Statistical Insights: Analyzing Shock Models, Reliability Operations and Testing Exponentiality for NBRUmgf Class of Life Distributions

Abstract


Introduction
Classes of life distributions encompass various categories or types of probability distributions utilized to model lifetimes, durations, or intervals between occurrences across diverse fields like actuarial science, survival analysis, and reliability engineering.These distributions aid in comprehending and forecasting the lifespan or duration of entities or events.They establish a unified scientific framework, facilitating collaboration among scientists engaged in aging studies across different disciplines.As a result, statisticians have organized life distributions into distinct classes, delineating aging characteristics.These classes find widespread application in multiple domains, including medicine, engineering, industry, agriculture, among others.
However, before delving into specifics, let's briefly recap some standard notions of stochastic orderings and aging concepts that are under consideration in this paper.
Another significant ordering method extensively utilized in life and reliability testing is as follows: a) Moment generating function order (denoted by  ≤  ), if which can be written as: The mentioned ordering involves comparing the ratios of survival functions, ensuring that this ratio either increases or remains constant as the variable 'x' increases.This comparison indicates the superiority of one distribution over the other in terms of reliability properties.For further details regarding multiple classes and their testing, please refer to works by Gadallah et al. [11], Mahmoud et al. [12], Abu-Youssef et al.
Fortunately, the previously mentioned orderings have been utilized in the analysis of lifespan distributions, providing new definitions and descriptions of aging classes.When discussing aging, we refer to the statistical phenomenon where an older system typically exhibits a comparatively shorter remaining lifetime than a younger one.

Definitions and Preliminaries
Join us on this journey as we navigate the terrain of renewal classes, revealing their role in shaping the reliability landscape.
From defining the core concepts to exploring their applications in diverse fields, this article is your compass through the captivating The renewal survival function is provided by  ̅ () = It's clear that equation ( 6) is synonymous with, Then, we have the following implication: NBRU ⊂ NBRUmgf ⊂ NBRUE

Preservation results
An effective approach for leveraging reliability class properties in systems analysis involves conducting a RAMS analysis.By employing the attributes inherent in reliability classes, it becomes possible to assess the evolution of a system's failure rate over time and comprehend its impact on availability and maintainability.
The processes of convolution, blending, and constructing cohesive systems within a specific category of life distributions are often highly regarded as essential reliability measures.
Studies have shown the closure of NBRUmgf under these operations.

a) Convolution
The convolution property within reliability classes asserts that a system exhibiting a specific reliability class, such as NBRUmgf, will maintain that class when convolved with other systems sharing the same reliability class.

Theorem 1:
The convolution operation maintains closure within the NBRUmgf class of life distributions.

Proof:
If  1 and  2 belong to the NBRUmgf class, then we obtain the following : This demonstration illustrates that the NBRUmgf does not maintain closure under the convolution property.

b) Mixture of NWRUmgf:
A system characterized as a mixture comprises diverse components selected randomly based on a specified probability distribution.Reliability classes serve as a means to articulate changes in a system's failure rate over time.An application of the mixture attribute within reliability classes is in the analysis of systems comprising varied components, each with distinct lifetimes and failure rates.The mixing property of reliability classes asserts that a system with a specific reliability class, such as NWRUmgf will retain that classification when formed by any combination of systems sharing the same reliability class.
Using Chebyshev's inequality Then we conclude that the NWRUmgf class is preserved under mixture.

c) Mixing
It is clear that the NBRUmgf class is not preserved under mixing.

d) Building coherent systems:
When every element within the system holds importance and the structural function, indicating the system's performance concerning each element's function, is on the rise, the system is identified as coherent.Engineers in design prioritize the establishment of coherent systems.
For more insights on coherent systems, refer to (Barlow and Proschan [5]).
The theorem below establishes the closure property of the NBRUmgf class under employing the operation to construct a coherent system.Proof: Let  1 ,  2 , … ,   be independent NBRUmgf then we have The proof is now concluded .

