39 research outputs found
Efficient nonparametric estimation of Toeplitz covariance matrices
A new nonparametric estimator for Toeplitz covariance matrices is proposed.
This estimator is based on a data transformation that translates the problem of
Toeplitz covariance matrix estimation to the problem of mean estimation in an
approximate Gaussian regression. The resulting Toeplitz covariance matrix
estimator is positive definite by construction, fully data-driven and
computationally very fast. Moreover, this estimator is shown to be minimax
optimal under the spectral norm for a large class of Toeplitz matrices. These
results are readily extended to estimation of inverses of Toeplitz covariance
matrices. Also, an alternative version of the Whittle likelihood for the
spectral density based on the Discrete Cosine Transform (DCT) is proposed. The
method is implemented in the R package vstdct that accompanies the paper.Comment: 58 pages, 6 figures, 9 table
Adaptive empirical Bayesian smoothing splines
In this paper we develop and study adaptive empirical Bayesian smoothing
splines. These are smoothing splines with both smoothing parameter and penalty
order determined via the empirical Bayes method from the marginal likelihood of
the model. The selected order and smoothing parameter are used to construct
adaptive credible sets with good frequentist coverage for the underlying
regression function. We use these credible sets as a proxy to show the superior
performance of adaptive empirical Bayesian smoothing splines compared to
frequentist smoothing splines
Uniformly Valid Inference Based on the Lasso in Linear Mixed Models
Linear mixed models (LMMs) are suitable for clustered data and are common in
e.g. biometrics, medicine, or small area estimation. It is of interest to
obtain valid inference after selecting a subset of available variables. We
construct confidence sets for the fixed effects in Gaussian LMMs that are
estimated via a Lasso-type penalization which allows quantifying the joint
uncertainty of both variable selection and estimation. To this end, we exploit
the properties of restricted maximum likelihood (REML) estimators to separate
the estimation of the regression coefficients and covariance parameters. We
derive an appropriate normalizing sequence to prove the uniform Cramer
consistency of the REML estimators. We then show that the resulting confidence
sets for the fixed effects are uniformly valid over the parameter space of both
the regression coefficients and the covariance parameters. Their superiority to
naive post-selection least-squares confidence sets is validated in simulations
and illustrated with a study of the acid neutralization capacity of U.S. lakes.Comment: 22 pages, 1 figur
On threshold estimation in threshold vector error correction models
Resource /Energy Economics and Policy,