The most direct purpose of collecting warranty data is tracking associated costs. However, they are also useful for quantifying a relationship between use rate and product time-to-first-failure and for estimating the distribution of product time-to-first-failure (which is modeled in this article as depending on use rate and a unit potential life length under continuous use). Employing warranty data for such reliability analysis purposes is typically complicated by the fact that some parts of some warranty data records are missing. A pseudo-likelihood methodology is introduced to deal with some kinds of incomplete warranty data (such as that available in a motivating real case from a machine manufacturer). A use rate distribution, the distribution of time to first failure, and the time associated with a cumulative probability of first failure are estimated, based on the proposed approach and available data.