297 research outputs found
Semiparametric models
Much empirical research is concerned with estimating conditional mean, median, or hazard functions. For example, labor economists are interested in estimating the mean wages of employed individuals conditional on characteristics such as years of work experience and education. The most frequently used estimation methods assume that the function of interest is known up to a set of constant parameters that can be estimated from data. Models in which the only unknown quantities are a finite set of constant parameters are called parametric. The use of a parametric model greatly simplifies estimation, statistical inference, and interpretation of the estimation results but is rarely justified by theoretical or other a priori considerations. Estimation and inference based on convenient but incorrect assumptions about the form of the conditional mean function can be highly misleading. --
Nonparametric methods for inference in the presence of instrumental variables
We suggest two nonparametric approaches, based on kernel methods and
orthogonal series to estimating regression functions in the presence of
instrumental variables. For the first time in this class of problems, we derive
optimal convergence rates, and show that they are attained by particular
estimators. In the presence of instrumental variables the relation that
identifies the regression function also defines an ill-posed inverse problem,
the ``difficulty'' of which depends on eigenvalues of a certain integral
operator which is determined by the joint density of endogenous and
instrumental variables. We delineate the role played by problem difficulty in
determining both the optimal convergence rate and the appropriate choice of
smoothing parameter.Comment: Published at http://dx.doi.org/10.1214/009053605000000714 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonparametric Estimation of an Additive Model With a Link Function
This paper describes an estimator of the additive components of a
nonparametric additive model with a known link function. When the additive
components are twice continuously differentiable, the estimator is
asymptotically normally distributed with a rate of convergence in probability
of n^{-2/5}. This is true regardless of the (finite) dimension of the
explanatory variable. Thus, in contrast to the existing asymptotically normal
estimator, the new estimator has no curse of dimensionality. Moreover, the
estimator has an oracle property. The asymptotic distribution of each additive
component is the same as it would be if the other components were known with
certainty.Comment: Published at http://dx.doi.org/10.1214/009053604000000814 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Methodology and convergence rates for functional linear regression
In functional linear regression, the slope ``parameter'' is a function.
Therefore, in a nonparametric context, it is determined by an infinite number
of unknowns. Its estimation involves solving an ill-posed problem and has
points of contact with a range of methodologies, including statistical
smoothing and deconvolution. The standard approach to estimating the slope
function is based explicitly on functional principal components analysis and,
consequently, on spectral decomposition in terms of eigenvalues and
eigenfunctions. We discuss this approach in detail and show that in certain
circumstances, optimal convergence rates are achieved by the PCA technique. An
alternative approach based on quadratic regularisation is suggested and shown
to have advantages from some points of view.Comment: Published at http://dx.doi.org/10.1214/009053606000000957 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonparametric Estimation of an Additive Quantile Regression Model
This paper is concerned with estimating the additive components of a nonparametric additive quantile regression model. We develop an estimator that is asymptotically normally distributed with a rate of convergence in probability of when the additive components are -times continuously differentiable for some . This result holds regardless of the dimension of the covariates and, therefore, the new estimator has no curse of dimensionality. In addition, the estimator has an oracle property and is easily extended to a generalized additive quantile regression model with a link function. The numerical performance and usefulness of the estimator are illustrated by Monte Carlo experiments and an empirical exampleAdditive models, local polynomial estimation, nonparametric regression, quantile regression, series estimation.
Variable selection in nonparametric additive models
We consider a nonparametric additive model of a conditional mean function in
which the number of variables and additive components may be larger than the
sample size but the number of nonzero additive components is "small" relative
to the sample size. The statistical problem is to determine which additive
components are nonzero. The additive components are approximated by truncated
series expansions with B-spline bases. With this approximation, the problem of
component selection becomes that of selecting the groups of coefficients in the
expansion. We apply the adaptive group Lasso to select nonzero components,
using the group Lasso to obtain an initial estimator and reduce the dimension
of the problem. We give conditions under which the group Lasso selects a model
whose number of components is comparable with the underlying model, and the
adaptive group Lasso selects the nonzero components correctly with probability
approaching one as the sample size increases and achieves the optimal rate of
convergence. The results of Monte Carlo experiments show that the adaptive
group Lasso procedure works well with samples of moderate size. A data example
is used to illustrate the application of the proposed method.Comment: Published in at http://dx.doi.org/10.1214/09-AOS781 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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