Bootstrap and empirical likelihood methods in statistical shape analysis
The aim of this thesis is to propose bootstrap and empirical likelihood confidence regions and hypothesis tests for use in statistical shape analysis. Bootstrap and empirical likelihood methods have some advantages when compared to conventional methods. In particular, they are nonparametric methods and so it is not necessary to choose a family of distribution for building confidence regions or testing hypotheses. There has been very little work on bootstrap and empirical likelihood methods in statistical shape analysis. Only one paper (Bhattacharya and Patrangenaru, 2003) has considered bootstrap methods in statistical shape analysis, but just for constructing confidence regions. There are no published papers on the use of empirical likelihood methods in statistical shape analysis. Existing methods for building confidence regions and testing hypotheses in shape analysis have some limitations. The Hotelling and Goodall confidence regions and hypothesis tests are not appropriate for data sets with low concentration. The main reason is that these methods are designed for data with high concentration, and if this hypothesis is violated, the methods do not perform well. On the other hand, simulation results have showed that bootstrap and empirical likelihood methods developed in this thesis are appropriate to the statistical shape analysis of low concentrated data sets. For highly concentrated data sets all the methods show similar performance. Theoretical aspects of bootstrap and empirical likelihood methods are also considered. Both methods are based on asymptotic results and those results are explained in this thesis. It is proved that the bootstrap methods proposed in this thesis are asymptotically pivotal. Computational aspects are discussed. All the bootstrap algorithms are implemented in “R”. An algorithm for computing empirical likelihood tests for several populations is also implemented in “R”.