Likelihood testing with censored and missing duration data

Duration data of a component, product, or system are often incomplete. We give guidelines for likelihood ratio testing for small samples with some data imperfections, that is, when some information is missing and some information is censored. We introduce the model of the missing time-to-failure mechanism, which still enables the exact likelihood ratio testing of the scale and homogeneity. Such a model encompasses well both the generalized gamma and Pareto duration time models with missing individual times to failure. The exact distribution of the likelihood ratio test for the censored sample fromWeibull distribution with known shape parameter is derived and discussed.We discuss separately Type I, Type II, and progressively censored samples. We show that the Type I censoring and Type II censoring differ substantially. The construction of the pivotal quantity is possible for Type II and progressively Type II censoring; however, it is not available for the case of Type I censoring. Thus, for the latter case the usage of the exact likelihood ratio test is a natural option. We also discuss the case of the generalized gamma distribution. For the case with nuisance shape parameters we use an integrated likelihood approach. Convenient examples illustrate the methods developed in the article.