Existing approaches to timing analysis under uncertainty are based on restrictive assumptions. Statistical STA techniques assume that the full probabilistic distribution of parameter uncertainty is available; in reality, the complete probabilistic description often cannot be obtained. In this paper, a new paradigm for parameter uncertainty description is proposed as a way to consistently and rigorously handle partially available descriptions of parameter uncertainty. The paradigm is based on a theory of interval probabilistic models that permit handling uncertainty that is described in a distribution-free mode - just via the range, the mean, and the variance. This permits effectively handling multiple real-life challenges, including imprecise and limited information about the distributions of process parameters, parameters coming from different populations, and the sources of uncertainty that are too difficult to handle via full probabilistic measures (e.g. on-chip supply voltage vari...