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## ‣ A nonparametric approach to the estimation of lengths and surface areas

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2007
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The Minkowski content $L_0(G)$ of a body $G\subset{\mathbb{R}}^d$ represents
the boundary length (for $d=2$) or the surface area (for $d=3$) of $G$. A
method for estimating $L_0(G)$ is proposed. It relies on a nonparametric
estimator based on the information provided by a random sample (taken on a
rectangle containing $G$) in which we are able to identify whether every point
is inside or outside $G$. Some theoretical properties concerning strong
consistency, $L_1$-error and convergence rates are obtained. A practical
application to a problem of image analysis in cardiology is discussed in some
detail. A brief simulation study is provided.; Comment: Published at http://dx.doi.org/10.1214/009053606000001532 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## ‣ Statistical inferences for functional data

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2007
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With modern technology development, functional data are being observed
frequently in many scientific fields. A popular method for analyzing such
functional data is ``smoothing first, then estimation.'' That is, statistical
inference such as estimation and hypothesis testing about functional data is
conducted based on the substitution of the underlying individual functions by
their reconstructions obtained by one smoothing technique or another. However,
little is known about this substitution effect on functional data analysis. In
this paper this problem is investigated when the local polynomial kernel (LPK)
smoothing technique is used for individual function reconstructions. We find
that under some mild conditions, the substitution effect can be ignored
asymptotically. Based on this, we construct LPK reconstruction-based estimators
for the mean, covariance and noise variance functions of a functional data set
and derive their asymptotics. We also propose a GCV rule for selecting good
bandwidths for the LPK reconstructions. When the mean function also depends on
some time-independent covariates, we consider a functional linear model where
the mean function is linearly related to the covariates but the covariate
effects are functions of time. The LPK reconstruction-based estimators for the
covariate effects and the covariance function are also constructed and their
asymptotics are derived. Moreover...

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## ‣ On the $\mathbb{L}_p$-error of monotonicity constrained estimators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2007
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We aim at estimating a function $\lambda:[0,1]\to \mathbb {R}$, subject to
the constraint that it is decreasing (or increasing). We provide a unified
approach for studying the $\mathbb {L}_p$-loss of an estimator defined as the
slope of a concave (or convex) approximation of an estimator of a primitive of
$\lambda$, based on $n$ observations. Our main task is to prove that the
$\mathbb {L}_p$-loss is asymptotically Gaussian with explicit (though unknown)
asymptotic mean and variance. We also prove that the local $\mathbb {L}_p$-risk
at a fixed point and the global $\mathbb {L}_p$-risk are of order $n^{-p/3}$.
Applying the results to the density and regression models, we recover and
generalize known results about Grenander and Brunk estimators. Also, we obtain
new results for the Huang--Wellner estimator of a monotone failure rate in the
random censorship model, and for an estimator of the monotone intensity
function of an inhomogeneous Poisson process.; Comment: Published at http://dx.doi.org/10.1214/009053606000001497 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## ‣ Fast learning rates for plug-in classifiers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/08/2007
Português

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It has been recently shown that, under the margin (or low noise) assumption,
there exist classifiers attaining fast rates of convergence of the excess Bayes
risk, that is, rates faster than $n^{-1/2}$. The work on this subject has
suggested the following two conjectures: (i) the best achievable fast rate is
of the order $n^{-1}$, and (ii) the plug-in classifiers generally converge more
slowly than the classifiers based on empirical risk minimization. We show that
both conjectures are not correct. In particular, we construct plug-in
classifiers that can achieve not only fast, but also super-fast rates, that is,
rates faster than $n^{-1}$. We establish minimax lower bounds showing that the
obtained rates cannot be improved.; Comment: Published at http://dx.doi.org/10.1214/009053606000001217 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## ‣ Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Despite the fact that the Euler allocation principle has been adopted by many
financial institutions for their internal capital allocation process, a
comprehensive description of Euler allocation seems still to be missing. We try
to fill this gap by presenting the theoretical background as well as practical
aspects. In particular, we discuss how Euler risk contributions can be
estimated for some important risk measures. We furthermore investigate the
analysis of CDO tranche expected losses by means of Euler's theorem and suggest
an approach to measure the impact of risk factors on non-linear portfolios.; Comment: 21 pages, 4 figures

