Modelling to Engineering Data Using a New Class of Continuous Models

In this paper, a new flexible generator of continuous lifespan models referred to as the Topp-Leone Weibull G (TLWG) family is developed and studied. Several mathematical characteristics have been investigated. The new hazard rate of the new model can be “monotonically increasing,” “monotonically decreasing,” “bathtub,” and “J shape.” The Farlie Gumbel Morgenstern (FGM) and the modified FGM (MFGM) families and Clayton Copula (CCO) are used to describe and display simple type Copula. We discuss the estimation of the model parameters by the maximum likelihood (MLL) estimations. Simulations are carried out to show the consistency and efficiency of parameter estimates, and finally, real data sets are used to demonstrate the flexibility and potential usefulness of the proposed family of algorithms by using the TLW exponential model as example of the new suggested family.


Introduction and Motivation
There has already been a great emphasis on building more flexible distributions in the recent past. To simulate reallife data in various practical disciplines, including finance, engineering, medical sciences, biological research, environmental studies, and insurance, over the last few decades, a variety of G families of distributions has been constructed and researched. We have generated numerous kinds of distributions by generalizing G families. With these new families, at least one shape parameter is merged with the baseline one, allowing for greater versatility, for instance, the generalized transmuted exponentiated G [1], Weibull (W) G (WG) [2], the Burr type X-G by [3], Type II half logistic G [4], exponentiated transmuted G by [5], a new compound G family [6], the beta W G by [7], the generalized odd W G [8], the transmuted W G by [9], a new W G [10], TL G [11], a special generalized mixture class of probabilistic models [12], sine Topp-Leone G [13], Type 2 power Topp-Leone G [14], a new version of Power Topp-Leone G [15], and Type 2 generalized Topp-Leone G [16], among others.
According to [11], the cumulative distribution function (CDF) of the TL G (TLG) class could well be found with where G Φ ðzÞ refers to the CDF of the baseline model, and the corresponding density function (PDF) of (1) can be derived as where G Φ ðzÞ = dG Φ ðzÞ/dz refers to the PDF of the baseline model. According to [2], the WG family's CDF may be computed via where O Φ ðzÞ = G Φ ðzÞ/ G Φ ðzÞ and G Φ ðzÞ = 1 − G Φ ðzÞ: Then, using (3) and (1), the CDF of the TL W G (TLWG) class may indeed be expressed via The corresponding PDF is The hazard rate function (HRF) can be easily derived using h α,β,Φ ðzÞ = f α,β,Φ ðzÞ/½1 − F α,β,Φ ðzÞ: The function h α,β,Φ ðzÞ is called the failure rate of F α,β,Φ ðzÞ and often referred to as the rate function or the intensity function or failure rate or instantaneous failure rate; in actuarial mathematics, it is named the "force of mortality," and in demographic disciplines, it is called the "mortality rate." It represents the failure intensity of an x-year-old equipment. It denotes the likelihood of an operational item failing in the next time period or the likelihood of a failure in a tiny unit interval of time ðz, z+Δz given that no failure has occurred in ½0, z and satisfies h α,β,Φ ðzÞ > 0 and Ð ∞ 0 h α,β,Φ ðz Þdx = 0. The HRF is critical because it intuitively translates as the level of risk associated with an object that has lived to time x. In life (death) tables, h α,β,Φ ðzÞ is approximated by the probability that a certain individual of age x will die during the next year. Some notions of aging refer to the HR such as h / α,β,Φ ðzÞ > 0 means positive aging, h / α,β,Φ ðzÞ = 0 means no aging, and h / α,β,Φ ðzÞ < 0 means negative aging. The idea of aging in statistical lifetime and reliability analysis does not imply that the unit grows older in the sense of time. Rather, it is a concept associated with residual life.

Important Expansions
Take a look at the binomial series expansion provided as Then, the PDF in (5) can be expressed as Applying the power series expansion to AðzÞ, we have where but Then, the f α,β,Φ ðzÞ can be written as Journal of Function Spaces and h β * ðzÞ = β * g Φ ðzÞG Φ ðzÞ β * −1 depicts the PDF of the exponentiated G (ExG) distribution with parameter β * .

Via FGM Family. Starting with the joint CDF for the FGM family of random variables (RVrs
where    [17][18][19][20][21][22]) where _ ΦðuÞ = uΦðuÞ and _ ΨðvÞ = vΨðvÞ. Where ΦðuÞ and ΨðvÞ are two absolutely continuous CDFs on ð0, 1Þ where Φð0Þ = Ψð0Þ = Φð1Þ = Ψð1Þ = 0, let Then, mlnðεβ, ξηÞ ≥ 1, where ∂ _ ΦðuÞ/∂u = ΦðuÞ + ðu∂Φ ðuÞ/∂uÞ, Type I MFGM: Consider ΦðuÞ and ΨðvÞ as defined above, then where Type II MFGM: Let Then, the corresponding bivariate Copula can be derived directly from Type III MFGM: The CDF of the bivariate Type III MFGM model can be derived from 3.3. Via CCO. The CCO is a weighted variant of the CCO, which has the following form: Then, setting Then, A simple d-dimensional expansion of the above will be as follows: Recently, many new articles are allocated to study some of these types, see [23,24].

