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## Estimação de cópulas via ondaletas; Copula estimation through wavelets

Silva, Francyelle de Lima e
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
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Cópulas tem se tornado uma importante ferramenta para descrever e analisar a estrutura de dependência entre variáveis aleatórias e processos estocásticos. Recentemente, surgiram alguns métodos de estimação não paramétricos, utilizando kernels e ondaletas. Neste contexto, sabendo que cópulas podem ser escritas como expansão em ondaletas, foi proposto um estimador não paramétrico via ondaletas para a função cópula para dados independentes e de séries temporais, considerando processos alfa-mixing. Este estimador tem como característica principal estimar diretamente a função cópula, sem fazer suposição alguma sobre a distribuição dos dados e sem ajustes prévios de modelos ARMA - GARCH, como é feito em ajuste paramétrico para cópulas. Foram calculadas taxas de convergência para o estimador proposto em ambos os casos, mostrando sua consistência. Foram feitos também alguns estudos de simulação, além de aplicações a dados reais.; Copulas are important tools for describing the dependence structure between random variables and stochastic processes. Recently some nonparametric estimation procedures have appeared, using kernels and wavelets. In this context, knowing that a copula function can be expanded in a wavelet basis...

## Nonparametric estimation of the mean function for recurrent event data with missing event category

LIN, FENG-CHANG; CAI, JIANWEN; FINE, JASON P.; LAI, HUICHUAN J.
Tipo: Artigo de Revista Científica
Português
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Recurrent event data frequently arise in longitudinal studies when study subjects possibly experience more than one event during the observation period. Often, such recurrent events can be categorized. However, part of the categorization may be missing due to technical difficulties. If the event types are missing completely at random, then a complete case analysis may provide consistent estimates of regression parameters in certain regression models, but estimates of the baseline event rates are generally biased. Previous work on nonparametric estimation of these rates has utilized parametric missingness models. In this paper, we develop fully nonparametric methods in which the missingness mechanism is completely unspecified. Consistency and asymptotic normality of the nonparametric estimators of the mean event functions accommodate nonparametric estimators of the event category probabilities, which converge more slowly than the parametric rate. Plug-in variance estimators are provided and perform well in simulation studies, where complete case estimators may exhibit large biases and parametric estimators generally have a larger mean squared error when the model is misspecified. The proposed methods are applied to data from a cystic fibrosis registry.

## Do Markov-switching models capture nonlinearities in the data? tests using nonparametric methods

Tipo: Artigo de Revista Científica Formato: 364772 bytes; 363 bytes; application/pdf; application/octet-stream
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Markov-switching models have become popular alternatives to linear autoregressive models. Many papers which estimate nonlinear models make little attempt to demonstrate whether the non-linearities they capture are of interest or if the models differ substantially from the linear option. By simulating the models and nonparametrically estimating functions of the simulated data, we can evaluate if and how the nonlinear and linear models differ.; [1] Ait-Sahalia, Y., Testing Continuous-Time Models of the Spot Interest Rate, Review of Financial Studies 9 (1996) 385-426. [2] Bodman, P., Asymmetry and Duration Dependence in Australian GDP and Unemployment, Economic Record 74 (1998) 399-411. [3] Breunig, R. and A.R. Pagan, Some Simple Methods for Assessing Markov Switching Models, mimeo, Australian National University (available at http://econrsss.anu.edu.au/~breunig/), 2001. [4] Clements, M.P. and A.B.C. Galvao, Conditional Mean Functions of Non-Linear Models of US Output, mimeo, University of Warwick, 2000. [5] Garcia, R., Asymptotic Null Distribution of the Likelihood Ratio Test in Markov Switching Models, International Economic Review 39(3) (1998) 763-788. [6] Goodwin. T.H., Business-Cycle Analysis with a Markov-Switching Model...