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## ‣ Variable selection through CART

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/01/2011
Português

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This paper deals with variable selection in the regression and binary
classification frameworks. It proposes an automatic and exhaustive procedure
which relies on the use of the CART algorithm and on model selection via
penalization. This work, of theoretical nature, aims at determining adequate
penalties, i.e. penalties which allow to get oracle type inequalities
justifying the performance of the proposed procedure. Since the exhaustive
procedure can not be executed when the number of variables is too big, a more
practical procedure is also proposed and still theoretically validated. A
simulation study completes the theoretical results.; Comment: 33 pages

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## ‣ Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/08/2009
Português

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We present theoretical properties of the log-concave maximum likelihood
estimator of a density based on an independent and identically distributed
sample in $\mathbb{R}^d$. Our study covers both the case where the true
underlying density is log-concave, and where this model is misspecified. We
begin by showing that for a sequence of log-concave densities, convergence in
distribution implies much stronger types of convergence -- in particular, it
implies convergence in Hellinger distance and even in certain exponentially
weighted total variation norms. In our main result, we prove the existence and
uniqueness of a log-concave density that minimises the Kullback--Leibler
divergence from the true density over the class all log-concave densities, and
also show that the log-concave maximum likelihood estimator converges almost
surely in these exponentially weighted total variation norms to this minimiser.
In the case of a correctly specified model, this demonstrates a strong type of
consistency for the estimator; in a misspecified model, it shows that the
estimator converges to the log-concave density that is closest in the
Kullback--Leibler sense to the true density.; Comment: 20 pages, 0 figures

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## ‣ Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2009
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We reconsider the existing kernel estimators for a copula function, as
proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990)
445--464], Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860]
and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of these
estimators have as a drawback that they can suffer from a corner bias problem.
A way to deal with this is to impose rather stringent conditions on the copula,
outruling as such many classical families of copulas. In this paper, we propose
improved estimators that take care of the typical corner bias problem. For
Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464] and
Chen and Huang [Canad. J. Statist. 35 (2007) 265--282], the improvement
involves shrinking the bandwidth with an appropriate functional factor; for
Fermanian, Radulovi\v{c} and Wegkamp [Bernoulli 10 (2004) 847--860], this is
done by using a transformation. The theoretical contribution of the paper is a
weak convergence result for the three improved estimators under conditions that
are met for most copula families. We also discuss the choice of bandwidth
parameters, theoretically and practically, and illustrate the finite-sample
behaviour of the estimators in a simulation study. The improved estimators are
applied to goodness-of-fit testing for copulas.; Comment: Published in at http://dx.doi.org/10.1214/08-AOS666 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## ‣ Wavelet block thresholding for samples with random design: a minimax approach under the $L^p$ risk

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/08/2007
Português

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We consider the regression model with (known) random design. We investigate
the minimax performances of an adaptive wavelet block thresholding estimator
under the $\mathbb{L}^p$ risk with $p\ge 2$ over Besov balls. We prove that it
is near optimal and that it achieves better rates of convergence than the
conventional term-by-term estimators (hard, soft,...).; Comment: Published at http://dx.doi.org/10.1214/07-EJS067 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## ‣ Breakdown and groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/08/2005
Português

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The concept of breakdown point was introduced by Hampel [Ph.D. dissertation
(1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896]
and developed further by, among others, Huber [Robust Statistics (1981). Wiley,
New York] and Donoho and Huber [In A Festschrift for Erich L. Lehmann (1983)
157-184. Wadsworth, Belmont, CA]. It has proved most successful in the context
of location, scale and regression problems. Attempts to extend the concept to
other situations have not met with general acceptance. In this paper we argue
that this is connected to the fact that in the location, scale and regression
problems the translation and affine groups give rise to a definition of
equivariance for statistical functionals. Comparisons in terms of breakdown
points seem only useful when restricted to equivariant functionals and even
here the connection between breakdown and equivariance is a tenuous one.; Comment: This paper discussed in: [math.ST/0508499], [math.ST/0508500],
[math.ST/0508501], [math.ST/0508502], [math.ST/0508503], [math.ST/0508504].
Rejoinder in [math.ST/0508505]

