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## "Regressão beta"; Beta regression

Fonte: Biblioteca Digitais de Teses e Dissertações da USP
Publicador: Biblioteca Digitais de Teses e Dissertações da USP

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 29/03/2007
Português

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26.37%

Muitos estudos em diferentes áreas examinam como um conjunto de variáveis influencia algum tipo de percentagem, proporção ou frações. Modelos de regressão lineares não são satisfatórios para modelar tais dados. Uma classe de modelos de regressão beta que em muitos aspectos é semelhante aos modelos lineares generalizados foi proposto por Ferrari e Cribari--Neto~(2004). A resposta média é relacionada com um predictor linear por uma função de ligação e o predictor linear envolve covariáveis e parâmetros de regressão desconhecidos. O modelo também é indexado por um parâmetro de precisão. Smithson e Verkuilen,(2005), entre outros, consideram o modelo de regressão beta em que esse parâmetro varia ao longo das observações. Nesta tese foram desenvolvidas técnicas de diagnóstico para os modelos regressão beta com dispersão constante e com dispersão variável, sendo que o método e influência local (Cook,~1986) mostrou-se decisivo, inclusive no sentido de identificar dispersão variável nos dados. Adicionalmente, avaliamos através de estudos de simulação o desempenho de estimadores de máxima verossimilhança para o modelo de regressão beta com dispersão variável, as conseqüências de estimar o modelo supondo dispersão constante quando de fato ela é variável e de testes assintóticos para testar a hipótese de dispersão constante. Finalmente...

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## Escore clínico-patológico para predizer o risco de metástases e recorrência local em pacientes com carcinoma cortical adrenal e papel do algoritmo da reticulina na distinção entre adenomas e carcinomas corticais adrenais; Clinicopathological score for predicting the risk of metastases and local recurrence in patients with adrenal cortical carcinoma and role of the reticulin algorithm in distinguishing between adrenal cortical adenomas and carcinomas

Fonte: Biblioteca Digitais de Teses e Dissertações da USP
Publicador: Biblioteca Digitais de Teses e Dissertações da USP

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 05/05/2014
Português

Relevância na Pesquisa

26.25%

#Adenoma adrenocortical/diagnóstico#Adenoma adrenocortical/patologia#Adrenal cortex neoplasms/diagnosis#Adrenal cortex neoplasms/pathology#Adrenocortical adenoma/diagnosis#Adrenocortical adenoma/pathology#Adrenocortical carcinoma/diagnosis#Adrenocortical carcinoma/pathology#Algorithms#Algoritmos#Carcinoma adrenocortical/diagnóstico

INTRODUÇÃO: o padrão-ouro para o diagnóstico histológico dos tumores corticais adrenais (TCAs) e sua diferenciação entre adenomas e carcinomas é o sistema de Weiss, cuja aplicação é limitada pela baixa reprodutibilidade de alguns dos critérios que o compõe. Recentemente foi proposto e validado um algoritmo diagnóstico para os TCAs baseado na integridade do arcabouço de reticulina e da membrana basal. Os carcinomas adrenais são tumores raros e apresentam prognóstico reservado, mesmo nos pacientes com doença aparentemente localizada. Além do estadiamento e da extensão da ressecção cirúrgica, outros dados foram reportados na literatura como tendo importância prognóstica, tais como idade ao diagnóstico, padrão funcional, tamanho tumoral, extensão local do tumor primário e alguns achados histológicos e imuno-histoquímicos, com destaque à taxa mitótica e ao índice de Ki-67. O sistema de Weiss, embora permita o diagnóstico diferencial entre adenomas e carcinomas, não foi testado completamente como uma ferramenta para distinguir os carcinomas com boa evolução clínica daqueles com desfecho desfavorável. OBJETIVOS: o presente estudo teve como objetivo primário construir um nomograma para estimar o risco de metástases e recorrência local em portadores de carcinoma adrenal...

