309 research outputs found
Dimension reduction-based significance testing in nonparametric regression
A dimension reduction-based adaptive-to-model test is proposed for
significance of a subset of covariates in the context of a nonparametric
regression model. Unlike existing local smoothing significance tests, the new
test behaves like a local smoothing test as if the number of covariates were
just that under the null hypothesis and it can detect local alternatives
distinct from the null at the rate that is only related to the number of
covariates under the null hypothesis. Thus, the curse of dimensionality is
largely alleviated when nonparametric estimation is inevitably required. In the
cases where there are many insignificant covariates, the improvement of the new
test is very significant over existing local smoothing tests on the
significance level maintenance and power enhancement. Simulation studies and a
real data analysis are conducted to examine the finite sample performance of
the proposed test.Comment: 49 pages, 2 figure
Specification testing for regressions: an approach bridging between local smoothing and global smoothing methods
For regression models, most of existing specification tests can be
categorized into the class of local smoothing tests and of global smoothing
tests. Compared with global smoothing tests, local smoothing tests can only
detect local alternatives distinct from the null hypothesis at a much slower
rate when the dimension of predictor vector is high, but can be more sensitive
to oscillating alternatives. In this paper, we suggest a projection-based test
to bridge between the local and global smoothing-based methodologies such that
the test can benefit from the advantages of these two types of tests. The test
construction is based on a kernel estimation-based method and the resulting
test becomes a distance-based test with a closed form. The asymptotic
properties are investigated. Simulations and a real data analysis are conducted
to evaluate the performance of the test in finite sample cases.Comment: 31 page
Adaptive-to-model hybrid of tests for regressions
In model checking for regressions, nonparametric estimation-based tests
usually have tractable limiting null distributions and are sensitive to
oscillating alternative models, but suffer from the curse of dimensionality. In
contrast, empirical process-based tests can, at the fastest possible rate,
detect local alternatives distinct from the null model, but is less sensitive
to oscillating alternative models and with intractable limiting null
distributions. It has long been an issue on how to construct a test that can
fully inherit the merits of these two types of tests and avoid the
shortcomings. We in this paper propose a generic adaptive-to-model hybrid of
moment and conditional moment-based test to achieve this goal. Further, a
significant feature of the method is to make nonparametric estimation-based
tests, under the alternatives, also share the merits of existing empirical
process-based tests. This methodology can be readily applied to other kinds of
data and constructing other hybrids. As a by-product in sufficient dimension
reduction field, the estimation of residual-related central subspace is used to
indicate the underlying models for model adaptation. A systematic study is
devoted to showing when alternative models can be indicated and when cannot.
This estimation is of its own interest and can be applied to the problems with
other kinds of data. Numerical studies are conducted to verify the powerfulness
of the proposed test.Comment: 35pages, 6figure
Model checking for generalized linear models: a dimension-reduction model-adaptive approach
Local smoothing testing that is based on multivariate nonparametric
regression estimation is one of the main model checking methodologies in the
literature. However, relevant tests suffer from the typical curse of
dimensionality resulting in slow convergence rates to their limits under the
null hypotheses and less deviation from the null under alternatives. This
problem leads tests to not well maintain the significance level and to be less
sensitive to alternatives. In this paper, a dimension-reduction model-adaptive
test is proposed for generalized linear models. The test behaves like a local
smoothing test as if the model were univariate, and can be consistent against
any global alternatives and can detect local alternatives distinct from the
null at a fast rate that existing local smoothing tests can achieve only when
the model is univariate. Simulations are carried out to examine the performance
of our methodology. A real data analysis is conducted for illustration. The
method can readily be extended to global smoothing methodology and other
testing problems
Integrated conditional moment test and beyond: when the number of covariates is divergent
The classic integrated conditional moment test is a promising method for
testing regression model misspecification. However, it severely suffers from
the curse of dimensionality. To extend it to handle the testing problem for
parametric multi-index models with diverging number of covariates, we
investigate three issues in inference in this paper. First, we study the
consistency and asymptotically linear representation of the least squares
estimator of the parameter matrix at faster rates of divergence than those in
the literature for nonlinear models. Second, we propose, via sufficient
dimension reduction techniques, an adaptive-to-model version of the integrated
conditional moment test. We study the asymptotic properties of the new test
under both the null and alternative hypothesis to examine its ability of
significance level maintenance and its sensitivity to the global and local
alternatives that are distinct from the null at the fastest possible rate in
