We propose an alternative estimator of the heavy tailed mean under censored data relies with the threshold t which divides it into two parts for x ≤ t and x > t respectively. In the first parts we present an empirical distribution function under random censoring (rc). Then, we check the convergence of Lyapunov condition to ensuring the asymptotic normality of this parts. For the second parts we use the Peaks-Over-Threshold (POT) method of the tail distribution and fitting to the generalized Pareto distribution (GPD). Intentionally to give an alternative estimator of the tail expected value for the heavy tailed distribution when x > t under random censoring relies the KIB estimators of the GPD parameters. Ditto, we ensuring the asymptotic normality property of the KIB estimators with bias reduced and we present the asymptotic normality of the second parts when x > t. An asymptotically normally distributed estimate for the heavy tailed mean estimation with infinite variance under censored data is introduced by combining the two parts. Finally, numerical examples are presented at the end of the paper to demonstrate the reliability of this estimation procedure and to better illustrate the results of this paper.