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## Bootstrapping high frequency data

Fonte: Université de Montréal
Publicador: Université de Montréal

Tipo: Thèse ou Mémoire numérique / Electronic Thesis or Dissertation

Português

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26.66%

#Données haute fréquence#Volatilité réalisée#Bruit de microstructure#Pré-moyennement#Betas réalisé#Bootstrap#Expansions d’Edgeworth#Realized betas#High frequency data#Realized volatility#Market microstructure noise

Nous développons dans cette thèse, des méthodes de bootstrap pour les données financières de hautes fréquences. Les deux premiers essais focalisent sur les méthodes de bootstrap appliquées à l’approche de "pré-moyennement" et robustes à la présence d’erreurs de microstructure. Le "pré-moyennement" permet de réduire l’influence de l’effet de microstructure avant d’appliquer la volatilité réalisée. En se basant sur cette ap- proche d’estimation de la volatilité intégrée en présence d’erreurs de microstructure, nous développons plusieurs méthodes de bootstrap qui préservent la structure de dépendance et l’hétérogénéité dans la moyenne des données originelles. Le troisième essai développe une méthode de bootstrap sous l’hypothèse de Gaussianité locale des données financières de hautes fréquences.
Le premier chapitre est intitulé: "Bootstrap inference for pre-averaged realized volatility based on non-overlapping returns". Nous proposons dans ce chapitre, des méthodes de bootstrap robustes à la présence d’erreurs de microstructure. Particulièrement nous nous sommes focalisés sur la volatilité réalisée utilisant des rendements "pré-moyennés" proposés par Podolskij et Vetter (2009)...

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## Nonparametric Testing Methods for Treatment-Biomarker Interaction based on Local Partial-Likelihood

Fonte: Quens University
Publicador: Quens University

Tipo: Relatório

Português

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36.3%

A fair amount of research has been done on the interactions between treatment and
biomarkers hoping to avoid failure to recognize effective agents which benefit only a subset of patients in traditional clinical designs and analysis, such as (Bonetti, 2004), (Bonetti et al., 2009), and (Royston and Sauerbrei, 2004). Particularly, Fan et al. (Fan et al., 2006) assumed the treatment effect is an unknown function of a putative biomarker, and proposed techniques to give the local partial likelihood estimation (LPLE) of this treatment effect function using local linear techniques (Fan and Chen, 1999). However, no methods were developed for assessing whether the treatment
effect is indeed a function of the biomarker (interaction exists) or just a constant (no
interactions).
Based on the idea of LPLE, a new nonparametric hypothesis testing methodology,
which we call local partial likelihood bootstrap (LPLB) test, is proposed in this work to identify the differences in treatment effects among subgroups of patients with different values of biomarkers in a Phase III clinical trials study. A bootstrap technique is used to evaluate the significance of the test. Meanwhile, the proposed method can also be applied to identify the interactions between a putative biomarker and a collection of covariates (covariate vectors) that are discrete or continuous. Numerical studies show that the LPLB test can provide a substantial improvement in the power of the interaction detection compared with the commonly used method...

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## Assessing extrema of empirical principal component functions

Fonte: Institute of Mathematical Statistics
Publicador: Institute of Mathematical Statistics

Tipo: Artigo de Revista Científica

Português

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#bootstrap#bootstrap likelihood#functional data analysis#mode#perturbation#principal components analysis#shoulder point#subsample#turning point

The difficulties of estimating and representing the distributions of
functional data mean that principal component methods play a substantially
greater role in functional data analysis than in more conventional
finite-dimensional settings. Local maxima and minima in principal component
functions are of direct importance; they indicate places in the domain of a
random function where influence on the function value tends to be relatively
strong but of opposite sign. We explore statistical properties of the
relationship between extrema of empirical principal component functions, and
their counterparts for the true principal component functions. It is shown that
empirical principal component funcions have relatively little trouble capturing
conventional extrema, but can experience difficulty distinguishing a
``shoulder' in a curve from a small bump. For example, when the true principal
component function has a shoulder, the probability that the empirical principal
component function has instead a bump is approximately equal to 1/2. We suggest
and describe the performance of bootstrap methods for assessing the strength of
extrema. It is shown that the subsample bootstrap is more effective than the
standard bootstrap in this regard. A ``bootstrap likelihood' is proposed for
measuring extremum strength. Exploratory numerical methods are suggested.

