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"additionalType": "doctoralThesis",
"name": "Consistency and bandwidth selection for dependent data in non-parametric functional data analysis",
"author": {
"name": "Simon Peter Müller",
"givenName": "Simon Peter",
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"description": "Besides a introduction this dissertation contains three more chapters. In the following paragraphs we will give a short summary for each of them. In Chapter 2 we examine non-parametric regression for α-mixing functional data. A method for estimating the regression function m(x) is the k-nearest neighbour kernel estimate. We prove that the k-NN kernel estimate is pointwise almost complete consistent for α-mixing data and we present, for two different assumptions on the covariance term, the almost complete convergence rate. The results are obtained on the one hand by using results of the functional kernel estimate, where a deterministic bandwidth sequence is used, and on the other hand by applying lemmas from Bradley and Burba et al.. Finally, we give an outline on how to avoid the drawback of susceptibility of the k-NN kernel estimate to outliers. We adumbrate on how to construct such a robust kernel estimate and on how get almost complete convergence. Chapter 3 is focused on uniform convergence rates on a compact set SE of non-parametric estimates for α-mixing random variables of various conditional quantities, such as the conditional expectation, the conditional distribution function, and the conditional density function. It turns out in our proofs that there is a link between the covering number of the set SE and the type of α-mixing. Indeed, there are many functional spaces on which a compact set has a covering number that grows exponentially. For such sets SE it is not possible to get uniform almost complete rates for general α-mixing random variable, there we have to restrict on geometric α-mixing random variables. Instead, if the covering number grows polynomially, we get almost complete rates for general α-mixing random variables. Furthermore, we present two results for the kernel estimate of the regression function, where we get with some additional conditions similar rates as in the independent case. With slightly modified assumptions, not listed in this thesis, we get similar results for the kernel estimate of the conditional distribution function and the conditional density function. Moreover, we comment on the uniform almost complete rate for the estimate of the non-parametric regression function and outline how to possibly prove the validity of a cross-validation bandwidth selection procedure for α-mixing functional data. In the last Chapter 4 we discuss the issue of a local adaptive bandwidth selection procedure for the kernel estimate of the regression function. Here, an obvious measure for the optimality of the parameter selection is the pointwise mean squared error. As the regression function m(x) is unknown, we cannot calculate it. In the literature different approximation methods as cross-validation or bootstrapping are presented. We pick up a bootstrap method for approximating this pointwise mean squared error for non-parametric functional regression. We prove that our approximation converges against the true error and afterwards we compare our method on simulated and real world data with a global and local version of a cross-validation method. The simulated data is constructed such that we have different nuances between homogenous and heterogenous data. The results differ then in the following way. On the one hand if the data is more homogenous, global and local methods perform similarly, on the other hand if the data gets more heterogenous, the local methods outperform the global bandwidth selection procedure more and more. In addition, we notice that in all examples the bootstrap method performs better or equal than the local cross-validation procedure. Moreover, it is possible to calculate confidence intervals from the bootstrapped data. As we need a pilot kernel estimate for bootstrapping, more calculation time is needed for that bootstrap procedure.",
"keywords": "Funktionale Datenanalyse , Nichtparametrische Regression, 510, Zeitreihen , Bandweitenwahl , Bootstrapping, functional data analysis , non-parametric regression , time series , bandwidth selection , bootstrapping",
"inLanguage": "en",
"datePublished": "2011-11-17",
"dateModified": "2014-12-08",
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