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Summary Discrete power-law distributions have significant consequences for understanding many phenomena in practice, and have attracted much attention in recent decades. However, in many practical applications, there exists a natural upper bound for the probability tail. In this paper, we develop maximum likelihood estimates for truncated discrete power-law distributions based on the upper order statistics, and large sample properties are mentioned as well. Monte Carlo simulation is carried out to examine the finite sample performance of the estimates. Applications in real cyber attack data and peak gamma-ray intensity of solar flares are highlighted.