Nonparametric regression and mixture models

• Main Author: Polsen, Orathai
• Published: University of Leeds 2011
• Subjects: 519.536
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.578651

Nonparametric regression estimation has become popular in the last 50 years. A commonly used nonparametric method for estimating the regression curve is the kernel estimator, exemplified by the Nadaraya- Watson estimator. The first part of thesis concentrates on the important issue of how to make a good choice of smoothing parameter for the Nadaraya- Watson estimator. In this study three types of smoothing parameter selectors are investigated: cross-validation, plug-in and bootstrap. In addition, two situations are examined: the same smoothing parameter and different smoothing parameters are employed for the estimates of the numerator and the denominator. We study the asymptotic bias and variance of the Nadaraya- Watson estimator when different smoothing parameters are used. We propose various plug-in methods for selecting smoothing parameter including a bootstrap smoothing parameter selector. The performances of the proposed selectors are investigated and also compared with cross-validation via a simulation study. Numerical results demonstrate that the proposed plug-in selectors outperform cross-validation when data is bivariate normal distributed. Numerical results also suggest that the proposed bootstrap selector with asymptotic pilot smoothing parameter compares favourably with cross-validation. We consider a circular-circular parametric regression model proposed by Taylor (2009), including parameter estimation and inference. In addition, we investigate diagnostic tools for circular regression which can be generally applied. A final thread is related to mixture models, in particular a mixture of linear regression models and a mixture of circular-circular regression models where there is unobserved group membership of the observation. We investigate methods for selecting starting values for EM algorithm which is used to fit mixture models and also the distributions of these values. Our experiments suggest that the proposed method compares favourably with the common method in mixture linear regression models.