For discrete-time survival data, conditional likelihood inference in Cox's hazard odds model is theoretically desirable but exact calculation is numerical intractable with a moderate to large number of tied events. Unconditional maximum likelihood estimation over both regression coefficients and baseline hazard probabilities can be problematic with a large number of time intervals. We develop new methods and theory using numerically simple estimating functions, along with model-based and model-robust variance estimation, in hazard probability and odds models. For the probability hazard model, we derive as a consistent estimator the Breslow-Peto estimator, previously known as an approximation to the conditional likelihood estimator in the hazard odds model. For the odds hazard model, we propose a weighted Mantel-Haenszel estimator, which satisfies conditional unbiasedness given the numbers of events in addition to the risk sets and covariates, similarly to the conditional likelihood estimator. Our methods are expected to perform satisfactorily in a broad range of settings, with small or large numbers of tied events corresponding to a large or small number of time intervals. The methods are implemented in the R package dSurvival.