## Moment inequalities for spatial processes

Gao, J.; Lu, Z.; Tjostheim, D.
Fonte: Elsevier Science BV Publicador: Elsevier Science BV
Tipo: Artigo de Revista Científica
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This paper establishes a general moment inequality for spatial processes satisfying the α-mixing condition [cf., Tran, 1990. Kernel density estimation on random fields. J. Multivariate Analy. 34, 37–53]. Such a general moment inequality is a nontrivial extension of the corresponding result established in Cox and Kim [1995. Moment bounds for mixing random variables useful in nonparametric function estimation. Stochastic Process. Appl. 56, 151–158] for the time series case. As is the case for the Cox–Kim inequality for nonparametric estimation of time series, the new inequality is useful in nonparametric kernel estimation of spatial processes.; Jiti Gao, Zudi Lu and Dag Tjøstheim

## Nonparametric forecasting in time series: a comparative study

Vilar Fernández, Juan Manuel; Cao Abad, Ricardo
Fonte: Taylor & Francis Publicador: Taylor & Francis
Tipo: Artigo de Revista Científica
Português
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The problem of predicting a future value of a time series is considered in this paper. If the series follows a stationary Markov process, this can be done by nonparametric estimation of the autoregression function. Two forecasting algorithms are introduced. They only differ in the nonparametric kernel-type estimator used: the Nadaraya-Watson estimator and the local linear estimator. There are three major issues in the implementation of these algorithms: selection of the autoregressor variables; smoothing parameter selection and computing prediction intervals. These have been tackled using recent techniques borrowed from the nonparametric regression estimation literature under dependence. The performance of these nonparametric algorithms has been studied by applying them to a collection of 43 well-known time series. Their results have been compared to those obtained using classical Box-Jenkins methods. Finally, the practical behaviour of the methods is also illustrated by a detailed analysis of two data sets.

## Nonparametric estimation of a polarization measure

Anderson, Gordon; Oliver, Linton; Whang, Yoon-Jae
Tipo: Trabalho em Andamento Formato: application/pdf
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This paper develops methodology for nonparametric estimation of a polarization measure due to Anderson (2004) and Anderson, Ge, and Leo (2006) based on kernel estimation techniques. We give the asymptotic distribution theory of our estimator, which in some cases is nonstandard due to a boundary value problem. We also propose a method for conducting inference based on estimation of unknown quantities in the limiting distribution and show that our method yields consistent inference in all cases we consider. We investigate the finite sample properties of our methods by simulation methods. We give an application to the study of polarization within China in recent years

## Nonparametric estimation and inference for Granger causality measures

Taamouti, Abderrahim; Bouezmarni, Taoufik; El Ghouch, Anouar
Tipo: info:eu-repo/semantics/draft; info:eu-repo/semantics/workingPaper Formato: application/pdf
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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).

## Estimação não paramétrica da função de covariância para dados funcionais agregados; Nonparametric estimation of the covariance function for aggregated functional data

Guilherme Vieira Nunes Ludwig
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
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O objetivo desta dissertação é desenvolver estimadores não paramétricos para a função de covariância de dados funcionais agregados, que consistem em combinações lineares de dados funcionais que não podem ser observados separadamente. Estes métodos devem ser capazes de produzir estimativas que separem a covariância típica de cada uma das subpopulações que geram os dados, e que sejam funções não negativas definidas. Sob estas restrições, foi definida uma classe de funções de covariância não estacionarias, à qual resultados da teoria de estimação de covariância de processos estacionários podem ser estendidos. Os métodos desenvolvidos foram ilustrados com a aplicação em dois problemas reais: a estimação do perfil de consumidores de energia elétrica, em função do tempo, e a estimação da transmitância de substâncias puras em espectroscopia de infravermelho, através da inspeção de misturas, em função do espectro da luz.; The goal of this dissertation is to develop nonparametric estimators for the covariance function of aggregated functional data, which consists into linear combinations of functional data that cannot be sampled separately. Such methods must be able to produce estimates that not only separate the typical covariance of the subpopulations composing the data...

