Asymptotic properties of a class of test statistics when applied to hazard-based residuals arising in survival and reliability models are presented. These test statistics are useful in goodness-of-fit tests and model validation. The properties are obtained by examining the asymptotic properties of generalized residual processes, which are (possibly random) time transformations of the processes associated with the incomplete failure times. Since the time transformations depend on unknown model parameters, the residual processes are obtained by replacing the unknown parameters by their estimators. The results shed light on the effects of estimating parameters to obtain the residual processes. Implications concerning possible pitfalls of some existing model validation procedures utilizing hazard-based residuals and ways to correct these problems are discussed.