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

Cuevas, Antonio; Fraiman, Ricardo; Rodríguez-Casal, Alberto
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/08/2007 Português
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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)

‣ Statistical inferences for functional data

Zhang, Jin-Ting; Chen, Jianwei
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/08/2007 Português
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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...

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

Durot, Cécile
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/08/2007 Português
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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)

‣ Fast learning rates for plug-in classifiers

Audibert, Jean-Yves; Tsybakov, Alexandre B.
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)

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

Tasche, Dirk
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

‣ Variable selection through CART

Sauvé, Marie; Tuleau-Malot, Christine
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

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

Cule, Madeleine; Samworth, Richard
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

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

Omelka, Marek; Gijbels, Irène; Veraverbeke, Noël
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 31/08/2009 Português
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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)

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

Chesneau, Christophe
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)

‣ Breakdown and groups

Davies, P. Laurie; Gather, Ursula
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]

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

Bull, Adam D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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.

‣ Honest adaptive confidence bands and self-similar functions

Bull, Adam D.
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.

‣ Upper bounds for spatial point process approximations

Schuhmacher, Dominic
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)

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

Xing, Yang
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.

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

Giné, Evarist; Sang, Hailin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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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

‣ Smooth tail index estimation

Müller, Samuel; Rufibach, Kaspar
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

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

Viallon, Vivian
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)

‣ Adapting to Unknown Smoothness by Aggregation of Thresholded Wavelet Estimators

Chesneau, Christophe; Lecué, Guillaume
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.

‣ Deconvolution with unknown error distribution

Johannes, Jan
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)

‣ Asymptotically minimax Bayes predictive densities

Aslan, Mihaela
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)