Mathematics Ph.D. Dissertations


Bootstrap Methods for Estimation in Linear Mixed Models with Heteroscedasticity

Date of Award


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)


Mathematics/Mathematical Statistics

First Advisor

Junfeng Shang (Advisor)

Second Advisor

Hanfeng Chen (Committee Member)

Third Advisor

Wei Ning (Committee Member)

Fourth Advisor

Hrishikesh Joshi (Other)


Bootstrap is a widely applicable computational statistical method and introduced by Efron (1979). The idea of bootstrap has extended to linear mixed models (LMMs) and is well established for the LMMs with homoscedasticity assumption. Thus, our specific focus in this dissertation lies on developing a bootstrap method for LMMs under homoscedasticity violation (heteroscedasticity).

In the study, we assume that the form of heteroscedasticity is unknown. Thus, to generate bootstrap response data as close as possible to the actual response data, we obtain the marginal residuals and transform them to ensure that the variance of the modified marginal residuals is an unbiased estimator of the variance of the error terms. The idea of the proposed bootstrap is inspired by Wu (1986). Furthermore, we prove the consistency of the bootstrap procedure, which demonstrates that the proposed bootstrap method has asymptotically valid inferences in the estimation of parameters in LMMs.

Simulations are conducted under different scenarios by varying error covariance structures and sample sizes. We generate data with four covariance structures and three sample size settings. Moreover, to show the effectiveness of the proposed method over the other methods, we compare the results of the proposed method with existing bootstrap methods: parametric, residual, REB, and wild.

By considering the above scenarios, we carry out a series of simulations under five different objectives. In the first two objectives, we observe the bootstrap distributions of model coefficients and set out the number of bootstrap replications as suitable for the upcoming simulations. In the third and fourth objectives, we study the parameter estimation performance and assess parameter estimation accuracy. Finally, we compute the empirical coverage probability of the parameters.

The simulation results with the heteroscedastic errors demonstrate that the accuracy of the estimation process of the proposed bootstrap outperforms the wild bootstrap under the small sample settings. In contrast, the estimation performance, accuracy, and coverage of both methods become similar in the large sample setting. Moreover, the proposed and existing bootstrap methods have similar performance under homoscedasticity.

Furthermore, two applications with real-life examples are illustrated to evaluate the effectiveness of the proposed bootstrap method.