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## ‣ A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Statistics Theory#Mathematics - Numerical Analysis#62G20 (Primary), 62G07, 62G08, 62G15, 65T60 (Secondary)

In nonparametric statistical problems, we wish to find an estimator of an
unknown function f. We can split its error into bias and variance terms;
Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel
estimate, the supremum norm of the variance term is asymptotically distributed
as a Gumbel random variable. In the following, we prove a version of this
result for estimators using compactly-supported wavelets, a popular tool in
nonparametric statistics. Our result relies on an assumption on the nature of
the wavelet, which must be verified by provably-good numerical approximations.
We verify our assumption for Daubechies wavelets and symlets, with N = 6, ...,
20 vanishing moments; larger values of N, and other wavelet bases, are easily
checked, and we conjecture that our assumption holds also in those cases.

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## ‣ Honest adaptive confidence bands and self-similar functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Confidence bands are confidence sets for an unknown function f, containing
all functions within some sup-norm distance of an estimator. In the density
estimation, regression, and white noise models, we consider the problem of
constructing adaptive confidence bands, whose width contracts at an optimal
rate over a range of H\"older classes.
While adaptive estimators exist, in general adaptive confidence bands do not,
and to proceed we must place further conditions on f. We discuss previous
approaches to this issue, and show it is necessary to restrict f to
fundamentally smaller classes of functions.
We then consider the self-similar functions, whose H\"older norm is similar
at large and small scales. We show that such functions may be considered
typical functions of a given H\"older class, and that the assumption of
self-similarity is both necessary and sufficient for the construction of
adaptive bands. Finally, we show that this assumption allows us to resolve the
problem of undersmoothing, creating bands which are honest simultaneously for
functions of any H\"older norm.

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## ‣ Upper bounds for spatial point process approximations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/03/2005
Português

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We consider the behavior of spatial point processes when subjected to a class
of linear transformations indexed by a variable T. It was shown in Ellis [Adv.
in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the
transformed processes behave approximately like Poisson processes for large T.
In this article, under very similar assumptions, explicit upper bounds are
given for the d_2-distance between the corresponding point process
distributions. A number of related results, and applications to kernel density
estimation and long range dependence testing are also presented. The main
results are proved by applying a generalized Stein-Chen method to discretized
versions of the point processes.; Comment: Published at http://dx.doi.org/10.1214/105051604000000684 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org)

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## ‣ Convergence rates of posterior distributions for observations without the iid structure

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/11/2008
Português

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The classical condition on the existence of uniformly exponentially
consistent tests for testing the true density against the complement of its
arbitrary neighborhood has been widely adopted in study of asymptotics of
Bayesian nonparametric procedures. Because we follow a Bayesian approach, it
seems to be more natural to explore alternative and appropriate conditions
which incorporate the prior distribution. In this paper we supply a new
prior-dependent integration condition to establish general posterior
convergence rate theorems for observations which may not be independent and
identically distributed. The posterior convergence rates for such observations
have recently studied by Ghosal and van der Vaart \cite{ghv1}. We moreover
adopt the Hausdorff $\alpha$-entropy given by Xing and Ranneby
\cite{xir1}\cite{xi1}, which is also prior-dependent and smaller than the
widely used metric entropies. These lead to extensions of several existing
theorems. In particular, we establish a posterior convergence rate theorem for
general Markov processes and as its application we improve on the currently
known posterior rate of convergence for a nonlinear autoregressive model.

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## ‣ On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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It is shown that the variable bandwidth density estimator proposed by McKay
(1993a and b) following earlier findings by Abramson (1982) approximates
density functions in $C^4(\mathbb R^d)$ at the minimax rate in the supremum
norm over bounded sets where the preliminary density estimates on which they
are based are bounded away from zero. A somewhat more complicated estimator
proposed by Jones McKay and Hu (1994) to approximate densities in $C^6(\mathbb
R)$ is also shown to attain minimax rates in sup norm over the same kind of
sets. These estimators are strict probability densities.; Comment: 29 pages

