The cumulative residual extropy (CRJ) is a measure of uncertainty that serves as an alternative to extropy. It replaces the probability density function with the survival function in the expression of extropy. This work introduces a new concept called normalized dynamic survival extropy (NDSE), a dynamic variation of CRJ. We observe that NDSE is equivalent to CRJ of the random variable of interest $X_{[t]}$ in the age replacement model at a fixed time $t$. Additionally, we have demonstrated that NDSE remains constant exclusively for exponential distribution at any time. We categorize two classes, INDSE and DNDSE, based on their increasing and decreasing NDSE values. Next, we present a non-parametric test to assess whether a distribution follows an exponential pattern against INDSE. We derive the exact and asymptotic distribution for the test statistic $\widehat{\Delta}^*$. Additionally, a test for asymptotic behavior is presented in the paper for right censoring data. Finally, we determine the critical values and power of our exact test through simulation. The simulation demonstrates that the suggested test is easy to compute and has significant statistical power, even with small sample sizes. We also conduct a power comparison analysis among other tests, which shows better power for the proposed test against other alternatives mentioned in this paper. Some numerical real-life examples validating the test are also included.

翻译：累积残差外熵（CRJ）是一种不确定性度量，可作为外熵的替代指标。它在外熵的表达式中用生存函数替换了概率密度函数。本文引入了一个称为归一化动态生存外熵（NDSE）的新概念，这是CRJ的一种动态变体。我们观察到，在固定时间$t$的年龄替换模型中，NDSE等价于目标随机变量$X_{[t]}$的CRJ。此外，我们证明了NDSE在任何时间仅对指数分布保持恒定。根据NDSE值的递增和递减特性，我们将其分为INDSE和DNDSE两类。接着，我们提出了一种非参数检验方法，用于评估分布是否遵循相对于INDSE的指数模式。我们推导了检验统计量$\widehat{\Delta}^*$的精确分布和渐近分布。此外，本文还针对右删失数据提出了渐近行为的检验方法。最后，通过模拟计算确定了精确检验的临界值和功效。模拟结果表明，即使在小样本情况下，所提出的检验方法也易于计算且具有显著的统计功效。我们还与其他检验方法进行了功效比较分析，结果显示本文提出的检验方法在面对文中提及的其他备择假设时具有更优的功效。文中还包含了一些验证该检验方法的实际数值案例。