Some Applications of Divergences to Robust Inference with Mixed Data

Abstract

This thesis focuses on the application of the density power divergence to studies involving mixed-data problems. It also develops a unified the- ory of two-sample nonparametric tests for a general class of divergence measures. The main content of the thesis is divided into three parts. The first part explores parameter estimation in ordinal response mod- els, which are prevalent in many scientific studies. A typical data set generated through an ordinal response model includes continuous, non- stochastic regressors and a response variable with ordinal outcomes. The theory of non-homogeneous density power divergence is applicable here, provided appropriate conditions on the regressors and link functions are satisfied. The roles of different link functions in estimation are thor- oughly analyzed, and the robustness of the estimators is evaluated using the influence function, the (explosive) breakdown point, and the implosive breakdown point. The latter two measures are found to be very high, en- suring the robustness of the minimum density power divergence method against various types of outliers. The second part focuses on the estimation and development of Wald- type tests for polychoric correlation. Initially, the standard density power divergence is applied. Subsequently, a two-step approach is introduced, which, while theoretically more complex, substantially reduces the com- putational burden. The results from the two-step approach are highly consistent with those from the initial method. Additionally, a new divergence measure involving two tuning parame- ters, derived from the density power divergence (DPD), is proposed. These estimates perform at least as good as the DPD under pure data condi- tions, up to a threshold defined by the tuning parameters. Moreover, the proposed estimates exhibit enhanced robustness compared to the DPD. Given the effectiveness of polychoric correlation in quantifying associa- tions between categorical variables, this research provides valuable tools for applied scientists. The third part introduces a class of two-sample nonparametric tests based on the class of extended Bregman divergences to assess the equality of two completely unstructured absolutely continuous distributions. The asymptotic distributions of the test statistics are derived under both the null hypothesis and contiguous alternatives. The robustness of the pro- posed method is studied through the influence function and the asymp- totic breakdown point. Numerical studies are conducted for two spe- cific divergence families: the generalized S-Bregman divergence and the Exponential-Polynomial divergence measures. Notably, divergences out- side the power divergence family often perform better within this frame- work. Finally, a generic tuning parameter selection strategy is proposed, en- abling the application of the method to real-world data. The theoretical developments presented in this part hold the potential for extension to various other research areas in the future.