## Estimação não-parametrica para função de covariancia de processos gaussianos espaciais; Nonparametric estimation for covariance function of spatial gaussian processes

Jose Clelto Barros Gomes
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Dissertação de Mestrado Formato: application/pdf
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O desafio na modelagem de processos espaciais está na descrição da estrutura de covariância do fenômeno sob estudo. Um estimador não-paramétrico da função de covariância foi construído de forma a usar combinações lineares de funções B-splines. Estas bases são usadas com muita frequência na literatura graças ao seu suporte compacto e a computação tão rápida quanto a habilidade de criar aproximações suaves e apropriadas. Verificouse que a função de covariância estimada era definida positiva por meio do teorema de Bochner. Para a estimação da função de covariância foi implementado um algoritmo que fornece um procedimento completamente automático baseado no número de funções bases. Então foram realizados estudos numéricos que evidenciaram que assintoticamente o procedimento é consistente, enquanto que para pequenas amostras deve-se considerar as restrições das funções de covariância. As funções de covariâncias usadas na estimação foram as de exponencial potência, gaussiana, cúbica, esférica, quadrática racional, ondular e família de Matérn. Foram estimadas ainda covariâncias encaixadas. Simulações foram realizadas também a fim de verificar o comportamento da distribuição da afinidade. As estimativas apresentaram-se satisfatórias; The challenge in modeling of spatials processes is in description of the framework of covariance of the phenomenon about study. The estimation of covariance functions was done using a nonparametric linear combinations of basis functions B-splines. These bases are used frequently in literature thanks to its compact support and fast computing as the ability to create smooth and appropriate approaches There was positive definiteness of the estimator proposed by the Bochner's theorem. For the estimation of the covariance functions was implemented an algorithm that provides a fully automated procedure based on the number of basis functions. Then numerical studies were performed that showed that the procedure is consistent assynthotically. While for small samples should consider the restrictions of the covariance functions...

## Understanding nonparametric estimation for clustered data

Huggins, Richard
Fonte: Biometrika Trust Publicador: Biometrika Trust
Tipo: Artigo de Revista Científica
Português
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In this note we give an alternative formulation of the nonparametric estimators of Wang (2003) with the identity link. This results in a closed form of the estimator that has computational advantages and gives insight into the rationale behind the estimat

## Bayesian nonparametric estimation and consistency of mixed multinomial logit choice models

De Blasi, Pierpaolo; James, Lancelot F.; Lau, John W.
Tipo: Artigo de Revista Científica
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This paper develops nonparametric estimation for discrete choice models based on the mixed multinomial logit (MMNL) model. It has been shown that MMNL models encompass all discrete choice models derived under the assumption of random utility maximization, subject to the identification of an unknown distribution $G$. Noting the mixture model description of the MMNL, we employ a Bayesian nonparametric approach, using nonparametric priors on the unknown mixing distribution $G$, to estimate choice probabilities. We provide an important theoretical support for the use of the proposed methodology by investigating consistency of the posterior distribution for a general nonparametric prior on the mixing distribution. Consistency is defined according to an $L_1$-type distance on the space of choice probabilities and is achieved by extending to a regression model framework a recent approach to strong consistency based on the summability of square roots of prior probabilities. Moving to estimation, slightly different techniques for non-panel and panel data models are discussed. For practical implementation, we describe efficient and relatively easy-to-use blocked Gibbs sampling procedures. These procedures are based on approximations of the random probability measure by classes of finite stick-breaking processes. A simulation study is also performed to investigate the performance of the proposed methods.; Comment: Published in at http://dx.doi.org/10.3150/09-BEJ233 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

## Nonparametric estimation of scalar diffusions based on low frequency data

Gobet, Emmanuel; Hoffmann, Marc; Reiss, Markus
Tipo: Artigo de Revista Científica
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We study the problem of estimating the coefficients of a diffusion (X_t,t\geq 0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The sampling frequency \Delta^{-1} is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a, respectively, first- and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X_{n\Delta},n=0,1,...,N) in a suitable Sobolev norm, together with an estimation of its invariant density.; Comment: Published at http://dx.doi.org/10.1214/009053604000000797 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

