4,219 research outputs found
Parametric versus nonparametric: the fitness coefficient
The fitness coefficient, introduced in this paper, results from a competition
between parametric and nonparametric density estimators within the likelihood
of the data. As illustrated on several real datasets, the fitness coefficient
generally agrees with p-values but is easier to compute and interpret. Namely,
the fitness coefficient can be interpreted as the proportion of data coming
from the parametric model. Moreover, the fitness coefficient can be used to
build a semiparamteric compromise which improves inference over the parametric
and nonparametric approaches. From a theoretical perspective, the fitness
coefficient is shown to converge in probability to one if the model is true and
to zero if the model is false. From a practical perspective, the utility of the
fitness coefficient is illustrated on real and simulated datasets
On the acceleration of some empirical means with application to nonparametric regression
Let be an i.i.d. sequence of random variables in ,
, for some function , under regularity conditions,
we show that \begin{align*} n^{1/2} \left(n^{-1} \sum_{i=1}^n
\frac{\varphi(X_i)}{\w f^{(i)}(X_i)}-\int_{} \varphi(x)dx \right)
\overset{\P}{\lr} 0, \end{align*} where \w f^{(i)} is the classical
leave-one-out kernel estimator of the density of . This result is striking
because it speeds up traditional rates, in root , derived from the central
limit theorem when \w f^{(i)}=f. As a consequence, it improves the classical
Monte Carlo procedure for integral approximation. The paper mainly addressed
with theoretical issues related to the later result (rates of convergence,
bandwidth choice, regularity of ) but also interests some statistical
applications dealing with random design regression. In particular, we provide
the asymptotic normality of the estimation of the linear functionals of a
regression function on which the only requirement is the H\"older regularity.
This leads us to a new version of the \textit{average derivative estimator}
introduced by H\"ardle and Stoker in \cite{hardle1989} which allows for
\textit{dimension reduction} by estimating the \textit{index space} of a
regression
Bootstrap Testing of the Rank of a Matrix via Least Squared Constrained Estimation
In order to test if an unknown matrix has a given rank (null hypothesis), we
consider the family of statistics that are minimum squared distances between an
estimator and the manifold of fixed-rank matrix. Under the null hypothesis,
every statistic of this family converges to a weighted chi-squared
distribution. In this paper, we introduce the constrained bootstrap to build
bootstrap estimate of the law under the null hypothesis of such statistics. As
a result, the constrained bootstrap is employed to estimate the quantile for
testing the rank. We provide the consistency of the procedure and the
simulations shed light one the accuracy of the constrained bootstrap with
respect to the traditional asymptotic comparison. More generally, the results
are extended to test if an unknown parameter belongs to a sub-manifold locally
smooth. Finally, the constrained bootstrap is easy to compute, it handles a
large family of tests and it works under mild assumptions
Integral approximation by kernel smoothing
Let be an i.i.d. sequence of random variables in
, . We show that, for any function , under regularity conditions, where
is the classical kernel estimator of the density of . This
result is striking because it speeds up traditional rates, in root , derived
from the central limit theorem when . Although this paper
highlights some applications, we mainly address theoretical issues related to
the later result. We derive upper bounds for the rate of convergence in
probability. These bounds depend on the regularity of the functions
and , the dimension and the bandwidth of the kernel estimator
. Moreover, they are shown to be accurate since they are used as
renormalizing sequences in two central limit theorems each reflecting different
degrees of smoothness of . As an application to regression modelling
with random design, we provide the asymptotic normality of the estimation of
the linear functionals of a regression function. As a consequence of the above
result, the asymptotic variance does not depend on the regression function.
Finally, we debate the choice of the bandwidth for integral approximation and
we highlight the good behavior of our procedure through simulations.Comment: Published at http://dx.doi.org/10.3150/15-BEJ725 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin
note: text overlap with arXiv:1312.449
Stock Prices, Total Factor Productivity and Economic Fluctuations; Some Further Evidence from Japanese and U.S. Sectoral Data
Stock Prices - Total Factor Productivity
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