Cosmogenic exposure dating provides a method for estimating the ages of glacial moraines deposited in the last ~105 years. Cosmic rays break atoms in surface rocks at predictable rates. Thus, the ages of moraines are directly related to the concentrations of cosmic ray-produced nuclides in rocks on the moraine surfaces, under ideal circumstances. However, many geomorphic processes may interfere with cosmogenic exposure dating. Because of these processes, boulders sometimes arrive at the moraines with preexisting concentrations of cosmogenic nuclides, or else the boulders are partly shielded from cosmic rays following deposition. Many methods for estimating moraine ages from cosmogenic exposure dates exist in the literature, but we cannot assess the appropriateness of these methods without knowing the parent distribution from which the dates were drawn on each moraine. Here, we make two contributions. First, we describe numerical models of two geomorphic processes, moraine degradation and inheritance, and their effects on cosmogenic exposure dating. Second, we assess the robustness of various simple methods for estimating the ages of moraines from collections of cosmogenic exposure dates. Our models estimate the probability distributions of cosmogenic exposure dates that we would obtain from moraine boulders with specified geomorphic histories, using Monte Carlo methods. We expand on pioneering modeling efforts to address this problem by placing these models into a common framework. We also evaluate the sensitivity of the models to changes in their input parameters. The sensitivity tests show that moraine degradation consistently produces left-skewed distributions of exposure dates; that is, the distributions have long tails toward the young end of the distribution. In contrast, inheritance produces right-skewed distributions that have long tails toward the old side of the distribution. Given representative distributions from these two models, we can determine which methods of estimating moraine ages are most successful in recovering the correct age for test cases where this value is known. The mean is a poor estimator of moraine age for data sets drawn from skewed parent distributions, and excluding outliers before calculating the mean does not improve this mismatch. The extreme estimators (youngest date and oldest date) perform well under specific circumstances, but fail in other cases. We suggest a simple estimator that uses the skewnesses of individual data sets to determine whether the youngest date, mean, or oldest date will provide the best estimate of moraine age. Although this method is perhaps the most globally robust of the estimators we tested, it sometimes fails spectacularly. The failure of simple methods to provide accurate estimates of moraine age points toward a need for more sophisticated statistical treatments. We present improved methods for estimating moraine ages in a companion paper.