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.

中文翻译：

---

基于相干系统寿命的第k个较低的记录值，扩展了累积熵

Kayal（Probab。Eng。Inf。Sci。30（4）：640-662，2016）首先提出了基于较低记录值的广义累积熵。由Kayal（Probab。Eng。Inf.Sci.30（4）：640–662，2016）推动，最近，Tahmasebi和Eskandarzadeh（Stat。Probab。Lett。126：164–172，2017）提出了扩展的累积熵（ ECE）基于第k个较低的记录值。在本文中，我们获得了ECE的一些属性。我们研究具有相同分布组件的相干系统生命周期中的这种信息度量。我们为系统寿命定义了条件ECE，并讨论了它的一些属性。我们还使用此想法提出了一种扩展累积的过去误差的方法。最后，我们使用经验方法提出这些措施的估计量。此外，我们研究了这些估计量的大样本属性。