Statistical Depth Functions and Depth-Based Robustness Diagnosis

This dissertation, "Statistical Depth Functions and Depth-based Robustness Diagnosis" by Wing-sze, Lok, 駱穎思, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of the thesis entitled STATISTICAL DEPTH FUNCTIONS AND DEPTH-BASED ROBUSTNESS DIAGNOSIS submitted by LOK Wing Sze for the degree of Master of Philosophy at The University of Hong Kong in December 2005 Robustnessofastatisticalprocedureconcernsthee(R)ectontheinferenceorprob- abilistic assertion induced by perturbations to the model or data. Due to the diverse types of perturbations and model departures available for investigation, no denitive notion of robustness has yet been recognized. This study aimed at developing a new depth-based diagnostic scheme for measuring robustness spe- cic to the actually observed data in di(R)erent problem settings. Its depth-based nature motivates our study of notions of data depth that can rank multivariate data in multidimensional spaces, which is the focus of the rst part of this disser- tation. Conventional depth functions do not extend to innite-dimensional data andfailtocaterformultimodalityfeatures. Weproposetwonewdepthfunctions toovercomethesetwodrawbacks. Theirdenitionsarebasedonsizesspannedby datasubsetsandinterpointdistancesrespectively. Applicationsofthenewdepth functionsin di(R)erentproblemsettings were illustrated, yielding veryencouragingresults. The second part developed and investigated a new diagnostic scheme. It consists of ve stages: (i) data perturbations, (ii) resampling, (iii) simulation of inferential output, (iv) calculation of p-values and (v) display of p-values. Based on a proper choice of depth function, we can calculate the p-value to assess extremeness of the perturbed inferential output relative to the original one. The robustness properties were then summarized and represented by the plots of p- values against degrees of perturbations. This scheme is data-driven and provides a standardized robustness measure for comparing di(R)erent statistical procedures. Incontrastwiththeconventionalrobustnessconcepts, itassessestherobustnessof the distribution of the inferential output rather than the output itself. A number of examples were used to illustrate the diverse applicability of our scheme to di(R)erent statistical procedures. DOI: 10.5353/th_b3483826 Subjects: Statistical hypothesis testing Confidence intervals