73 research outputs found
Non-crossing nonparametric estimates of quantile curves
--Quantile estimation , conditional distribution , local linear estimate , Nadaraya Watson estimate , crossing quantile curves
Testing for Homogeneity in Mixture Models
Statistical models of unobserved heterogeneity are typically formalized as
mixtures of simple parametric models and interest naturally focuses on testing
for homogeneity versus general mixture alternatives. Many tests of this type
can be interpreted as tests, as in Neyman (1959), and shown to be
locally, asymptotically optimal. These tests will be contrasted
with a new approach to likelihood ratio testing for general mixture models. The
latter tests are based on estimation of general nonparametric mixing
distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimator.
Recent developments in convex optimization have dramatically improved upon
earlier EM methods for computation of these estimators, and recent results on
the large sample behavior of likelihood ratios involving such estimators yield
a tractable form of asymptotic inference. Improvement in computation efficiency
also facilitates the use of a bootstrap methods to determine critical values
that are shown to work better than the asymptotic critical values in finite
samples. Consistency of the bootstrap procedure is also formally established.
We compare performance of the two approaches identifying circumstances in which
each is preferred
Weak convergence of the empirical copula process with respect to weighted metrics
The empirical copula process plays a central role in the asymptotic analysis
of many statistical procedures which are based on copulas or ranks. Among other
applications, results regarding its weak convergence can be used to develop
asymptotic theory for estimators of dependence measures or copula densities,
they allow to derive tests for stochastic independence or specific copula
structures, or they may serve as a fundamental tool for the analysis of
multivariate rank statistics. In the present paper, we establish weak
convergence of the empirical copula process (for observations that are allowed
to be serially dependent) with respect to weighted supremum distances. The
usefulness of our results is illustrated by applications to general bivariate
rank statistics and to estimation procedures for the Pickands dependence
function arising in multivariate extreme-value theory.Comment: 39 pages + 7 pages of supplementary material, 1 figur
A subsampled double bootstrap for massive data
The bootstrap is a popular and powerful method for assessing precision of
estimators and inferential methods. However, for massive datasets which are
increasingly prevalent, the bootstrap becomes prohibitively costly in
computation and its feasibility is questionable even with modern parallel
computing platforms. Recently Kleiner, Talwalkar, Sarkar, and Jordan (2014)
proposed a method called BLB (Bag of Little Bootstraps) for massive data which
is more computationally scalable with little sacrifice of statistical accuracy.
Building on BLB and the idea of fast double bootstrap, we propose a new
resampling method, the subsampled double bootstrap, for both independent data
and time series data. We establish consistency of the subsampled double
bootstrap under mild conditions for both independent and dependent cases.
Methodologically, the subsampled double bootstrap is superior to BLB in terms
of running time, more sample coverage and automatic implementation with less
tuning parameters for a given time budget. Its advantage relative to BLB and
bootstrap is also demonstrated in numerical simulations and a data
illustration
A test for Archimedeanity in bivariate copula models
We propose a new test for the hypothesis that a bivariate copula is an
Archimedean copula. The test statistic is based on a combination of two
measures resulting from the characterization of Archimedean copulas by the
property of associativity and by a strict upper bound on the diagonal by the
Fr\'echet-upper bound. We prove weak convergence of this statistic and show
that the critical values of the corresponding test can be determined by the
multiplier bootstrap method. The test is shown to be consistent against all
departures from Archimedeanity if the copula satisfies weak smoothness
assumptions. A simulation study is presented which illustrates the finite
sample properties of the new test.Comment: 18 pages, 2 figure
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