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## Bootstrap prediction intervals in State Space models

Fonte: Universidade Carlos III de Madrid
Publicador: Universidade Carlos III de Madrid

Tipo: Trabalho em Andamento
Formato: application/pdf

Publicado em /03/2008
Português

Relevância na Pesquisa

36.34%

Prediction intervals in State Space models can be obtained by assuming Gaussian innovations and
using the prediction equations of the Kalman filter, where the true parameters are substituted by
consistent estimates. This approach has two limitations. First, it does not incorporate the
uncertainty due to parameter estimation. Second, the Gaussianity assumption of future innovations
may be inaccurate. To overcome these drawbacks, Wall and Stoffer (2002) propose to obtain
prediction intervals by using a bootstrap procedure that requires the backward representation of
the model. Obtaining this representation increases the complexity of the procedure and limits its
implementation to models for which it exists. The bootstrap procedure proposed by Wall and
Stoffer (2002) is further complicated by fact that the intervals are obtained for the prediction errors
instead of for the observations. In this paper, we propose a bootstrap procedure for constructing
prediction intervals in State Space models that does not need the backward representation of the
model and is based on obtaining the intervals directly for the observations. Therefore, its
application is much simpler, without loosing the good behavior of bootstrap prediction intervals.
We study its finite sample properties and compare them with those of the standard and the Wall
and Stoffer (2002) procedures for the Local Level Model. Finally...

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## Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters

Fonte: Universidade Carlos III de Madrid
Publicador: Universidade Carlos III de Madrid

Tipo: Trabalho em Andamento
Formato: application/pdf

Publicado em /01/2010
Português

Relevância na Pesquisa

26.24%

Prediction intervals in State Space models can be obtained by assuming Gaussian innovations
and using the prediction equations of the Kalman filter, where the true parameters are
substituted by consistent estimates. This approach has two limitations. First, it does not
incorporate the uncertainty due to parameter estimation. Second, the Gaussianity assumption of
future innovations may be inaccurate. To overcome these drawbacks, Wall and Stoffer (2002)
propose to obtain prediction intervals by using a bootstrap procedure that requires the backward
representation of the model. Obtaining this representation increases the complexity of the
procedure and limits its implementation to models for which it exists. The bootstrap procedure
proposed by Wall and Stoffer (2002) is further complicated by fact that the intervals are
obtained for the prediction errors instead of for the observations. In this paper, we propose a
bootstrap procedure for constructing prediction intervals in State Space models that does not
need the backward representation of the model and is based on obtaining the intervals directly
for the observations. Therefore, its application is much simpler, without loosing the good
behavior of bootstrap prediction intervals. We study its finite sample properties and compare
them with those of the standard and the Wall and Stoffer (2002) procedures for the Local Level
Model. Finally...

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## Nonparametric estimation and inference for Granger causality measures

Fonte: Universidade Carlos III de Madrid
Publicador: Universidade Carlos III de Madrid

Tipo: info:eu-repo/semantics/draft; info:eu-repo/semantics/workingPaper
Formato: application/pdf

Publicado em 29/03/2012
Português

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#Causality measures#Nonparametric estimation#Time series#Copulas#Bernstein copula density#Local bootstrap#Conditional distribution function#Stock returns#C12#C14#C15

We propose a nonparametric estimator and a nonparametric test for Granger causality measures that quantify linear and nonlinear Granger causality in distribution between random variables. We first show how to write the Granger causality measures in terms of copula densities. We suggest a consistent estimator for these causality measures based on nonparametric estimators of copula densities. Further, we prove that the nonparametric estimators are asymptotically normally distributed and we discuss the validity of a local smoothed bootstrap that we use in finite sample settings to compute a bootstrap bias-corrected estimator and test for our causality measures. A simulation study reveals that the bias-corrected bootstrap estimator of causality measures behaves well and the corresponding test has quite good finite sample size and power properties for a variety of typical data generating processes and different sample sizes. Finally, we illustrate the practical relevance of nonparametric causality measures by quantifying the Granger causality between S&P500 Index returns and many exchange rates (US/Canada, US/UK and US/Japen exchange rates).

