Statistical prediction is of fundamental importance in statistics theory and plays an important role in practices, especially in industrial reliability applications.In this dissertation, we focus on methods to compute prediction intervals with focus on reliability applications.

In Chapter 2, we review two main types of prediction interval methods under the parametric framework: methods based on an (approximate) pivotal quantity and methods based on a predictive distribution.The former one includes plug-in, pivotal, and calibration methods while the latter one includes Bayesian, fiducial, and direct-bootstrap methods. We discuss the connection between these two types of methods and provide examples involving both continuous and discrete distributions. We also review prediction interval methods for dependent data.

In Chapter 3, we focus on prediction interval methods for a specific prediction problem---predicting the number of future events from a population of units associated with an on-going time-to-events process---that is often called the "within-sample prediction''.We first show that the plug-in method is not asymptotically correct (i.e., for a large amount of data, the coverage probability always fails to converge to the nominal confidence level). Then we show that a commonly used calibration prediction method is asymptotically correct. In addition, two alternative predictive-distribution-based methods that perform better than the calibration method are presented and justified. We also discuss the extension from the single cohort within sample prediction to the multiple cohort one.

In Chapter 4, we propose a general prediction method that uses a likelihood ratio test to construct a statistic involving both the data and a future random variable.The proposed general approach can often identify extract prediction interval procedures without having to identify a pivotal quantity. For applications where a pivotal quantity does not exist, the proposed method also has good coverage properties.