hypothesis testing. Third, we derive the consistency of the bootstrap
approximation for the new test in the diverging dimension setting. The
numerical studies show that the new test can very much enhance the performance
of the original ICM test in high-dimensional scenarios. We also apply the test
to a real data set for illustrations
An adaptive-to-model test for partially parametric single-index models
Residual marked empirical process-based tests are commonly used in regression
models. However, they suffer from data sparseness in high-dimensional space
when there are many covariates. This paper has three purposes. First, we
suggest a partial dimension reduction adaptive-to-model testing procedure that
can be omnibus against general global alternative models although it fully use
the dimension reduction structure under the null hypothesis. This feature is
because that the procedure can automatically adapt to the null and alternative
models, and thus greatly overcomes the dimensionality problem. Second, to
achieve the above goal, we propose a ridge-type eigenvalue ratio estimate to
automatically determine the number of linear combinations of the covariates
under the null and alternatives. Third, a Monte-Carlo approximation to the
sampling null distribution is suggested. Unlike existing bootstrap
approximation methods, this gives an approximation as close to the sampling
null distribution as possible by fully utilising the dimension reduction model
structure under the null. Simulation studies and real data analysis are then
conducted to illustrate the performance of the new test and compare it with
existing tests.Comment: 35 pages, 2 figure
Estimation and adaptive-to-model testing for regressions with diverging number of predictors
The research described in this paper is motivated by model checking for
parametric single-index models with diverging number of predictors. To
construct a test statistic, we first study the asymptotic property of the
estimators of involved parameters of interest under the null and alternative
hypothesis when the dimension is divergent to infinity as the sample size goes
to infinity. For the testing problem, we study an adaptive-to-model
residual-marked empirical process as the basis for constructing a test
statistic. By modifying the approach in the literature to suit the diverging
dimension settings, we construct a martingale transformation. Under the null,
local and global alternative hypothesis, the weak limits of the empirical
process are derived and then the asymptotic properties of the test statistic
are investigated. Simulation studies are carried out to examine the performance
of the test
A projection-based adaptive-to-model test for regressions
A longstanding problem of existing empirical process-based tests for
regressions is that when the number of covariates is greater than one, they
either have no tractable limiting null distributions or are not omnibus. To
attack this problem, we in this paper propose a projection-based
adaptive-to-model approach. When the hypothetical model is parametric
single-index, the method can fully utilize the dimension reduction model
structure under the null hypothesis as if the covariate were one-dimensional
such that the martingale transformation-based test can be asymptotically
distribution-free. Further, the test can automatically adapt to the underlying
model structure such that the test can be omnibus and thus detect alternative
models distinct from the hypothetical model at the fastest possible rate in
hypothesis testing. The method is examined through simulation studied and is
illustrated by a real data analysis
Bounds smaller than the Fisher information for generalized linear models
In this paper, we propose a parameter space augmentation approach that is
based on "intentionally" introducing a pseudo-nuisance parameter into
generalized linear models for the purpose of variance reduction. We first
consider the parameter whose norm is equal to one. By introducing a
pseudo-nuisance parameter into models to be estimated, an extra estimation is
asymptotically normal and is, more importantly, non-positively correlated to
the estimation that asymptotically achieves the Fisher/quasi Fisher
information. As such, the resulting estimation is asymptotically with smaller
variance-covariance matrices than the Fisher/quasi Fisher information. For
general cases where the norm of the parameter is not necessarily equal to one,
two-stage quasi-likelihood procedures separately estimating the scalar and
direction of the parameter are proposed. The traces of the limiting
variance-covariance matrices are in general smaller than or equal to that of
the Fisher/quasi-Fisher information. We also discuss the pros and cons of the
new methodology, and possible extensions. As this methodology of parameter
space augmentation is general, and then may be readily extended to handle, say,
cluster data and correlated data, and other models.Comment: 32 page
A robust adaptive-to-model enhancement test for parametric single-index models
In the research on checking whether the underlying model is of parametric
single-index structure with outliers in observations, the purpose of this paper
is two-fold. First, a test that is robust against outliers is suggested. The
Hampel's second-order influence function of the test statistic is proved to be
bounded. Second, the test fully uses the dimension reduction structure of the
hypothetical model and automatically adapts to alternative models when the null
hypothesis is false. Thus, the test can greatly overcome the dimensionality
problem and is still omnibus against general alternative models. The
performance of the test is demonstrated by both Monte Carlo simulation studies
and an application to a real dataset
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