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## Bootstrap prediction intervals in state space models

Fonte: Wiley-Blackwell
Publicador: Wiley-Blackwell

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article
Formato: application/pdf

Publicado em //2009
Português

Relevância na Pesquisa

36.29%

Prediction intervals in state space models can be obtained by assuming Gaussian innovations and using the prediction equations of the Kalman filter, with the true parameters substituted by consistent estimates. This approach has two limitations.
First, it does not incorporate the uncertainty caused by parameter estimation. Second, the Gaussianity of future innovations assumption may be inaccurate. To overcome these drawbacks, Wall and Stoffer [Journal of Time Series Analysis (2002) Vol. 23, pp. 733 751] propose a bootstrap procedure for evaluating conditional forecast errors that requires the backward representation of the model. Obtaining this representation increases the complexity of the procedure and limits its implementation to models for which it exists. In this article, we propose a bootstrap procedure for constructing prediction intervals directly for the observations, which does not need the backward representation of the model. Consequently, its application is much simpler, without losing the good behaviour of bootstrap prediction intervals. We study its finite sample properties and compare them with those of the standard and the Wall and Stoffer procedures for the local level model. Finally, we illustrate the results by implementing the new procedure to obtain prediction intervals for future values of a real time series.; Financial support from Project SEJ2006-03919 by the Spanish Government is
gratefully acknowledged

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## Local Bootstrap Percolation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/06/2008
Português

Relevância na Pesquisa

36.02%

We study a variant of bootstrap percolation in which growth is restricted to
a single active cluster. Initially there is a single active site at the origin,
while other sites of Z^2 are independently occupied with small probability p,
otherwise empty. Subsequently, an empty site becomes active by contact with 2
or more active neighbors, and an occupied site becomes active if it has an
active site within distance 2. We prove that the entire lattice becomes active
with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p
and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions
for the scaling of the correction term.; Comment: 19 pages, 2 figures

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## Form factors in finite volume I: form factor bootstrap and truncated conformal space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#High Energy Physics - Theory#Condensed Matter - Other Condensed Matter#High Energy Physics - Lattice

We describe the volume dependence of matrix elements of local fields to all
orders in inverse powers of the volume (i.e. only neglecting contributions that
decay exponentially with volume). Using the scaling Lee-Yang model and the
Ising model in a magnetic field as testing ground, we compare them to matrix
elements extracted in finite volume using truncated conformal space approach to
exact form factors obtained using the bootstrap method. We obtain solid
confirmation for the form factor bootstrap, which is different from all
previously available tests in that it is a non-perturbative and direct
comparison of exact form factors to multi-particle matrix elements of local
operators, computed from the Hamiltonian formulation of the quantum field
theory. We also demonstrate that combining form factor bootstrap and truncated
conformal space is an effective method for evaluating finite volume form
factors in integrable field theories over the whole range in volume.; Comment: 43 pages, 31 eps figures, LaTeX2e file. v2: main theoretical argument
substantially expanded and clarified, typos and references corrected

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## Local and global Fokker-Planck neoclassical calculations showing flow and bootstrap current modification in a pedestal

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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26.34%

In transport barriers, particularly H-mode edge pedestals, radial scale
lengths can become comparable to the ion orbit width, causing neoclassical
physics to become radially nonlocal. In this work, the resulting changes to
neoclassical flow and current are examined both analytically and numerically.
Steep density gradients are considered, with scale lengths comparable to the
poloidal ion gyroradius, together with strong radial electric fields sufficient
to electrostatically confine the ions. Attention is restricted to relatively
weak ion temperature gradients (but permitting arbitrary electron temperature
gradients), since in this limit a delta-f (small departures from a Maxwellian
distribution) rather than full-f approach is justified. This assumption is in
fact consistent with measured inter-ELM H-Mode edge pedestal density and ion
temperature profiles in many present experiments, and is expected to be
increasingly valid in future lower collisionality experiments. In the numerical
analysis, the distribution function and Rosenbluth potentials are solved for
simultaneously, allowing use of the exact field term in the linearized
Fokker-Planck collision operator. In the pedestal, the parallel and poloidal
flows are found to deviate strongly from the best available conventional
neoclassical prediction...

