In many failure mechanisms, most subjects under study deteriorate physically over time, and thus a depreciation in health may precede failure. A latent stochastic process, called degradation process, may be assumed for modeling such depreciation whereby an event occurs when the process first crosses a threshold. A class of survival regression models can be constructed from the first hitting time of a latent accelerating degradation process, which turns out to be a transformation model in the literature. To characterize these models, we propose to use first‐hitting‐time models for the baseline distribution, specifically inverse Gaussian, Birnbaum–Saunders and gamma distributions, among others. The proposed models have many desirable features, such as a wide variety of shapes of hazard rates, analytical tractability and, most of all, its motivation from a plausible stochastic setting for failure. We estimate the model parameters by the non‐parametric maximum likelihood approach. The estimators are shown to be consistent, asymptotically normal and asymptotically efficient. Simple and stable numerical algorithms are provided to calculate the parameter estimators and estimate their variances. Simulation studies show that the proposed approach is appropriate for practical use. The methodology is illustrated with two real examples. Copyright © 2013 John Wiley & Sons Ltd