176,029 research outputs found
Testing When a Parameter Is on the Boundary of the Maintained Hypothesis
This paper considers testing problems where several of the standard regularity conditions fail to hold. We consider the case where (i) parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and (ii) there may be a nuisance parameter that appears under the alternative hypothesis, but not under the null. The paper establishes the asymptotic null and local alternative distributions of quasi-likelihood ratio, rescaled quasi-likelihood ratio, Wald, and score tests in this case. The results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models. Examples treated in the paper are: (1) tests of the null hypothesis of no conditional heteroskedasticity in a GARCH(1, 1) regression model and (2) tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with (possibly) correlated random coefficients.Asymptotic distribution, boundary, conditional heteroskedasticity, extremum estimator, GARCH model, inequality restrictions, likelihood ratio test, local power, maximum likelihood estimator, parameter restrictions, random coefficients regression, quasi-maximum likelihood estimator, quasi-likelihood ratio test, restricted estimator, score test, Wald test
Inference Under Convex Cone Alternatives for Correlated Data
In this research, inferential theory for hypothesis testing under general
convex cone alternatives for correlated data is developed. While there exists
extensive theory for hypothesis testing under smooth cone alternatives with
independent observations, extension to correlated data under general convex
cone alternatives remains an open problem. This long-pending problem is
addressed by (1) establishing that a "generalized quasi-score" statistic is
asymptotically equivalent to the squared length of the projection of the
standard Gaussian vector onto the convex cone and (2) showing that the
asymptotic null distribution of the test statistic is a weighted chi-squared
distribution, where the weights are "mixed volumes" of the convex cone and its
polar cone. Explicit expressions for these weights are derived using the
volume-of-tube formula around a convex manifold in the unit sphere.
Furthermore, an asymptotic lower bound is constructed for the power of the
generalized quasi-score test under a sequence of local alternatives in the
convex cone. Applications to testing under order restricted alternatives for
correlated data are illustrated.Comment: 31 page
Asymptotic regime for impropriety tests of complex random vectors
Impropriety testing for complex-valued vector has been considered lately due
to potential applications ranging from digital communications to complex media
imaging. This paper provides new results for such tests in the asymptotic
regime, i.e. when the vector dimension and sample size grow commensurately to
infinity. The studied tests are based on invariant statistics named impropriety
coefficients. Limiting distributions for these statistics are derived, together
with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in
the Gaussian case. This characterization in the asymptotic regime allows also
to identify a phase transition in Roy's test with potential application in
detection of complex-valued low-rank subspace corrupted by proper noise in
large datasets. Simulations illustrate the accuracy of the proposed asymptotic
approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS
Global testing under sparse alternatives: ANOVA, multiple comparisons and the higher criticism
Testing for the significance of a subset of regression coefficients in a
linear model, a staple of statistical analysis, goes back at least to the work
of Fisher who introduced the analysis of variance (ANOVA). We study this
problem under the assumption that the coefficient vector is sparse, a common
situation in modern high-dimensional settings. Suppose we have covariates
and that under the alternative, the response only depends upon the order of
of those, . Under moderate sparsity levels, that
is, , we show that ANOVA is essentially optimal under some
conditions on the design. This is no longer the case under strong sparsity
constraints, that is, . In such settings, a multiple comparison
procedure is often preferred and we establish its optimality when
. However, these two very popular methods are suboptimal, and
sometimes powerless, under moderately strong sparsity where .
We suggest a method based on the higher criticism that is powerful in the whole
range . This optimality property is true for a variety of designs,
including the classical (balanced) multi-way designs and more modern ""
designs arising in genetics and signal processing. In addition to the standard
fixed effects model, we establish similar results for a random effects model
where the nonzero coefficients of the regression vector are normally
distributed.Comment: Published in at http://dx.doi.org/10.1214/11-AOS910 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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