-covariance matrices,
these tests remain valid under arbitrary elliptically symmetric innovation
densities, including those with infinite variance and heavy-tails. We show that
the AREs of our fixed-score procedures, with respect to traditional (Gaussian)
methods, are the same as for the tests of randomness proposed in Hallin and
Paindaveine [Bernoulli 8 (2002b) 787-815]. The multivariate serial extensions
of the classical Chernoff-Savage and Hodges-Lehmann results obtained there thus
also hold here; in particular, the van der Waerden versions of our tests are
uniformly more powerful than those based on cross-covariances. As for our
estimated-score procedures, they are fully adaptive, hence, uniformly optimal
over the class of innovation densities satisfying the required technical
assumptions.Comment: Published at http://dx.doi.org/10.1214/009053604000000724 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Performance of Statistical Tests for Single Source Detection using Random Matrix Theory
This paper introduces a unified framework for the detection of a source with
a sensor array in the context where the noise variance and the channel between
the source and the sensors are unknown at the receiver. The Generalized Maximum
Likelihood Test is studied and yields the analysis of the ratio between the
maximum eigenvalue of the sampled covariance matrix and its normalized trace.
Using recent results of random matrix theory, a practical way to evaluate the
threshold and the -value of the test is provided in the asymptotic regime
where the number of sensors and the number of observations per sensor
are large but have the same order of magnitude. The theoretical performance of
the test is then analyzed in terms of Receiver Operating Characteristic (ROC)
curve. It is in particular proved that both Type I and Type II error
probabilities converge to zero exponentially as the dimensions increase at the
same rate, and closed-form expressions are provided for the error exponents.
These theoretical results rely on a precise description of the large deviations
of the largest eigenvalue of spiked random matrix models, and establish that
the presented test asymptotically outperforms the popular test based on the
condition number of the sampled covariance matrix.Comment: 45 p. improved presentation; more proofs provide
Computational barriers in minimax submatrix detection
This paper studies the minimax detection of a small submatrix of elevated
mean in a large matrix contaminated by additive Gaussian noise. To investigate
the tradeoff between statistical performance and computational cost from a
complexity-theoretic perspective, we consider a sequence of discretized models
which are asymptotically equivalent to the Gaussian model. Under the hypothesis
that the planted clique detection problem cannot be solved in randomized
polynomial time when the clique size is of smaller order than the square root
of the graph size, the following phase transition phenomenon is established:
when the size of the large matrix , if the submatrix size
for any , computational complexity
constraints can incur a severe penalty on the statistical performance in the
sense that any randomized polynomial-time test is minimax suboptimal by a
polynomial factor in ; if for any
, minimax optimal detection can be attained within
constant factors in linear time. Using Schatten norm loss as a representative
example, we show that the hardness of attaining the minimax estimation rate can
crucially depend on the loss function. Implications on the hardness of support
recovery are also obtained.Comment: Published at http://dx.doi.org/10.1214/14-AOS1300 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states
Hypothesis testing is a fundamental issue in statistical inference and has
been a crucial element in the development of information sciences. The Chernoff
bound gives the minimal Bayesian error probability when discriminating two
hypotheses given a large number of observations. Recently the combined work of
Audenaert et al. [Phys. Rev. Lett. 98, 160501] and Nussbaum and Szkola
[quant-ph/0607216] has proved the quantum analog of this bound, which applies
when the hypotheses correspond to two quantum states. Based on the quantum
Chernoff bound, we define a physically meaningful distinguishability measure
and its corresponding metric in the space of states; the latter is shown to
coincide with the Wigner-Yanase metric. Along the same lines, we define a
second, more easily implementable, distinguishability measure based on the
error probability of discrimination when the same local measurement is
performed on every copy. We study some general properties of these measures,
including the probability distribution of density matrices, defined via the
volume element induced by the metric, and illustrate their use in the
paradigmatic cases of qubits and Gaussian infinite-dimensional states.Comment: 16 page
Limits on Support Recovery with Probabilistic Models: An Information-Theoretic Framework
The support recovery problem consists of determining a sparse subset of a set
of variables that is relevant in generating a set of observations, and arises
in a diverse range of settings such as compressive sensing, and subset
selection in regression, and group testing. In this paper, we take a unified
approach to support recovery problems, considering general probabilistic models
relating a sparse data vector to an observation vector. We study the
information-theoretic limits of both exact and partial support recovery, taking
a novel approach motivated by thresholding techniques in channel coding. We
provide general achievability and converse bounds characterizing the trade-off
between the error probability and number of measurements, and we specialize
these to the linear, 1-bit, and group testing models. In several cases, our
bounds not only provide matching scaling laws in the necessary and sufficient
number of measurements, but also sharp thresholds with matching constant
factors. Our approach has several advantages over previous approaches: For the
achievability part, we obtain sharp thresholds under broader scalings of the
sparsity level and other parameters (e.g., signal-to-noise ratio) compared to
several previous works, and for the converse part, we not only provide
conditions under which the error probability fails to vanish, but also
conditions under which it tends to one.Comment: Accepted to IEEE Transactions on Information Theory; presented in
part at ISIT 2015 and SODA 201
Optimal Comparison of Misspecified Moment Restriction Models under a Chosen Measure of Fit
Abstract Suppose that the econometrician is interested in comparing two misspecified moment restriction models, where the comparison is performed in terms of some chosen measure of fit. This paper is concerned with describing an optimal test of the Vuong (1989) and Rivers and Vuong (2002) type null hypothesis that the two models are equivalent under the given measure of fit (the ranking may vary for different measures). We adopt the generalized Neyman-Pearson optimality criterion, which focuses on the decay rates of the type I and II error probabilities under fixed non-local alternatives, and derive an optimal but practically infeasible test. Then, as an illustration, by considering the model comparison hypothesis defined by the weighted Euclidean norm of moment restrictions, we propose a feasible approximate test statistic to the optimal one and study its asymptotic properties. Local power properties, one-sided test, and comparison under the generalized empirical likelihood-based measure of fit are also investigated. A simulation study illustrates that our approximate test is more powerful than the Rivers-Vuong test.Moment restriction; Model comparison; Misspecification; Generalized Neyman-Pearson optimality; Empirical likelihood; GMM
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