## Nonparametric estimation of composite functions

Juditsky, Anatoli B.; Lepski, Oleg V.; Tsybakov, Alexandre B.
Tipo: Artigo de Revista Científica
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We study the problem of nonparametric estimation of a multivariate function $g:\mathbb {R}^d\to\mathbb{R}$ that can be represented as a composition of two unknown smooth functions $f:\mathbb{R}\to\mathbb{R}$ and $G:\mathbb{R}^d\to \mathbb{R}$. We suppose that $f$ and $G$ belong to known smoothness classes of functions, with smoothness $\gamma$ and $\beta$, respectively. We obtain the full description of minimax rates of estimation of $g$ in terms of $\gamma$ and $\beta$, and propose rate-optimal estimators for the sup-norm loss. For the construction of such estimators, we first prove an approximation result for composite functions that may have an independent interest, and then a result on adaptation to the local structure. Interestingly, the construction of rate-optimal estimators for composite functions (with given, fixed smoothness) needs adaptation, but not in the traditional sense: it is now adaptation to the local structure. We prove that composition models generate only two types of local structures: the local single-index model and the local model with roughness isolated to a single dimension (i.e., a model containing elements of both additive and single-index structure). We also find the zones of ($\gamma$, $\beta$) where no local structure is generated...

## Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials

Marcon, G.; Padoan, S. A.; Naveau, P.; Muliere, P.
Tipo: Artigo de Revista Científica
Português
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Many applications in risk analysis, especially in environmental sciences, require the estimation of the dependence among multivariate maxima. A way to do this is by inferring the so-called Pickands dependence function used in multivariate Extreme Value Theory. In this context, a clear advantage of a nonparametric approach over a parametric one is its flexibility and theoretical generality. Beyond the bivariate case, nonparametric estimation of the dependence function remains a challenging task and an active research field. In this article, we propose a new nonparametric approach for estimating the Pickands dependence function and we insure that it obeys all Pickands' constraints by taking advantage of a specific type of Bernstein polynomials representation. We discuss the properties of the proposed estimation method and illustrate its performance with a simulation study. For moderate dimension sizes, we illustrate our approach by analyzing clusters made of seven weather stations that have recorded weekly maxima of hourly rainfall in France from 1993 to 2011.

## A Robbins Monro algorithm for nonparametric estimation of NAR process with Markov-Switching: consistency

Fermín, Lisandro; Ríos, Ricardo; Rodríguez, Luis-Angel
Tipo: Artigo de Revista Científica
Português
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We consider nonparametric estimation for functional autoregressive processes with Markov switching. First, we study the case where complete data is available; i.e. when we observe the Markov switching regime. Then we estimate the regression function in each regime using a Nadaraya-Watson type estimator. Second, we introduce a nonparametric recursive algorithm in the case of hidden Markov switching regime. Our algorithm restores the missing data by means of a Monte-Carlo step and estimate the regression function via a Robbins-Monro step. Consistency of the estimators are proved in both cases. Finally, we present some numerical experiments on simulated data illustrating the performances of our nonparametric estimation procedure.

## Nonparametric estimation of the division rate of an age dependent branching process

Tipo: Artigo de Revista Científica
Português
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We study the nonparametric estimation of the branching rate $B(x)$ of a supercritical Bellman-Harris population: a particle with age $x$ has a random lifetime governed by $B(x)$; at its death time, it gives rise to $k \geq 2$ children with lifetimes governed by the same division rate and so on. We observe in continuous time the process over $[0,T]$. Asymptotics are taken as $T \rightarrow \infty$; the data are stochastically dependent and one has to face simultaneously censoring, bias selection and non-ancillarity of the number of observations. In this setting, under appropriate ergodicity properties, we construct a kernel-based estimator of $B(x)$ that achieves the rate of convergence $\exp(-\lambda_B \frac{\beta}{2\beta+1}T)$, where $\lambda_B$ is the Malthus parameter and $\beta >0$ is the smoothness of the function $B(x)$ in a vicinity of $x$. We prove that this rate is optimal in a minimax sense and we relate it explicitly to classical nonparametric models such as density estimation observed on an appropriate (parameter dependent) scale. We also shed some light on the fact that estimation with kernel estimators based on data alive at time $T$ only is not sufficient to obtain optimal rates of convergence, a phenomenon which is specific to nonparametric estimation and that has been observed in other related growth-fragmentation models.

