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

Authors: Nagla Yahia; Najwan Alsadat
Journal: Journal of Mathematics
 (2022)

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## Abstract

A four-parameter Type II Topp Leone generalized inverse Weibull (TIITLGINW) model is suggested. The 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. The maximum likelihood (ML) estimate is employed for TIITLGINW parameters. A simulation study is conducted to estimate the model parameters of the TIITLGINW model. A single real-world collection of data is used to analyze TIITLGINW’s significance and accessibility.


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## Body

## 1. Introduction

Because of its failure rate, the inverse Weibull (INW) model has a broader applicability in the field 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). The INW model fits numerous datasets through terms of the time required for an insulated fluid 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]. The flexibility of the INW model was investigated [6]. Bayesian and maximum likelihood estimates of the INW parameters with progressive type-II censoring were examined.The probability density function (pdf) and cumulative FUN (cdf) of the generalized INW (GINW) model are given by [7](1)fz;μ,η,γ=γημηzη+1e−γμ/zη,z,μ,η,γ>0,and(2)Fz;μ,η,γ=e−γμ/zη,z,μ,η,γ>0.The INW model has recently been introduced in statistical theory literature [8]. A modified 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]. The Topp Leone (TL) INW model was presented [13]. The 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. The TIITL class cdf is supplied via(3)Fz;μ,ω=1−1−Gx;ω2μ.The equivalent pdf to (3) is produced via(4)fz;μ,ω=2μgz;ωGz;ω1−Gz;ω2μ−1,z∈R,μ>0.The 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 modelThe suggested model is very flexible and contains many submodelsThe suggested model has closed form of quantileThe pdf of the suggested model can be unimodal and right skewness. Also, the hazard rate function can be increasing and J-shaped.The following is how this document is structured. The Section2 defines the new model (which is a broad model). The 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.


## 2. The New Model

Inside this section, we develop the TIITLGINW model, a novel lifespan model. The cdf of the TIITLGINW model with set of parametersφ=μ,η,ρ,γ is computed by inserting (2) into (3) as follows:(5)Gz;φ=1−1−e−2γμ/znρ.Inserting (1) and (2) into (4) yields the matching pdf to (5):(6)gz;φ=2ργημηz−η−1e−2γμ/zη1−e−2γμ/zηρ−1,z,ρ,μ,η,γ>0,where μ and γ are the scale parameters and ρ,η are the two shape parameters.The TIITLGINW model is a highly adaptable model that contains several additional models. The submodels of the TIITLGINW model are given in Table1.Table 1
Submodels of the TIITLGINW model.
ModelρμηγcdfAuthor1TIITLGINR———1Fz;ρ,μ,γ=1−1−e−2γμ/z2ρNew2TIITLGINE———2Fz;ρ,μ,γ=1−1−e−2γμ/zρNew3TIITLINW—1——Fz;ρ,μ=1−1−e−2μ/z2ρNew4TIITLINR—1—1Fz;ρ,μ=1−1−e−2μ/z2ρNew5TIITLINE—1—2Fz;ρ,μ=1−1−e−2μ/zρNewFigure1 shows different TIITLGINW pdf graphs for appropriate parameter combinations.Figure 1
TIITLGINW pdf graphs for appropriate parameter combinations.X’s survival FUN (sf), hrf, inverted hrf, and cumulative hrf are described as follows:(7)Rz;φ=1−e−2γμ/zηρ,z,ρ,μ,η,γ>0,(8)hz;φ=2ργημηz−η−1e−2γμ/zη1−e−2γμ/zη,(9)τz;φ=2ργημηz−η−1e−2γμ/zη1−e−2γμ/zηρ−11−1−e−2γμ/zηρ,(10)Hz;φ=−ln1−e−2γμ/zηρ.Figure2 shows different TIITLGINW hrf graphs for appropriate parameter combinations.Figure 2
TIITLGINW hrf graphs for appropriate parameter combinations.