Testing exponentiality
In order to construct our testing exponentiality, we will integrate both sides of Eq. ( 6) as defined in definition (4) concerning t across the interval [0, ∞), Following several computations, we obtain The test that follows is predicated on a sample  1 ,  2 , … ,   from a population with distribution F, we test Using Δ() as a deviation measure from  0 yields, Take note that whereas Δ() > 0 under  1 and 0 under  0 , we use the following to guarantee the scale invariance of the test As stated in Eq. ( 8), the empirical estimate of Δ ̂() is Let, and define the symmetric kernel as

𝑅
Where the total includes all of   and   arrangements.This demonstrates that the   -statistic provided by Δ ̂() is identical to ∑ ( 1 ,  2 ).

The Pitman Asymptotic Efficiencies (PAE's) of 𝜹(𝒔)
The Pitman asymptotic efficiencies (PAEs) for the Makeham, Weibull, and Linear failure rate families (LFR) are calculated in this section.

Monte Carlo Null Distribution Critical Points
In this section, we calculate the lower and upper percentiles of Δ ̂() given in Eq. ( 9) based on 10000 simulated samples of size n = 5(50)5, as in Tables 3 and 4.  it's noticeable that the behavior of critical values tends to approach a normal distribution as the sample size increases.

Applications
In order to showcase the relevance of the study's conclusions, we utilize distinct real-world datasets at 95% confidence level.
The following results illustrate the effectiveness of our test across various types of real data.Data-set #1.Take into consideration the data set in Murthy et al. [27], which represents the time takes for thirty repairable components to fail (see Figure 3).We obtain Δ ̂() = 0.46555 at  = 0.1 and Δ ̂() = 1.06107 at  = 0.05 in this instance.
These results fall in the reject region of  0 .Then, we can reject the exponential property of this data, at  = 0.05.

Theorem 2 :
An NBRUmgf series, composed of n independent components belonging to new better than renewal used in moment generating function order (NBRUmgf) class, collectively constitutes an NBRUmgf.

.Theorem 3 :
NBRUmgf class of life distribution, then we obtainThe proof is now concluded.e)Applications: Shock model applicationThe stochastic model known as the homogeneous Poisson shock model delineates system failures resulting from random shocks conforming to the homogeneous Poisson process.This process involves counting independent events occurring at a constant rate within a specified time period.In the shock model, each event induces damage to the system, and system failure ensues when the cumulative damage surpasses a predetermined threshold.Applications of Poisson homogeneous shock models extend to various domains, including modeling phenomena such as insurance claims, health deterioration, machinery breakdowns, or automobile accidents arising from random mechanical failures occurring consistently over time.Consider a device subjected to shocks, where N(t) represents the count of shocks occurring within the time interval (0,  ].The arrival of the  ℎ shock is denoted by the time   .Let   =  +1 −   denote the time be the time between the  ℎ and ( + )  shocks.We consider that  1 ,  2 , … are iid distributed according to .Let   () = (() = ),  = 1, 2, … and define ̅  as the device's chance of surviving  shocks.Subsequently, the system's survival probability up to time  is  ̅ () = ∑   () ̅  ∞ =0  is NBRUmgf implies H is NBRUmgf.Proof: Note that  ̅ () can be expressed as  ̅ () = ∑  ̅  ()   ∞ =1 Where   =  ̅ −1 −  ̅  ,  = 1, 23, … and   is the distribution function of   , and ∫ ∫    ̅ ( + )

Table 1 .
Includes the asymptotic efficiencies of Δ() at different values of s.

Table 1
[26]s that the PAE's of Δ(s) decrease as s increases, and that the Makeham distribution has higher PAE's than the LFR, Weibull, and Gamma distributions.Contrasting this values with others that could be relevant to this issue.Here, tests  ̂() which represented by Hassan and Said[26], at  = 0.04 , 0.2, is our choice.