## Nonparametric estimation for dependent data with an application to panel time series

Johannes, Jan; Rao, Suhasini Subba
Tipo: Artigo de Revista Científica
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In this paper we consider nonparametric estimation for dependent data, where the observations do not necessarily come from a linear process. We study density estimation and also discuss associated problems in nonparametric regression using the 2-mixing dependence measure. We compare the results under 2-mixing with those derived under the assumption that the process is linear. In the context of panel time series where one observes data from several individuals, it is often too strong to assume the joint linearity of processes. Instead the methods developed in this paper enable us to quantify the dependence through 2-mixing which allows for nonlinearity. We propose an estimator of the panel mean function and obtain its rate of convergence. We show that under certain conditions the rate of convergence can be improved by allowing the number of individuals in the panel to increase with time.

## Bayesian nonparametric estimation of Tsallis diversity indices under Gnedin-Pitman priors

Cerquetti, Annalisa
Tipo: Artigo de Revista Científica
Português
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Tsallis entropy is a generalized diversity index first derived in Patil and Taillie (1982) and then rediscovered in community ecology by Keylock (2005). Bayesian nonparametric estimation of Shannon entropy and Simpson's diversity under uniform and symmetric Dirichlet priors has been already advocated as an alternative to maximum likelihood estimation based on frequency counts, which is negatively biased in the undersampled regime. Here we present a fully general Bayesian nonparametric estimation of the whole class of Tsallis diversity indices under Gnedin-Pitman priors, a large family of random discrete distributions recently deeply investigated in posterior predictive species richness and discovery probability estimation. We provide both prior and posterior analysis. The results, illustrated through examples and an application to a real dataset, show the procedure is easily implementable, flexible and overcomes limitations of previous frequentist and Bayesian solutions.; Comment: 17 pages, new improved version

## Nonparametric Estimation of Multivariate Distributions with Given Marginals

Sancetta, Alessio
Tipo: Trabalho em Andamento Formato: 391353 bytes; application/pdf; application/pdf
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Nonparametric estimation of the copula function using Bernstein polynomials is studied. Convergence in the uniform topology is established. From the nonparametric Bernstein copula, the nonparametric Bernstein copula density is derived. It is shown that the nonparametric Bernstein copula density is closely related to the histogram estimator, but has the smoothing properties of kernel estimators. The optimal order of polynomial under the L2 norm is shown to be closely related to the inverse of the optimal smoothing factor for common nonparametric estimator. In order of magnitude, this estimator has variance equal to the square root of other common nonparametric estimators, e.g. kernel smoothers, but it is biased as a histogram estimator.

## Nonparametric estimation of nonhomogeneous Poisson processes using wavelets

Kuhl, Michael; Bhairgond, Prashant
Fonte: Rochester Instituto de Tecnologia Publicador: Rochester Instituto de Tecnologia
Tipo: Proceedings
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Nonhomogeneous Poisson processes (NHPPs) are frequently used in stochastic simulations to model nonstationary point processes. These NHPP models are often constructed by estimating the rate function from one or more observed realizations of the process. Both parametric and nonparametric models have been developed for the NHPP rate function. The current parametric models require prior knowledge of the behavior of the NHPP under study for model selection. The current nonparametric estimators, in general, require the storage of all of the observed data. Other hybrid approaches have also been developed. This paper focuses on the nonparametric estimation of the rate function of a nonhomogeneous Poisson process using wavelets. The advantages of wavelets include the flexibility of a nonparametric estimator enabling one to model the nonstationary rate function of an NHPP without prior knowledge or assumptions about the behavior of the process. Furthermore, this method has some advantages of current nonparametric techniques. Thus, using wavelets we can develop an efficient yet highly flexible NHPP rate function. In this paper, we develop the methodology required for constructing a wavelet estimator for the NHPP rate function. In addition, we present an experimental performance evaluation for this method.