## 3. Useful Expansion

Inside this part, we propose two useful pdf and cdf expansions for the TIITLGINW model. Now, examine the binomial series:(11)1−an=∑k=0∞−1kakak,0<a<1.As a result of using (11), the accompanying phrase within (6) can be indicated:(12)1−e−2γμ/ziηρ−1=∑i=0∞−1iρ−1ie−2γμ/ziη.After several simplifications, we arrive at(13)fz=ημη∑i=0∞Wiz−η−1e−2i+1γμ/ziη,where wi=2−1iρ−1iργ.Also, the expansion of cdf can be expressed as follows:(14)Fzh=1−1−e−2γμ/ziηρh.Then,(15)Fzh=∑k=0∞Wke−2kγμ/ziη,where wk=∑j=0h−1j+khjρjk.


## 4. Fundamental Mathematical Features

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


### 4.1. Quantile Function

The quantile FUN ofZ, denoted by Zu, is determined via(16)Zu=μ−1γln1−1−u1/ρ−1/η.


### 4.2. Different Types of Moments

The rth moment (Mom) of Z may be determined utilizing relation.(17)μr′=EZr=∫−∞∞zrgz;φdz.Simply replacing (13) within (17) results in(18)μr′=∑i=0∞Wi∫0∞zr−η−1e−2i+1γμ/ziηdz.Supposey=μ/ziη; after that,(19)μr′=μr∑i=0∞W∫i∞ye−2i+1γydy.Then,μr′ becomes(20)μr′=μr∑i=0∞WiΓ1−r/η2i+1γ1−r/η,rη<1.The TIITLGINW model’s Mom generating FUN is provided by(21)MZt=∑r=0∞trr!EZr=∑r,i=0∞trr!wiΓ1−r/β2i+1γ1−r/β,r/β<1.The probability weighted Mom (PrWMs) may be computed as follows:(22)τr,s=EZrGzs=∫−∞∞xrgzGzsdz.Trying to insert (13) and (6) within (22), we get(23)τr,s=∑i,k=0∞WiWk∫0∞zr−η−1e−2i+k+1γμ/ziηdz.As a result, the PrWM of the TIITLGINW model looks like(24)τr,s=μr∑i,k=0∞WiWkΓ1−r/η2i+k+1γ1−r/η,rη<1.


## 4.1. Quantile Function

The quantile FUN ofZ, denoted by Zu, is determined via(16)Zu=μ−1γln1−1−u1/ρ−1/η.


## 4.2. Different Types of Moments

The rth moment (Mom) of Z may be determined utilizing relation.(17)μr′=EZr=∫−∞∞zrgz;φdz.Simply replacing (13) within (17) results in(18)μr′=∑i=0∞Wi∫0∞zr−η−1e−2i+1γμ/ziηdz.Supposey=μ/ziη; after that,(19)μr′=μr∑i=0∞W∫i∞ye−2i+1γydy.Then,μr′ becomes(20)μr′=μr∑i=0∞WiΓ1−r/η2i+1γ1−r/η,rη<1.The TIITLGINW model’s Mom generating FUN is provided by(21)MZt=∑r=0∞trr!EZr=∑r,i=0∞trr!wiΓ1−r/β2i+1γ1−r/β,r/β<1.The probability weighted Mom (PrWMs) may be computed as follows:(22)τr,s=EZrGzs=∫−∞∞xrgzGzsdz.Trying to insert (13) and (6) within (22), we get(23)τr,s=∑i,k=0∞WiWk∫0∞zr−η−1e−2i+k+1γμ/ziηdz.As a result, the PrWM of the TIITLGINW model looks like(24)τr,s=μr∑i,k=0∞WiWkΓ1−r/η2i+k+1γ1−r/η,rη<1.


## 5. ML Method of Approach

The ML estimates (MLEs) of the unknown parameters for such TIITLGINW model are produced using complete samples. AssumeZ1,…,Zn be seen from the TIITLGINW model with something like a certain number of parameters φ=μ,η,ρ,γT. The total log-likelihood (LL) FUN for the vector of parameters φ may be phrased as(25)lnLφ=nln2ρ+nlnγ+nlnη+nηlnμ−η+1∑i=1nlnzi−2γ∑i=1nμziη+ρ−1∑i=1nln1−e−2γμ/ziη.The scoring FUNUφ=Uμ,Uη,Uρ,Uγ elements are specified by(26)Uμ=nημ−2γημη−1∑i=1nzi−η+2γημη−1ρ−1∑i=1nzi−ηe−2γμ/ziη1−e−2γμ/xiη,(27)Uη=nη−nlnμ−∑i=1nlnzi−2γ∑i=1nμziηlnμzi+ρ−1∑i=1n2γμ/ziηlnμ/zie−2γμ/ziη1−e−2γμ/ziziη,(28)Uρ=nρ+∑i=1nln1−e−2γμ/ziη,(29)Uγ=nγ−2∑i=1nμziη+2ρ−1∑i=1nμ/ziηe−2γμ/ziη1−e−2γμ/ziη.The MLEs of theφ parameters are then produced via assigning Uφ  = 0 and calculating them.


