A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy

TL;DR

This paper introduces a new goodness-of-fit test based on normalized dynamic survival extropy to assess exponentiality, demonstrating its effectiveness and power through theoretical derivations, simulations, and real-life data examples.

Contribution

It proposes the normalized dynamic survival extropy (NDSE) as a novel measure and develops a new statistical test for exponentiality based on NDSE, including its distribution and power analysis.

Findings

The test effectively detects deviations from exponential distribution.

The proposed test has higher power than existing methods in simulations.

It is computationally simple and performs well with small samples.

Abstract

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.