Degree Type


Date of Award


Degree Name

Doctor of Philosophy



First Advisor

William Q. Meeker

Second Advisor

Daniel J. Nordman


The linear mixed model is a very popular and powerful tool in many applications, such as engineering, biology and social science. Oftentimes it is of interest to make statistical inference on functions of parameters in a linear mixed model.

In Chapter 2 we constructed a canonical linear mixed-effects model with some nonlinear parametric functions of interest based on motivating examples in reliability and nondestructive evaluation. Several competing procedures that can be used to construct

confidence intervals for these nonlinear functions of parameters in a linear mixed model, such as likelihood, Wald, Bayesian and bootstrap procedures, are described in this chapter. Then we designed a simulation study to compare the coverage properties and computational cost for different interval estimation procedures.

The Markov chain Monte Carlo (MCMC) procedure introduced in Chapter 2 for Bayesian estimation is an efficient way to produce the credible intervals for statistical models. There is, however, always Monte Carlo error in the estimates because the MCMC procedure involves the use of random numbers. If it is of interest to determine the interval end points, which are quantified as quantile estimates, with certain degrees of repeatability, a large amount of MCMC draws may be required. It is especially true when strong autocorrelation exists in MCMC draws. In Chapter 3 we described a procedure

to estimate the number of MCMC draws needed for the quantile estimates with desired precision and confidence level. We also used several examples, where different MCMC procedures are involved in, to illustrate the use of the procedure.

In Chapter 4 we introduced an R function to implement the procedures of estimating the number of draws for either the MCMC sequences or the i.i.d. sequences described in Chapter 3. The R function takes a vector of pilot draws from either an MCMC sequence

or an i.i.d. sequence, the quantile probability, the desired precision and the confidence level as the input, and returns the required number of draws. Details about how to use the function are discussed in this chapter.

Copyright Owner

Jia Liu



File Format


File Size

75 pages