Modeling and predicting mortality rates are crucial for managing and mitigating longevity risk in pension funds. To address the impacts of extreme mortality events in forecasting, researchers suggest directly fitting a heavy-tailed distribution to the residuals in modeling mortality indexes. Since the true mortality indexes are unobserved, this fitting relies on the estimated mortality indexes containing measurement errors, leading to estimation biases in standard inferences within the actuarial literature. In this paper, the empirical characteristic function (ECF) technique is employed to fit heavy-tailed distributions to the time series residuals of mortality indexes and normal distributions to the measurement errors. Through a simulation study, we empirically validate the consistency of our proposed method and demonstrate the importance and challenges associated with making inferences in the presence of measurement errors. Upon analyzing publicly available mortality datasets, we observe instances where mortality indexes may follow highly heavy-tailed distributions, even exhibiting an infinite mean. This complexity adds a layer of difficulty to the statistical inference for mortality models.