Mathematics Ph.D. Dissertations

Title

Statistical Inferences for Missing Data/causal Inferences Based on Modified Empirical Likelihood

Date of Award

2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

Wei Ning (Advisor)

Second Advisor

Jonathan Kershaw (Other)

Third Advisor

John Chen (Committee Member)

Fourth Advisor

Junfeng Shang (Committee Member)

Abstract

In this dissertation we first modify profile empirical likelihood function conditioned on complete data to estimate the population mean in presence of missing values in the response variable. Also in Chapter 3 under the counterfactual potential outcome by Rubin (1974, 1976, 1977), we propose some methods to estimate causal effect. This dissertation specifically expands upon the work of Qin and Zhang (2007), as they fail to address two main shortcomings of their empirical likelihood utilization. The first flaw is when the estimation fails to exist. The second flaw is under- coverage probability of the confidence region. Both of these two flaws get exacerbated when the sample size is small.

In Chapter 2, we modify the associated empirical likelihood function to obtain consistent estimators which address each of the shortcomings. Our adjusted-empirical-likelihood-based consistent estimator, using similar strategy to Chen et al. (2008), adds a point to the convex hull of the data to ensure the algorithm converges. Furthermore, inspired by Jing et al.2017, we propose a quadratic transformation to the associated empirical likelihood ratio test statistic to yield a consistent estimator with greater coverage probability.

In Chapter 3 using the techniques developed in Chapter 2, adjusted empirical likelihood causal effect estimator which is consistent is developed.

In Chapter 2 simulation study for estimating the mean response under the presence of missing values, both of our proposed estimators show competitive results compared with other historical method. These modified estimators generally outperform historical estimators in terms of RMSE and coverage probability. Chapter 3 simulations exhibit that the consistent adjusted empirical likelihood causal effect estimator is competitive compared to the historical methods.

Along the way, we also propose a weighted adjusted empirical likelihood for both estimating the mean response, and causal effect, which is proved to be consistent under the presence of missing values in the response variable. This estimator exhibits competitive results compared with the empirical likelihood estimator proposed by Qin and Zhang (2007).

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