Robust data analysis based on characteristic functions

Abstract

Characteristic functions have heretofore mainly been a theoretical tool in the theory of probability and mathematical statistics and have been all but neglected in the field of data analysis. The research presented in this thesis indicates the usefulness of the characteristic function in the area of data analysis, specifically robust estimation. By investigating the minimization over the distributional parameters of the integral of the weighted squared modulus of the difference between the theoretical and empirical characteristic function, a relatively simple fixed point procedure was obtained for the estimation of the parameters of univariate sample data using the underlying assumption of normality. Sensitivity and influence curves and breakdown analysis indicate the robustness qualities of the estimates. The procedures are extended to the multivariate normal parameter estimation problem and are shown to obtain robust estimates of location, scale and correlation. Finally, the univariate procedure is applied to the general linear regression problem by minimizing the scale estimate of the residual under the hypothesized model. Diagnostics are provided to indicate the degree of an observation's compatibility with the underlying assumptions and the estimated parameters. Analyses of various examples are presented which indicate the use of the procedures and diagnostics in a data analytic framework.