# Reliability Analysis of the Inverse Modified Weibull Model with Applications

Authors: Mohamed Kayid; Salah Djemili
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper presents a novel flexible inverse modified Weibull model with a concave Weibull probability diagram that describes various reliability phenomena. We investigate some essential attributes and reliability properties of this model. The new model exhibits an increasing or unimodal form of the hazard rate function, making it more suitable for simulating certain aging classes of life distributions with appropriate choices of parameter values. We used the Weibull probability paper approach and the maximum likelihood method to estimate the model's parameters. In addition, we conduct a simulation study to investigate the behavior and efficiency of these estimators. Finally, we analyze a maintenance dataset to demonstrate the usefulness of the proposed model.


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## Body

## 1. Introduction

The well-known Weibull distribution plays an important role in many scientific fields, such as physics, electrical engineering, industrial engineering, extreme value theory, survival analysis, reliability theory, information technology, weather forecasting, communications, insurance, hydrology, and many others. Jonasz and Fournier [1], Lamon [2], and Ahmed and Chaurasiya [3] have presented examples of the applicability of the Weibull model in describing physical phenomena. In recent decades, the Weibull model has been the cornerstone of much research in reliability theory and survival analysis. The Weibull model accounts for increasing and decreasing hazard rate functions (HRF). Many modifications and extensions of the model have been presented and studied in the literature to adapt the Weibull model to situations where the HRF is not monotonic or to create more flexible models.Phani [4] applied a modified Weibull model to analyze the tensile strength of two groups of fused silica optical fibers. Mudholkar and Srivastava [5] proposed and studied an exponentiated Weibull distribution. Marshal and Olkin [6] presented a method to add a parameter to a family of distributions and applied it to extend the Weibull distribution. Mudholkar and Kollia [7] generalized the Weibull distribution by adding a parameter to the quantile function of the Weibull model. Sornette and others. [8], Malevergne and Sornette [9], and Nadarajah and Kotz [10] worked on a modification of the Weibull modification to describe asset return data. Xie and others. [11] introduced a modified Weibull extension with bathtub-shaped HRF.Lai and others. [12] proposed a modified Weibull (MW) distribution to capture bathtub-shaped HRF. Carrasco and others. [13] generalized the model of Lai and others. [12] to a four-parameter distribution. Almalki and Yuan [14] investigated a competing risk model of the Weibull and MW distributions proposed by Lai and others. [12]. Thach and Bris [15] urged that their improved distribution of the competing risk model of Almalki and Yuan [14] provides a better description for bathtub-shaped hazard rate data with sharp changes in the attrition phase. In addition, Almalki and Nadarajah [16] reviewed Weibull modifications and extensions found in the literature.This paper presents and studies a novel flexible inverse modified Weibull (IMW) model with an increasing and unimodal (upside-down bathtub shape) HRF. The Weibull probability paper (WPP) plot for a Weibull model shows a straight line, while the WPP plot corresponding to the MW distribution introduced by Lai and others. [12] offers a convex curve. However, the WPP plot of the proposed model shows a concave curve. This point can be used as a characteristic guide for modeling the data. In addition, the model benefits from its simple yet flexible form.The paper is organized as follows. In Section2, the new IMW distribution is developed and some of its properties are studied. It is shown that the HRF is increasing or unimodal. Also, the WPP plot is discussed. The parameters are estimated in Section 3 by two estimators, namely the WPP and the maximum likelihood estimator. Section 4 is devoted to a simulation study to track the behaviour and efficiency of these estimators. In Section 5, a maintenance data set is analyzed with this model to show its applicability. Finally, Section 6 concludes the paper with further extended explanations, descriptions, and recommendations for further research.


