A large class of financial models of costs associated with warranties involve functions that arise in renewal theory. The simplest interesting example is the renewal function which, for the purposes of this article, may be interpreted as the expected number of failures by a specific warranty duration. To estimate the renewal function, cross-sectional regression data is assumed to be available. The data is assumed to be semiparametric in that the observation may be expressed as a known function of a vector of covariates, a vector of unknown parameters and an unknown error term while no distributional assumptions are made on the error terms. An estimator of the renewal function is constructed and conditions are provided so that it is asymptotically normal, after suitable standardization. This estimator is a special case of a class of statistics introduced here called residual-based infinite order U-statistics. U-statistics are complex averages over functions of observations. In this article, the observations are allowed to be residuals from a complex regression model. By appealing to this large class of statistics, several other parameters of interest in warranty analysis and related fields may be estimated. Convolutions of distribution functions and a discounted renewal function are discussed in this article. Using cross-sectional regression data to estimate characteristics of a stochastic process, such as the renewal function, requires strong assumptions. In fact, an interesting aspect of this article is that this estimation can be accomplished using reasonable models for the contracts and data collection.