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## Bootstrap kernel for organic low dimensional systems; PPV, pentacene and picene

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/11/2013
Português

Relevância na Pesquisa

26.29%

We apply the bootstrap kernel within time dependent density functional theory
to study one-dimensional chain of organic polymer poly-phenylene-vinylene and
molecular crystals of picene and pentacene. The behaviour of this kernel in the
presence and absence of local field effects is studied. The absorption spectra
of poly-phenylene-vinylene has a bound excitonic peak which is well reproduced
by the bootstrap kernel. Pentacene and picene, electronically similar
materials, have remarkably different excitonic physics which is also captured
properly by the bootstrap kernel. Inclusion of local-field effects dramatically
change the spectra for both picene and pentacene. We highlight the reason
behind this change. This also sheds light on the reasons behind the discrepancy
in results between two different previous Bethe-Salpeter calculations.; Comment: 5 figs

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## Subcritical $\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

26.37%

We prove that there exist natural generalizations of the classical bootstrap
percolation model on $\mathbb{Z}^2$ that have non-trivial critical
probabilities, and moreover we characterize all homogeneous, local, monotone
models with this property.
Van Enter (in the case $d=r=2$) and Schonmann (for all $d \geq r \geq 2$)
proved that $r$-neighbour bootstrap percolation models have trivial critical
probabilities on $\mathbb{Z}^d$ for every choice of the parameters $d \geq r
\geq 2$: that is, an initial set of density $p$ almost surely percolates
$\mathbb{Z}^d$ for every $p>0$. These results effectively ended the study of
bootstrap percolation on infinite lattices.
Recently Bollob\'as, Smith and Uzzell introduced a broad class of percolation
models called $\mathcal{U}$-bootstrap percolation, which includes $r$-neighbour
bootstrap percolation as a special case. They divided two-dimensional
$\mathcal{U}$-bootstrap percolation models into three classes -- subcritical,
critical and supercritical -- and they proved that, like classical 2-neighbour
bootstrap percolation, critical and supercritical $\mathcal{U}$-bootstrap
percolation models have trivial critical probabilities on $\mathbb{Z}^2$. They
left open the question as to what happens in the case of subcritical families.
In this paper we answer that question: we show that every subcritical
$\mathcal{U}$-bootstrap percolation model has a non-trivial critical
probability on $\mathbb{Z}^2$. This is new except for a certain `degenerate'
subclass of symmetric models that can be coupled from below with oriented site
percolation. Our results re-open the study of critical probabilities in
bootstrap percolation on infinite lattices...

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## Bias Reduction of Long Memory Parameter Estimators via the Pre-filtered Sieve Bootstrap

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/02/2014
Português

Relevância na Pesquisa

26.3%

This paper investigates the use of bootstrap-based bias correction of
semi-parametric estimators of the long memory parameter in fractionally
integrated processes. The re-sampling method involves the application of the
sieve bootstrap to data pre-filtered by a preliminary semi-parametric estimate
of the long memory parameter. Theoretical justification for using the bootstrap
techniques to bias adjust log-periodogram and semi-parametric local Whittle
estimators of the memory parameter is provided. Simulation evidence comparing
the performance of the bootstrap bias correction with analytical bias
correction techniques is also presented. The bootstrap method is shown to
produce notable bias reductions, in particular when applied to an estimator for
which analytical adjustments have already been used. The empirical coverage of
confidence intervals based on the bias-adjusted estimators is very close to the
nominal, for a reasonably large sample size, more so than for the comparable
analytically adjusted estimators. The precision of inferences (as measured by
interval length) is also greater when the bootstrap is used to bias correct
rather than analytical adjustments.; Comment: 38 pages

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## Simultaneous likelihood-based bootstrap confidence sets for a large number of models

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/06/2015
Português

Relevância na Pesquisa

26.34%

The paper studies a problem of constructing simultaneous likelihood-based
confidence sets. We consider a simultaneous multiplier bootstrap procedure for
estimating the quantiles of the joint distribution of the likelihood ratio
statistics, and for adjusting the confidence level for multiplicity.
Theoretical results state the bootstrap validity in the following setting: the
sample size \(n\) is fixed, the maximal parameter dimension
\(p_{\textrm{max}}\) and the number of considered parametric models \(K\) are
s.t. \((\log K)^{12}p_{\max}^{3}/n\) is small. We also consider the situation
when the parametric models are misspecified. If the models' misspecification is
significant, then the bootstrap critical values exceed the true ones and the
simultaneous bootstrap confidence set becomes conservative. Numerical
experiments for local constant and local quadratic regressions illustrate the
theoretical results.

