A New Four-Parameter Inverse Weibull Model: Statistical Properties and Applications

A four-parameter Type II Topp Leone generalized inverse Weibull (TIITLGINW) model is suggested. e reliability study of the new model is provided. Quantiles, moments, moment generating function, and probability weighted moment are some of the mathematical properties being researched. e maximum likelihood (ML) estimate is employed for TIITLGINW parameters. A simulation study is conducted to estimate the model parameters of the TIITLGINWmodel. A single real-world collection of data is used to analyze TIITLGINW’s signicance and accessibility.


Introduction
Because of its failure rate, the inverse Weibull (INW) model has a broader applicability in the eld of dependability and biological investigations. Keller and Kanath [1] proposed the INW model to investigate the form of the density function (pdf ) and hazard rate function (FUN) (hrf ). e INW model ts numerous datasets through terms of the time required for an insulated uid to decompose, the topic of which led to the action of continuous tension. Nelson and Jiang et al. [2,3] presented Weibull (W) and INW mixing models [4]. Models with two INW models were examined [5]. e exibility of the INW model was investigated [6]. Bayesian and maximum likelihood estimates of the INW parameters with progressive type-II censoring were examined. e probability density function (pdf) and cumulative FUN (cdf) of the generalized INW (GINW) model are given by [7] f(z; μ, η, c) cημ η z η+1 e − c(μ/z) η , z, μ, η, c > 0, and F(z; μ, η, c) e − c(μ/z) η , z, μ, η, c > 0.
e INW model has recently been introduced in statistical theory literature [8]. A modi ed INW model was suggested, while Shahbaz et al. [9] proposed the Kumaraswamy INW model. Hanook et al. [10] developed the beta INW model, whereas Khan et al. [11] investigated features of the transmuted INW model [12]. e Topp Leone (TL) INW model was presented [13]. e beta generalized INW geometric model was investigated. Alkarni et al. [14] proposed the extended INW model. Even power weighted generalized INW distribution was studied by Mutlk and Al-Dubaicy [15], Algarni et al. [16] proposed classical and Bayesian estimation of the INW distribution under progressive type-I censoring scheme, Al-Moisheer et al. [17] discussed the odd inverse power generalized Weibull generated family of distributions, and Ahmadini et al. [18] studied estimation of the constant stress partially accelerated life test for INW distribution with type-I censoring.
Ahmadini et al. [18] investigated the Type II TL class (TIITL) class of models. In addition, Elgarhy et al. [19] developed a three-parameter TIITLINW model. e TIITL class cdf is supplied via e equivalent pdf to (3) is produced via e main goal of this study is as follows: To introduce a new four-parameter model which is called Type II Topp Leone generalized inverse Weibull model e suggested model is very flexible and contains many submodels e suggested model has closed form of quantile e pdf of the suggested model can be unimodal and right skewness. Also, the hazard rate function can be increasing and J-shaped. e following is how this document is structured. e Section 2 defines the new model (which is a broad model).
e Section 3 investigates the linear formulation of the TIITLGINW model's pdf. Section 4 investigates statistical characteristics. In the Section 5, the ML estimation approach is used to generate the estimates of the TIITLGINW parameters. In Section 6, a simulation study is carried out to determine the model parameters of the TIITLGINW model. Section 7 employs the study of a single real-world data collection. Section 8 has concluding observations.

The New Model
Inside this section, we develop the TIITLGINW model, a novel lifespan model. e cdf of the TIITLGINW model with set of parameters φ � (μ, η, ρ, c) is computed by inserting (2) into (3) as follows: Inserting (1) and (2) into (4) yields the matching pdf to (5): where μ and c are the scale parameters and ρ, η are the two shape parameters. e TIITLGINW model is a highly adaptable model that contains several additional models. e submodels of the TIITLGINW model are given in Table 1. Figure 1 shows different TIITLGINW pdf graphs for appropriate parameter combinations.

Useful Expansion
Inside this part, we propose two useful pdf and cdf expansions for the TIITLGINW model. Now, examine the binomial series: As a result of using (11), the accompanying phrase within (6) can be indicated: After several simplifications, we arrive at where Also, the expansion of cdf can be expressed as follows: en, where

Fundamental Mathematical Features
Numerous statistical features of the TIITLGINW model are obtained in this section.

Quantile Function.
e quantile FUN of Z, denoted by Z u , is determined via

Different Types of Moments.
e r th moment (Mom) of Z may be determined utilizing relation.
Simply replacing (13) within (17) results in Suppose y � (μ/z i ) η ; after that, en, μ r ′ becomes e TIITLGINW model's Mom generating FUN is provided by e probability weighted Mom (PrWMs) may be computed as follows: Trying to insert (13) and (6) within (22), we get As a result, the PrWM of the TIITLGINW model looks like

ML Method of Approach
e ML estimates (MLEs) of the unknown parameters for such TIITLGINW model are produced using complete samples. Assume Z 1 , . . . , Z n be seen from the TIITL-GINW model with something like a certain number of parameters φ � (μ, η, ρ, c) T . e total log-likelihood (LL) FUN for the vector of parameters φ may be phrased as ln L(φ) � n ln 2 ρ + n ln c + n ln η + nη ln μ e MLEs of the φ parameters are then produced via assigning U(φ) � 0 and calculating them.

Numerical Outcomes
Comparing the theoretical performances of alternative estimators MLE for the TIITLGINW model is extremely challenging. Mathematica 9 software is used to do a numerical analysis. e experiments take into account different sample sizes of n � 30, 50, and 200, and furthermore, the various values of the φ parameters. e study will indeed be repeated 5000 times in total. In each experiment, ML estimation techniques will be utilized to provide parameter estimates. As a consequence of these experiments, the MLEs and mean square errors (C1) for the various estimators will be reported.

Modelling
roughout this section, we test the adaptability of the TIITLGINW model by applying it to a real-world data collection.
e TIITLGINW model is compared to the TIITLGIR, TIITLIE, GINW, GIR, and IE models. e used data are reported in [20], and it is 2.7, 4.  Tables 2 and 3 provide the ML estimates as well as the standard errors (C2) of the model parameters. Analytical metrics such as 2LL (C3), Kolmogorov-Smirnov (C4), and p value (C5) are included in the identical tables.
e fits of the TIITLGINW model to the TIITLGIR, TII-TLIE, GINW, GIR, and IE models are compared and given in Table 3. e statistics in these tables demonstrate that the  TIITLGINW model has the lowest C3 and C4 values and the highest C5 of any fitted model. As a result, it may be picked as the best model. Figure 3 shows the fitted pdf and estimated cdf plots for the TIITLGINW model.

Summary and Conclusion
e TIITLGINW model, a novel four-parameter model, is presented throughout this study. Simply put, TIITLGINW pdfs are a linear combination of GINW densities. We compute accurate formulations for certain of its statistical properties. We look into estimation using machine learning. e proposed model outperforms certain rival models in terms of fit when tested on real data. Also, in the future work, many authors can use this model to generalize it or study it as statistical inference using censored schemes.

Data Availability
e data used to support this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.