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Kayal (Probab. Eng. Inf. Sci. 30(4):640–662, 2016) first proposed generalized cumulative entropy based on lower record values. Motivated by Kayal (Probab. Eng. Inf. Sci. 30(4):640–662, 2016), recently, Tahmasebi and Eskandarzadeh (Stat. Probab. Lett. 126:164–172, 2017) proposed an extended cumulative entropy (ECE) based on kth lower record values. In this paper, we obtain some properties of ECE. We study this measure of information for the coherent systems lifetime with identically distributed components. We define the conditional ECE for the system lifetime and discuss some properties of it. We also use this idea to propose a measure of extended cumulative past inaccuracy. Finally, we propose the estimators of these measures using empirical approach. In addition, we study large sample properties of these estimators.

Extended cumulative entropy based on kth lower record values for the coherent systems lifetime