Qingyang Liu

Date of Award

Spring 2023

Document Type

Open Access Dissertation



First Advisor

Xianzheng Huang


This dissertation considers statistical inference methods for parametric modal regression models. In Chapter 1, we motivate the mode as the measure of central tendency instead of the median or the mean with an example. Following the motivational example, we include an overview of existing modal regression models. Later, in the same chapter, we explain advantages of the parametric modal regression models over existing nonparametric modal regression models. In Chapter 2, we address issues in statistical inference brought in by data contaminated with measurement error. With measurement error in covariates, statistical inference methods designed for modal regression models with error-free covariates become inappropriate. We use an innovative Monte-Carlo based method to revise the original log-likelihood function that one uses in the absence of covariates measurement error. This revision leads to a new objective function adequately accounting for measurement error that one maximizes with respect to unknown parameters in the regression model. We also propose a model diagnostic method based on parametric bootstrap for the parametric modal regression with error in covariates. The proposed method for estimating regression parameters is applicable for any parametric modal regression models. However, there are only a handful of existing distributions that are suitable for the modal regression model for heavy-tailed response data. To allow for flexible modal regression, we propose a new unimodal distribution called flexible Gumbel distribution in Chapter 3. We present both frequentist and Bayesian inference methods for the flexible Gumbel distribution in the same chapter. Chapter 4 introduces the general unimodal distribution family that encompasses a range of unimodal asymmetric distributions and incorporates the flexible Gumbel distribution as a specific instance. Based on the general unimodal distribution family, we propose a unified framework for Bayesian modal regression that is well-suited for analyzing asymmetric and fat-tailed data. We propose the Gaussian process modal regression model in Chapter 5. Unlike the classic Gaussian process regression model where one assumes a Gaussian process for the conditional mean of the response, in our proposed Gaussian process regression model, we assume a Gaussian process for the conditional mode.