Non-parametric goodness-of-fit testing and applications
- The current state of statistical practice is dominated by the use of model-based methods. A natural basic question is whether the model fits the data. This is a central question and has been studied for over a century. There is a vast literature available; however, a majority of the literature is focused on the univariate case. Far less work has been done on the general case and the theory is very sparse. This thesis introduces a general methodology, based on (non-commutative) Fourier analysis, to construct tests of goodness of fit for distributions on relatively general spaces. The procedure is developed for a simple null hypothesis and has been carried out for several examples, including the normal distribution, the uniform distribution, and the uniform distribution on high-dimensional spheres. The method is used to construct two families of tests of uniformity on the compact classical groups. Carrying out the program for the compact groups involves a substantial use of the representation theory of Lie groups, including derivation of new group theoretic formulas. These tests are used to numerically study the mixing-time of a recently introduced Markov chain Monte Carlo sampler on the orthogonal group. The method is extended to the composite null hypothesis of parametric families of distributions. The asymptotic properties are studied and computational aspects are discussed. This has been carried out for several examples: testing for multivariate normality, testing for the beta family, testing for the gamma family, testing for the chi-square family, and testing for the exponential family. Motivated by an application in material sciences, the method has been successfully carried out for several parametric families of distributions on the group of three dimensional rotations. The power function against local alternatives is studied in details and various properties are established. In particular, these tests are asymptotically admissible. The tests have been tried out on several examples and show favorable performance compared to existing methods.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Statistics.
|Statement of responsibility
|Submitted to the Department of Statistics.
|Thesis (Ph.D.)--Stanford University, 2017.
- © 2017 by Amir Sepehri
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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