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## Biased bootstrap methods for reducing the effects of contamination

Fonte: Aiden Press
Publicador: Aiden Press

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

36.19%

#Keywords: Biased bootstrap#Empirical likelihood#Influence#Inlier#Local linear smoothing#Multivariate analysis#Nonparametric curve estimation#Outlier#Regression#Robust statistical methods#Trimming

Contamination of a sampled distribution, for example by a heavy-tailed distribution, can degrade the performance of a statistical estimator. We suggest a general approach to alleviating this problem, using a version of the weighted bootstrap. The idea is

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## Intentionally biased bootstrap methods

Fonte: Aiden Press
Publicador: Aiden Press

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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#Keywords: Bias reduction#Empirical likelihood#Hypothesis testing#Local linear smoothing#Nonparametric curve estimation#Variance stabilization#Weighted bootstrap

A class of weighted bootstrap techniques, called biased bootstrap or b-bootstrap methods, is introduced. It is motivated by the need to adjust empirical methods, such as the "uniform" bootstrap, in a surgical way to alter some of their features while leaving others unchanged. Depending on the nature of the adjustment, the b-bootstrap can be used to reduce bias, or to reduce variance or to render some characteristic equal to a predetermined quantity. Examples of the last application include a b-bootstrap approach to hypothesis testing in nonparametric contexts, where the b-bootstrap enables simulation "under the null hypothesis", even when the hypothesis is false, and a b-bootstrap competitor to Tibshirani's variance stabilization method. An example of the bias reduction application is adjustment of Nadaraya-Watson kernel estimators to make them competitive with local linear smoothing. Other applications include density estimation under constraints, outlier trimming, sensitivity analysis, skewness or kurtosis reduction and shrinkage.

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## Reducing bias in curve estimation by use of weights

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

Português

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26.27%

#Keywords: Bandwidth#Estimation#Regression analysis#Bootstrap#Kernel density estimation#Variable kernel methods#Data reduction#bootstrapping#error correction#estimation method Abramson's method#Bandwidth

A technique is suggested for reducing the order of bias of kernel estimators by weighting the contributions that different data values make to the estimator. The method is developed initially in the context of density estimation, where, unlike the 'variable kernel' method proposed by Abramson, our approach does not involve using different bandwidths at different data values. Rather, it is a weighted-bootstrap version of the standard uniform-bootstrap method that is used to construct traditional kernel density estimators. The reduction in bias is achieved by biasing the bootstrap appropriately, in a global rather than local way. Our technique has a variety of different forms, each of which reduces the order of bias from the square to the fourth power of bandwidth, but does not alter the order of variance. It has immediate application to nonparametric regression, where it allows bias to be reduced without prejudicing the one sign of an estimator.

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## Bandwidth choice for local polynomial estimation of smooth boundaries

Fonte: Academic Press
Publicador: Academic Press

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

36.04%

#Keywords: Bandwidth choice#Bootstrap#Global bandwidth#Local bandwidth#Local polynomial methods#Nonparametric curve estimation

Local polynomial methods hold considerable promise for boundary estimation, where they offer unmatched flexibility and adaptivity. Most rival techniques provide only a single order of approximation; local polynomial approaches allow any order desired. Their more conventional rivals, for example high-order kernel methods in the context of regression, do not have attractive versions in the case of boundary estimation. However, the adoption of local polynomial methods for boundary estimation is inhibited by lack of knowledge about their properties, in particular about the manner in which they are influenced by bandwidth; and by the absence of techniques for empirical bandwidth choice. In the present paper we detail the way in which bandwidth selection determines mean squared error of local polynomial boundary estimators, showing that it is substantially more complex than in regression settings. For example, asymptotic formulae for bias and variance contributions to mean squared error no longer decompose into monotone functions of bandwidth. Nevertheless, once these properties are understood, relatively simple empirical bandwidth selection methods can be developed. We suggest a new approach to both local and global bandwidth choice, and describe its properties.

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