Structural Properties of the TLWG Family
4.1. Quantile Function. The TLWG quantile function (QuF), say z = QðuÞ, might be obtained by filliping (4); we have We can easily generate z by taking u as a uniform RVr in ð0, 1Þ.

Moments.
The r th moment (MO) of TLWG could be acquired in the prescribed sequence: where I ð0,∞Þ ðβ * Þ = Ð ∞ 0 z r h β * ðzÞdz is the r th moment of the ExG model using parameter β * : where I ð0,tÞ ðβ * , s, tÞ = Ð t 0 z s h β * ðzÞdz is the s th lower ICMO of ExG model with exponential parameter β * . Similarly, the s th upper ICMO of TLWG is where I ðt,∞Þ ðβ * , s, tÞ = Ð ∞ t z s h β * ðzÞdz is the s th upper ICM of ExG model with exponential parameter β * :

Bonferroni and Lorenz Curves. A positive RVr Z is described by the following Lorenz curve
where q = G −1 ðpÞ: Also, the Bonferroni curve is defined by There are numerous uses for the Bonferroni curve in economics to analyze income and poverty, as well as dependability, medical, and insurance areas.

Special Cases
In this part, we will look at various TLWG family-specific situations. We provide six TLWG family special models equivalent to the baseline exponential (E), Weibull (W), Lomax (Lx), Burr-X (BX), log-logistic (LL), and Lindley (L) distributions. The odd ratio O Φ ðzÞ of these baseline models along with the new models are listed in Table 1.

Maximum Likelihood (MLL) Estimation
Suppose z 1 , ⋯, z n be an n th random sample from the TLWG class provided via (5). Take P = ðα, β, δÞ T become the vector of parameters. The total log-likelihood (LLL) function for P is L n P ð Þ = n log 2βα where The LLL can really be optimized immediately employing SAS software or the R-language, or implicitly through solving nonlinear LL formulas acquired through differentiating (34). The score function's related components U n ðψÞ = ð∂L n ðPÞ/∂α, ∂L n ðPÞ/∂β, ∂L n ðPÞ/∂ΦÞ T are where δ k is the kth member of the parameter vector δ.
The MLL estimation (MLLE) of P is achieved through solving the nonlinear equations U n ðPÞ = 0.

Graphical Simulations
(i) We could perform numerical simulations to visually analyze the finite sample performance of the MLLEs utilizing biases (Bs) and mean squared errors (MSEs). For the assessment, the basic procedure had been used (ii) Generate N = 1000 samples of size nj ðn=50,100,⋯,500Þ from the TLWE model using (7) (iii) Compute the MLLEs for N = 1000 samples (iv) Compute the standard errors (SErs) of the MLLEs for the 1000 samples (v) Compute the Bs and MSErs given for P = α, β, θ The biases (left boxes) and MSEs (right windows) for the parameters are shown in Figures 3-5. The plots on the left demonstrate how the three biases grow with large sample n , while the graphs on the right platform how the three MSEs change with n. The zero biases are depicted by the broken line as shown in Figure 1. From Figures 3-5, the biases with each parameter are typically negative and eventually drop to 0 as n tends to infinity the MSEs with each parameter decrease to 0 as tends to infinity. 10 Journal of Function Spaces Table 3 gives the MLLEs, SErs, and C.I. values for the survival time data. Table 4 illustrates the C 1 , C 2 , C 3 , C 4 , C 5 , C 6 , D1, and D2 for the relief time data. Table 5 refers to the C 1 , C 2 , C 3 , C 4 , C 5 , C 6 , D1, and D2 for the survival time data. Figure 7 gives the P-P plot, Kaplan-Meier survival (KMS) plot, fitted PDF (FPDF), and FCDF for the 1 st data. Figure 8 offers the P-P plot, KMS plot, fitted PDF (FPDF), and FCDF for the 2 nd data.
The TLWE model is much better than many common competitive models such as the exponential (standard version), MOE, OLE, LBHE, MomE, GMOE, KwE, MOKwE, and KwMOE models. As a result, the new lifespan model offers a practical alternate to all these models. For both data sets, we can see from Figures 7 and 8 that the TLWE model fits the two real data sets well.

Discussion and Concluding Remarks
We described and explored the TLWG family, a novel generator of continuous lifespan distributions, in this work. Statistical attributes of the family are offered, such as density function expansion, moments, incomplete moments, mean deviation, and Bonferroni and Lorenz curves. The new HRF might be described as "monotonically rising," "bathtub," "J shape," or "monotonically declining." FGM and MFGM families and CCO are often used to describe and visualize Copula of the basic kind. Regarding estimating model parameters, we glance at the MLL methodology. We conducted simulated studies to examine the limited sample behavior of MLL estimations utilizing graphs, biases, and mean squared errors. Two applications to actual data sets demonstrate the relevance and versatility of the intended family.
As a future work, we can apply many new useful goodness-of-fit tests for right-censored validation such as the Nikulin-Rao-Robson goodness-of-fit test, modified Nikulin-Rao-Robson goodness-of-fit test, Bagdonavicius-Nikulin goodness-of-fit test, and modified Bagdonavicius-Nikulin goodness-of-fit test. However, some bivariate versions could be studied in more details.

Data Availability
If you would like to get the quantitative data set utilized to conduct the investigation described in the publication, please notify the person author.

Conflicts of Interest
The authors declare no conflict of interest.