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Nonparametric Regression, Confidence Regions and Regularization
In this paper we offer a unified approach to the problem of nonparametric
regression on the unit interval. It is based on a universal, honest and
non-asymptotic confidence region which is defined by a set of linear
inequalities involving the values of the functions at the design points.
Interest will typically centre on certain simplest functions in that region
where simplicity can be defined in terms of shape (number of local extremes,
intervals of convexity/concavity) or smoothness (bounds on derivatives) or a
combination of both. Once some form of regularization has been decided upon the
confidence region can be used to provide honest non-asymptotic confidence
bounds which are less informative but conceptually much simpler
Most Likely Transformations
We propose and study properties of maximum likelihood estimators in the class
of conditional transformation models. Based on a suitable explicit
parameterisation of the unconditional or conditional transformation function,
we establish a cascade of increasingly complex transformation models that can
be estimated, compared and analysed in the maximum likelihood framework. Models
for the unconditional or conditional distribution function of any univariate
response variable can be set-up and estimated in the same theoretical and
computational framework simply by choosing an appropriate transformation
function and parameterisation thereof. The ability to evaluate the distribution
function directly allows us to estimate models based on the exact likelihood,
especially in the presence of random censoring or truncation. For discrete and
continuous responses, we establish the asymptotic normality of the proposed
estimators. A reference software implementation of maximum likelihood-based
estimation for conditional transformation models allowing the same flexibility
as the theory developed here was employed to illustrate the wide range of
possible applications.Comment: Accepted for publication by the Scandinavian Journal of Statistics
2017-06-1
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