In operational risk measurement, the estimation of severity distribution parameters is the main driver of capital estimates, yet this remains a nontrivial challenge for many reasons. Maximum likelihood estimation (MLE) does not adequately meet this challenge because of its well-documented nonrobustness to modest violations of idealized textbook model assumptions: specifically, that the data is independent and identically distributed (iid), which is clearly violated by operational loss data. Yet, even using iid data, capital estimates based on MLE are biased upward, sometimes dramatically, due to Jensen's inequality. This overstatement of the true risk profile increases as the heaviness of the severity distribution tail increases, so dealing with data collection thresholds by using truncated distributions, which have thicker tails, increases MLE-related capital bias considerably. Truncation also augments correlation between a distribution's parameters, and this exacerbates the nonrobustness of MLE. This paper derives influence functions for MLE for a number of severity distributions, both truncated and not, to analytically demonstrate its nonrobustness and its sometimes counterintuitive behavior under truncation. Empirical influence functions are then used to compare MLE with robust alternatives such as the optimally bias-robust estimator (OBRE) and the Cramér-von Mises estimator. The ultimate focus, however, is on the economic and regulatory capital estimates generated by these three estimators. The mean adjusted single-loss approximation is used to translate these parameter estimates into value-at-risk-based estimates of regulatory and economic capital. The results show that OBREs are very promising alternatives to MLE for use with actual operational loss event data, whether truncated or not, when the ultimate goal is to obtain accurate (nonbiased) and robust capital estimates.