675 research outputs found
Nonparametric inference in hidden Markov models using P-splines
Hidden Markov models (HMMs) are flexible time series models in which the
distributions of the observations depend on unobserved serially correlated
states. The state-dependent distributions in HMMs are usually taken from some
class of parametrically specified distributions. The choice of this class can
be difficult, and an unfortunate choice can have serious consequences for
example on state estimates, on forecasts and generally on the resulting model
complexity and interpretation, in particular with respect to the number of
states. We develop a novel approach for estimating the state-dependent
distributions of an HMM in a nonparametric way, which is based on the idea of
representing the corresponding densities as linear combinations of a large
number of standardized B-spline basis functions, imposing a penalty term on
non-smoothness in order to maintain a good balance between goodness-of-fit and
smoothness. We illustrate the nonparametric modeling approach in a real data
application concerned with vertical speeds of a diving beaked whale,
demonstrating that compared to parametric counterparts it can lead to models
that are more parsimonious in terms of the number of states yet fit the data
equally well
A comparison of alternative approaches to sup-norm goodness of fit tests with estimated parameters
Goodness of fit tests based on sup-norm statistics of empirical processes have nonstandard limiting distributions when the null hypothesis is composite-that is, when parameters of the null model are estimated. Several solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985) can be applied to compute appropriate critical values for tests based on sup-norm statistics. The resulting tests have quite accurate size, a fact which has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin's approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981) through the score function of the parametric model. Monte Carlo experiments suggest that these two testing strategies are roughly comparable to one another and more powerful than a simple bootstrap procedure.
Confidence Corridors for Multivariate Generalized Quantile Regression
We focus on the construction of confidence corridors for multivariate
nonparametric generalized quantile regression functions. This construction is
based on asymptotic results for the maximal deviation between a suitable
nonparametric estimator and the true function of interest which follow after a
series of approximation steps including a Bahadur representation, a new strong
approximation theorem and exponential tail inequalities for Gaussian random
fields. As a byproduct we also obtain confidence corridors for the regression
function in the classical mean regression. In order to deal with the problem of
slowly decreasing error in coverage probability of the asymptotic confidence
corridors, which results in meager coverage for small sample sizes, a simple
bootstrap procedure is designed based on the leading term of the Bahadur
representation. The finite sample properties of both procedures are
investigated by means of a simulation study and it is demonstrated that the
bootstrap procedure considerably outperforms the asymptotic bands in terms of
coverage accuracy. Finally, the bootstrap confidence corridors are used to
study the efficacy of the National Supported Work Demonstration, which is a
randomized employment enhancement program launched in the 1970s. This article
has supplementary materials
New L2-type exponentiality tests
We introduce new consistent and scale-free goodness-of-fit tests for the exponential distribution based on the Puri-Rubin characterization. For the construction of test statistics we employ weighted L2 distance between V-empirical Laplace transforms of random variables that appear in the characterization. We derive the asymptotic behaviour under the null hypothesis as well as under fixed alternatives. We compare our tests, in terms of the Bahadur efficiency, to the likelihood ratio test, as well as some recent characterization based goodness-of-fit tests for the exponential distribution. We also compare the power of our tests to the power of some recent and classical exponentiality tests. According to both criteria, our tests are shown to be strong and outperform most of their competitors.Peer Reviewe
Optimal Calibration for Multiple Testing against Local Inhomogeneity in Higher Dimension
Based on two independent samples X_1,...,X_m and X_{m+1},...,X_n drawn from
multivariate distributions with unknown Lebesgue densities p and q
respectively, we propose an exact multiple test in order to identify
simultaneously regions of significant deviations between p and q. The
construction is built from randomized nearest-neighbor statistics. It does not
require any preliminary information about the multivariate densities such as
compact support, strict positivity or smoothness and shape properties. The
properly adjusted multiple testing procedure is shown to be sharp-optimal for
typical arrangements of the observation values which appear with probability
close to one. The proof relies on a new coupling Bernstein type exponential
inequality, reflecting the non-subgaussian tail behavior of a combinatorial
process. For power investigation of the proposed method a reparametrized
minimax set-up is introduced, reducing the composite hypothesis "p=q" to a
simple one with the multivariate mixed density (m/n)p+(1-m/n)q as infinite
dimensional nuisance parameter. Within this framework, the test is shown to be
spatially and sharply asymptotically adaptive with respect to uniform loss on
isotropic H\"older classes. The exact minimax risk asymptotics are obtained in
terms of solutions of the optimal recovery
Inference on distribution functions under measurement error
This paper is concerned with inference on the cumulative distribution function (cdf) FXā in the classical measurement error model X = Xā + Īµ. We consider the case where the density of the measurement error Īµ is unknown and estimated by repeated measurements, and show validity of a bootstrap approximation for the distribution of the deviation in the sup-norm between the deconvolution cdf estimator and FXā. We allow the density of Īµ to be ordinary or super smooth. We also provide several theoretical results on the bootstrap and asymptotic Gumbel approximations of the sup-norm deviation for the case where the density of Īµ is known. Our approximation results are applicable to various contexts, such as confidence bands for FXā and its quantiles, and for performing various cdf-based tests such as goodness-of-fit tests for parametric models of Xā, two sample homogeneity tests, and tests for stochastic dominance. Simulation and real data examples illustrate satisfactory performance of the proposed methods
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