  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.