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## ‣ Smooth tail index estimation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Both parametric distribution functions appearing in extreme value theory -
the generalized extreme value distribution and the generalized Pareto
distribution - have log-concave densities if the extreme value index gamma is
in [-1,0]. Replacing the order statistics in tail index estimators by their
corresponding quantiles from the distribution function that is based on the
estimated log-concave density leads to novel smooth quantile and tail index
estimators. These new estimators aim at estimating the tail index especially in
small samples. Acting as a smoother of the empirical distribution function, the
log-concave distribution function estimator reduces estimation variability to a
much greater extent than it introduces bias. As a consequence, Monte Carlo
simulations demonstrate that the smoothed version of the estimators are well
superior to their non-smoothed counterparts, in terms of mean squared error.; Comment: 17 pages, 5 figures. Slightly changed Pickand's estimator, added some
more introduction and discussion

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## ‣ Functional limit laws for the increments of the quantile process; with applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We establish a functional limit law of the logarithm for the increments of
the normed quantile process based upon a random sample of size $n\to\infty$. We
extend a limit law obtained by Deheuvels and Mason (12), showing that their
results hold uniformly over the bandwidth $h$, restricted to vary in
$[h'_n,h''_n]$, where $\{h'_n\}_{n\geq1}$ and $\{h''_n\}_{n\geq 1}$ are
appropriate non-random sequences. We treat the case where the sample
observations follow possibly non-uniform distributions. As a consequence of our
theorems, we provide uniform limit laws for nearest-neighbor density
estimators, in the spirit of those given by Deheuvels and Mason (13) for
kernel-type estimators.; Comment: Published in at http://dx.doi.org/10.1214/07-EJS099 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## ‣ Adapting to Unknown Smoothness by Aggregation of Thresholded Wavelet Estimators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/12/2006
Português

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We study the performances of an adaptive procedure based on a convex
combination, with data-driven weights, of term-by-term thresholded wavelet
estimators. For the bounded regression model, with random uniform design, and
the nonparametric density model, we show that the resulting estimator is
optimal in the minimax sense over all Besov balls under the $L^2$ risk, without
any logarithm factor.

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## ‣ Deconvolution with unknown error distribution

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We consider the problem of estimating a density $f_X$ using a sample
$Y_1,...,Y_n$ from $f_Y=f_X\star f_{\epsilon}$, where $f_{\epsilon}$ is an
unknown density. We assume that an additional sample
$\epsilon_1,...,\epsilon_m$ from $f_{\epsilon}$ is observed. Estimators of
$f_X$ and its derivatives are constructed by using nonparametric estimators of
$f_Y$ and $f_{\epsilon}$ and by applying a spectral cut-off in the Fourier
domain. We derive the rate of convergence of the estimators in case of a known
and unknown error density $f_{\epsilon}$, where it is assumed that $f_X$
satisfies a polynomial, logarithmic or general source condition. It is shown
that the proposed estimators are asymptotically optimal in a minimax sense in
the models with known or unknown error density, if the density $f_X$ belongs to
a Sobolev space $H_{\mathbh p}$ and $f_{\epsilon}$ is ordinary smooth or
supersmooth.; Comment: Published in at http://dx.doi.org/10.1214/08-AOS652 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## ‣ Asymptotically minimax Bayes predictive densities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/08/2007
Português

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Given a random sample from a distribution with density function that depends
on an unknown parameter $\theta$, we are interested in accurately estimating
the true parametric density function at a future observation from the same
distribution. The asymptotic risk of Bayes predictive density estimates with
Kullback--Leibler loss function $D(f_{\theta}||{\hat{f}})=\int{f_{\theta}
\log{(f_{\theta}/ hat{f})}}$ is used to examine various ways of choosing prior
distributions; the principal type of choice studied is minimax. We seek
asymptotically least favorable predictive densities for which the corresponding
asymptotic risk is minimax. A result resembling Stein's paradox for estimating
normal means by the maximum likelihood holds for the uniform prior in the
multivariate location family case: when the dimensionality of the model is at
least three, the Jeffreys prior is minimax, though inadmissible. The Jeffreys
prior is both admissible and minimax for one- and two-dimensional location
problems.; Comment: Published at http://dx.doi.org/10.1214/009053606000000885 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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