Integral Transform Methods in Goodness-of-Fit Testing
Open Access
- Author:
- Hadjicosta, Elena
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 22, 2019
- Committee Members:
- Donald Richards, Dissertation Advisor/Co-Advisor
Bharath Sriperumbudur, Committee Chair/Co-Chair
Michael Akritas, Committee Member
Lynn Lin, Committee Member
Kirsten Eisentraeger, Outside Member - Keywords:
- Bessel function
Contiguous alternative
Gamma distribution
Gaussian random field
Hankel transform
Wishart distribution - Abstract:
- In the first part of this dissertation, we apply the method of Hankel transforms to develop goodness-of-fit tests for gamma distributions with given shape parameter and unknown rate parameter. We derive the limiting null distribution of the test statistic, obtain the corresponding covariance operator; also, we show that the eigenvalues of the operator are simple and satisfy an interlacing property. We apply our tests to two data sets, one consisting of waiting times for a Geiger counter to observe 100 alpha-particles and a second data set of failure times for some tractor brakes. Further, we establish the consistency of the test and obtain the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We investigate the approximate Bahadur slope of the test statistic, and the validity of the Wieand condition, under which the approaches through the approximate Bahadur efficiency and the Pitman efficiency are in accord. In the second part of this dissertation, we initiate a study of the problem of goodness-of-fit testing when the data consist of positive definite matrices. Motivated by the recent appearance of the cone of positive definite matrices in numerous areas of applied research, including diffusion tensor imaging, finance, wireless communication systems, and in the analysis of polarimetric radar images, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and the corresponding covariance operator, and we derive an interlacing property for the eigenvalues of the operator. In this setting, the determination of the multiplicity of the eigenvalues is an open problem. As an application, we test the hypothesis that a financial data set, consisting of the sample covariance matrices for the biweekly logarithmic returns of the stock prices of three corporations, has a Wishart distribution. We also establish the consistency of the test against a large class of alternative distributions and derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We establish the Bahadur and Pitman efficiency properties of the test statistic and we show the validity of a modified Wieand condition. The proof of Wieand's condition remains an open problem.