This paper focuses on nonparametric statistical inference of the hazard rate function of discrete distributions based on $\delta$-record data. We derive the explicit expression of the maximum likelihood estimator and determine its exact distribution, as well as some important characteristics such as its bias and mean squared error. We then discuss the construction of confidence intervals and goodness-of-fit tests. The performance of our proposals is evaluated using simulation methods. Applications to real data are given, as well. The estimation of the hazard rate function based on usual records has been studied in the literature, although many procedures require several samples of records. In contrast, our approach relies on a single sequence of $\delta$-records, simplifying the experimental design and increasing the applicability of the methods.

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