‎As an application of the improved Cauchy-Schwartz inequality due to Walker (Statist. Probab. Lett. (2017) 122:86-90)‎, ‎we obtain an improved version of the Cramer-Rao inequality for randomly censored data derived by Abdushukurov and Kim (J. Soviet. Math. (1987) pp. 2171-2185)‎. ‎We derive a lower bound of Bhattacharya type for the mean square error of a parametric function based on randomly censored data‎.
 

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