294 research outputs found
Bandwidth selection for smooth backfitting in additive models
The smooth backfitting introduced by Mammen, Linton and Nielsen [Ann.
Statist. 27 (1999) 1443-1490] is a promising technique to fit additive
regression models and is known to achieve the oracle efficiency bound. In this
paper, we propose and discuss three fully automated bandwidth selection methods
for smooth backfitting in additive models. The first one is a penalized least
squares approach which is based on higher-order stochastic expansions for the
residual sums of squares of the smooth backfitting estimates. The other two are
plug-in bandwidth selectors which rely on approximations of the average squared
errors and whose utility is restricted to local linear fitting. The large
sample properties of these bandwidth selection methods are given. Their finite
sample properties are also compared through simulation experiments.Comment: Published at http://dx.doi.org/10.1214/009053605000000101 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A simple smooth backfitting method for additive models
In this paper a new smooth backfitting estimate is proposed for additive
regression models. The estimate has the simple structure of Nadaraya--Watson
smooth backfitting but at the same time achieves the oracle property of local
linear smooth backfitting. Each component is estimated with the same asymptotic
accuracy as if the other components were known.Comment: Published at http://dx.doi.org/10.1214/009053606000000696 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Smooth backfitting in generalized additive models
Generalized additive models have been popular among statisticians and data
analysts in multivariate nonparametric regression with non-Gaussian responses
including binary and count data. In this paper, a new likelihood approach for
fitting generalized additive models is proposed. It aims to maximize a smoothed
likelihood. The additive functions are estimated by solving a system of
nonlinear integral equations. An iterative algorithm based on smooth
backfitting is developed from the Newton--Kantorovich theorem. Asymptotic
properties of the estimator and convergence of the algorithm are discussed. It
is shown that our proposal based on local linear fit achieves the same bias and
variance as the oracle estimator that uses knowledge of the other components.
Numerical comparison with the recently proposed two-stage estimator [Ann.
Statist. 32 (2004) 2412--2443] is also made.Comment: Published in at http://dx.doi.org/10.1214/009053607000000596 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Semi-parametric regression: Efficiency gains from modeling the nonparametric part
It is widely admitted that structured nonparametric modeling that circumvents
the curse of dimensionality is important in nonparametric estimation. In this
paper we show that the same holds for semi-parametric estimation. We argue that
estimation of the parametric component of a semi-parametric model can be
improved essentially when more structure is put into the nonparametric part of
the model. We illustrate this for the partially linear model, and investigate
efficiency gains when the nonparametric part of the model has an additive
structure. We present the semi-parametric Fisher information bound for
estimating the parametric part of the partially linear additive model and
provide semi-parametric efficient estimators for which we use a smooth
backfitting technique to deal with the additive nonparametric part. We also
present the finite sample performances of the proposed estimators and analyze
Boston housing data as an illustration.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ296 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Backfitting and smooth backfitting for additive quantile models
In this paper, we study the ordinary backfitting and smooth backfitting as
methods of fitting additive quantile models. We show that these backfitting
quantile estimators are asymptotically equivalent to the corresponding
backfitting estimators of the additive components in a specially-designed
additive mean regression model. This implies that the theoretical properties of
the backfitting quantile estimators are not unlike those of backfitting mean
regression estimators. We also assess the finite sample properties of the two
backfitting quantile estimators.Comment: Published in at http://dx.doi.org/10.1214/10-AOS808 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). With Correction
Flexible generalized varying coefficient regression models
This paper studies a very flexible model that can be used widely to analyze
the relation between a response and multiple covariates. The model is
nonparametric, yet renders easy interpretation for the effects of the
covariates. The model accommodates both continuous and discrete random
variables for the response and covariates. It is quite flexible to cover the
generalized varying coefficient models and the generalized additive models as
special cases. Under a weak condition we give a general theorem that the
problem of estimating the multivariate mean function is equivalent to that of
estimating its univariate component functions. We discuss implications of the
theorem for sieve and penalized least squares estimators, and then investigate
the outcomes in full details for a kernel-type estimator. The kernel estimator
is given as a solution of a system of nonlinear integral equations. We provide
an iterative algorithm to solve the system of equations and discuss the
theoretical properties of the estimator and the algorithm. Finally, we give
simulation results.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1026 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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Generalised additive dependency inflated models including aggregated covariates
Let us assume that X, Y and U are observed and that the conditional mean of U given X and Y can be expressed via an additive dependency of X, λ(X)Y and X + Y for some unspecified function . This structured regression model can be transferred to a hazard model or a density model when applied on some appropriate grid, and has important forecasting applications via structured marker dependent hazards models or structured density models including age-period-cohort relationships. The structured regression model is also important when the severity of the dependent variable has a complicated dependency on waiting times X, Y and the total waiting time X+Y . In case the conditional mean of U approximates a density, the regression model can be used to analyse the age-period-cohort model, also when exposure data are not available. In case the conditional mean of U approximates a marker dependent hazard, the regression model introduces new relevant age-period-cohort time scale interdependencies in understanding longevity. A direct use of the regression relationship introduced in this paper is the estimation of the severity of outstanding liabilities in non-life insurance companies. The technical approach taken is to use B-splines to capture the underlying one-dimensional unspecified functions. It is shown via finite sample simulation studies and an application for forecasting future asbestos related deaths in the UK that the B-spline approach works well in practice. Special consideration has been given to ensure identifiability of all models considered
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Asymptotics for In-Sample Density Forecasting
This paper generalizes recent proposals of density forecasting models and it develops theory for this class of models. In density forecasting the density of observations is estimated in regions where the density is not observed. Identification of the density in such regions is guaranteed by structural assumptions on the density that allows exact extrapolation. In this paper the structural assumption is made that the density is a product of one-dimensional functions. The theory is quite general in assuming the shape of the region where the density is observed. Such models naturally arise when the time point of an observation can be written as the sum of two terms (e.g. onset and incubation period of a disease). The developed theory also allows for a multiplicative factor of seasonal effects. Seasonal effects are present in many actuarial, biostatistical, econometric and statistical studies. Smoothing estimators are proposed that are based on backfitting. Full asymptotic theory is derived for them. A practical example from the insurance business is given producing a within year budget of reported insurance claims. A small sample study supports the theoretical results
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