Estimation Of Parameters Of Gamma And Generalized Gamma Distributions Based On Censored Experimental Data

In time-to-event data analysis, censoring is one of the unique features that restricts our ability to observe the time-to-events and poses difficulties for statistical analysis. Censoring occurs when the exact time-to-event cannot be observed for some or all observations. In this thesis, we study the parameter estimation methods for a two-parameter gamma distribution and a three-parameter generalized gamma distribution based on different kinds of censored data arising from life-testing experiments. We first study the parameter estimation of a three-parameter generalized gamma distribution based on left-truncated and right-censored data. It is well known that the maximum likelihood estimates of the parameters for the generalized gamma distribution may not be stable, especially when the data is incomplete. A stochastic version of the expectation-maximization (EM) algorithm is proposed as an alternative method to compute approximate maximum likelihood estimates. The proposed estimation procedure is compared with some existing estimation procedures based on the maximum likelihood method, such as the direct optimization method, the profile likelihood method, and the EM algorithm, in terms of accuracy and stability. Two different methods to obtain reliable initial estimates of the parameters required for the iterative algorithms are also proposed. Interval estimation based on a parametric bootstrap method is discussed. The proposed methodologies are illustrated with a numerical example. Then, a Monte Carlo simulation study is used to evaluate the performance of the proposed estimation procedures. Based on the simulation results, we make some recommendations about which estimation procedures are more appropriate in practice. Then, we consider the parameter estimation and reliability analysis based on one-shot device testing data in a realistic situation where defectives are produced in the manufacturing process. For one-shot device testing, all the lifetimes of the devices are either left-censored or right-censored due to the destructive nature of the test. Unlike non-destructive testing of products with continuous monitoring, defective one-shot devices will not be detected until the time of usage or testing. Moreover, in the presence of defectives, if a one-shot device does not work at the time of testing, we may not be able to distinguish whether the particular device is a defective or a device that has a lifetime smaller than the testing time. A maximum likelihood approach and a Bayesian approach are proposed for the point and interval estimation of the parameters and reliability indices under different scenarios. The proposed methodologies are illustrated with a numerical example when the lifetimes of the devices follow a two-parameter gamma distribution. A Monte Carlo simulation study is used to evaluate the performance of the proposed estimation procedures under different settings. Based on the simulation results, some practical guidelines are provided. Finally, some possible future research directions based on this thesis are discussed