## 6. 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. The experiments take into account different sample sizes ofn = 30, 50, and 200, and furthermore, the various values of the φ parameters.The 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.


## 7. Modelling

Throughout this section, we test the adaptability of the TIITLGINW model by applying it to a real-world data collection. The TIITLGINW model is compared to the TIITLGIR, TIITLIE, GINW, GIR, and IE models.The used data are reported in [20], and it is 2.7, 4.1, 1.8, 1.5, 1.1, 1.4, 1.8, 1.6, 2.2, 1.7, 1.2, 1.4, 3, 1.3, 1.7, 1.9, 1.7, 2.3, 1.6, 2.Tables2 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.Table 2
Numerous numerical values of MLEs and C1 of the TIITLGINW model.
nParMLEC1MLEC1Set 1: (0.5, 0.5, 0.5, 0.5)Set 2: (0.5, 0.5, 0.5, 0.5)30μ0.48700.00710.49910.0042η0.47110.02121.57400.2106ρ0.46640.01630.53860.0424γ0.37430.03290.89780.520450μ0.52170.00310.49170.0030η0.48620.01021.56090.1372ρ0.50950.00710.50540.0227γ0.38710.02450.77900.3332200μ0.50070.00110.50210.0016η0.48890.00331.54400.0447ρ0.50780.00270.51550.0117γ0.40730.01270.71200.0856Set 3: (0.5, 0.5, 0.5, 0.3)Set 4: (0.5, 1.5, 0.5, 0.3)30μ0.54100.01620.51180.0136η0.55250.05121.82930.6032ρ0.54550.03321.00482.5698γ0.43920.12880.45040.283350μ0.55260.00970.52800.0082η0.54780.02761.74700.2578ρ0.55510.01730.66030.1249γ0.39530.03910.35320.0251200μ0.48490.00120.48090.0017η0.44050.00611.64740.0560ρ0.47130.00280.46490.0086γ0.29320.00280.25480.0042Set 5: (0.3, 0.8, 0.3, 0.5)Set 6: (0.3, 0.5, 0.3, 0.3)30μ0.29610.00300.35160.0106η0.89370.26630.47910.0338ρ0.35630.03620.41230.0572γ0.59520.53130.17350.628650μ0.28880.00100.30490.0021η0.63980.04700.38000.0218ρ0.27080.00480.30280.0061γ0.26290.06330.19010.4063200μ0.30480.00050.30410.0003η0.73410.00920.39340.0131ρ0.31220.00170.30940.0014γ0.31840.03540.41730.0356Table 3
MLEs, C2, C3, C4, and C5 for the dataset.
ModelMLE and C2C3C4C5TIITLGINW (μ, η, ρ, γ)1.494 (195300)3.724 (2.265)1.146 (1.21)0.664 (323200)30.80.097020.99177TIITLGIR (μ, ρ, γ)1.112 (333800)3.61 (1.334)2.204 (1323000)31.7360.126420.90655GINW (μ, β, γ)1.868 (282500)4.017 (0.697)0.489 (296900)30.8170.101950.98545GIR (μ, γ)1.953 (244100)0.724 (181000)42.3650.256590.14358TIITLIE (μ, ρ)3.086 (0.543)20.766 (10.928)34.2090.154230.72826IE (μ)1.725 (0.386)65.3370.387250.00497The numerical values of MLEs, C2, C3, C4, and C5 are given in Table3.The fits of the TIITLGINW model to the TIITLGIR, TIITLIE, GINW, GIR, and IE models are compared and given in Table3. The 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.Figure 3
The empirical pdf and empirical cdf of the TIITLGINW model.


## 8. Summary and Conclusion

The 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. The 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.



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