Nonparametric variance estimation
For the heteroscedastic nonparametric regression model with unknown mean function f and variance function V, problems regarding the variance function V are considered. In the first part of this dissertation, assuming the variance function V is Hoelder continuous, an upper bound on uniform error r...
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Format:  Doctoral Thesis 
Language:  English 
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PhilippsUniversität Marburg
2022

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Summary:  For the heteroscedastic nonparametric regression model with unknown mean function f and variance function V, problems regarding the variance function V are considered.
In the first part of this dissertation, assuming the variance function V is Hoelder continuous, an upper bound on uniform error rate for a linear estimator is derived, using Gaussian approximation of partial sums under dependency results from Berkes et al. (2014), and Dudley's entropy bound as in van der Vaart and Wellner (1996). Bootstrap uniform confidence bands are also constructed and their asymptotically correct coverage property is verified through anticoncentration inequality by Chernozhukov et al. (2014). The asymptotic normality is established through the LindebergFeller central limit theorem for mdependent variables by Janson (2021). In the simulation study, the theorem of uniform rate and bootstrap confidence bands are first illustrated with oracle bandwidth. Using the twostep algorithm proposed by Bissantz et al. (2007) for bandwidth selection as well as an additional calibration process, one can obtain satisfactory performance of uniform confidence bands in finite samples.
In the second part, we consider the case where a discontinuity point (kink) on the variance function V or its derivative is present. The zerocrossingtime technique used in Bengs and Holzmann (2019a) is applied to the variance function V, to study the pointwise error of kink location and size. For the lower bounds, the twopoint testing argument in Tsybakov (2009) and the moment matching technique in Wang et al. (2008) are employed, to derive the optimal rate of convergence. The asymptotic normality for kink location and size estimators is established through the LindebergFeller central limit theorem for mdependent variables by Janson (2021).
In the third part, the heteroscedastic nonparametric regression model in functional data analysis (FDA) is considered, where independent copies of the random process Z are involved in the model. Using a linear estimator constructed through differencebased method from Wang et al. (2008), an upper bound on uniform error rate for random noise variance V is derived, using Gaussian approximation of partial sums under dependency results from Berkes et al. (2014), and Dudley's entropy bound as in van der Vaart and Wellner (1996). The uniform error rate is confirmed by numerical results with error decomposition using oracle bandwidth. Applying Kfold crossvalidation and the twostep algorithm proposed by Bissantz et al. (2007), the estimator shows satisfactory performance in finite samples. 

Physical Description:  140 Pages 
DOI:  10.17192/z2022.0223 