## 2. Novel IMW Model

The new IMW model with parametersa, b, and θ, IMW(a, b, θ), is defined by its reliability function(1)Rx=exp−axbexp−θx,a,b,θ>0,x≥0.It is constructed by including the expressionexp−θ/x, which depends on the inverse of the time, in the Weibull model. When θ=0, the IMW reduces to the well-known Weibull model.The probability density function (PDF) of the IMW is(2)fx=abxb−1+aθxb−2exp−θxexp−axbexp−θx,a,b,θ>0,x≥0,and its k th moment is(3)EXk=∫0∞kxk−1exp−axbexp−θxdx.This expression should be computed numerically. Clearly,EXk is finite for all parameter values and a lower bound for it is EYk where Y follows the Weibull model with reliability function Ry=exp−ayb,y≥0.LetF be the cumulative distribution function (CDF) of IMW. Then, the α-quantile of F, Qα=F−1α, can be obtained by solving the equation(4)ln−lnα¯=lna+blnQα−θQα.Since there is no algebraic solution forQα, this equation should be solved numerically.The HRF of IMW(a, b, θ) is(5)hx=abxb−1+aθxb−2exp−θx.Figure1, left side, draws this PDF and shows a unimodal shape. The right side draws the HRF, which starts from zero and eventually increases form to infinity or decreases to zero.Figure 1
The PDF (left side) and the HRF (right side) of the IMW for some parameter values. The following proposition explores different forms of the HRF.
(a)(b)Proposition 1.
The HRF of the IMW(a, b, θ) is increasing when b≥1 and unimodal (upside-down bathtub-shaped) when b<1. Moreover, when b<1, the HRF is maximized at the point(6)x∗=θb1−b−0.5−1.Proof.
In view of (5), we have(7)h′x=bb−1x2+2θb−1x+θ2xb−4exp−θx,
Thus, the HRF is proportional with the same sign to the expression(8)ηx;a,b,θ=bb−1x2+2θb−1x+θ2,which is a second degree algebraic expression with respect to x. Analysing the roots of the equation ηx:a,b,θ=0, we find that if b≥1, then ηx:a,b,θ>0 for all x≥0. But, if b<1, then ηx:a,b,θ>0 for 0≤x<x∗, ηx:a,b,θ=0, for x=x∗ and ηx:a,b,θ<0 for x>x∗ where x∗ is defined in (6).
The mean residual life (MRL) function of IMW is given as follows:(9)mx=expaxbexp−θx∫x∞exp−atbexp−θtdt.
Since the HRF is increasing or unimodal, the MRL function is suitable for decreasing or bathtub-shaped forms. Figure1, left side, shows the MRL shape for some parameter values and verifies different MRL shapes.
Theα-QRL function of the IMW is(10)qαx=F−11−α¯Rx−x,where F−1 is defined by (4). Figure 2, right side, plots the 0.5-QRL which is referred to the median residual life function, for some parameters. As the HRF is unimodal or increasing, the α-QRL function is bathtub shaped or decreasing.Figure 2
The MRL (left side) and the median residual life function (right side) of the IMW for some parameter values.
(a)(b)


### 2.1. The WPP Plot

From (1), we have(11)ln−lnRx=lna+blnx−θx.If we takey=ln−lnRx and t=lnx, we can write(12)y=α+bt−θexp−t.where α=lna. In the case of θ=0 (Weibull model), the relation between y and t shows a straight line. For θ>0, which is the case of IMW, the relation between y and t is a concave curve tending to a straight line at infinity. Figure 2, left side, shows a schematic form of the curve (11).The WPP plot can be used to characterize the IMW model. Letx1,x2,...,xn be a sample of IMW distribution, then the plot of yi=ln−lnR^xi versus ti=lnxi, i=1,2,...,n will show a concave plot. Here, R^ stands for the well-known empirical reliability function.Figure3, right side, shows the WPP plot for three simulated samples from the IMW distribution of size n=200. The figure shows that the WPP plot moves away from the corresponding straight Weibull line with increasing time as θ increases.Figure 3
The schematic form of the WPP plot for IMW(1.2, 1.2, 3) (left side) and WPP plot for three simulated samples from the IMW distribution (right side).
(a)(b)


## 2.1. The WPP Plot

From (1), we have(11)ln−lnRx=lna+blnx−θx.If we takey=ln−lnRx and t=lnx, we can write(12)y=α+bt−θexp−t.where α=lna. In the case of θ=0 (Weibull model), the relation between y and t shows a straight line. For θ>0, which is the case of IMW, the relation between y and t is a concave curve tending to a straight line at infinity. Figure 2, left side, shows a schematic form of the curve (11).The WPP plot can be used to characterize the IMW model. Letx1,x2,...,xn be a sample of IMW distribution, then the plot of yi=ln−lnR^xi versus ti=lnxi, i=1,2,...,n will show a concave plot. Here, R^ stands for the well-known empirical reliability function.Figure3, right side, shows the WPP plot for three simulated samples from the IMW distribution of size n=200. The figure shows that the WPP plot moves away from the corresponding straight Weibull line with increasing time as θ increases.Figure 3
The schematic form of the WPP plot for IMW(1.2, 1.2, 3) (left side) and WPP plot for three simulated samples from the IMW distribution (right side).
(a)(b)


## 3. Estimation

Letx1,x2,...,xn be an independent and identically distributed (iid) realization from IMWa,b,θ. In this section, we apply the maximum likelihood method and the WPP plot approach to estimate the parameters of the IMW distribution.


### 3.1. WPP Method

As explained in the previous section, takeyi=ln−lnR^xi and ti=lnxi, i=1,2,...,n where R^x=n−1∑j=1nlxi>x is the empirical reliability function. To estimate the parameters, we fit the following regression model to these data:(13)yi=α+bti−θe−ti+εi,i=1,2,...,n,and usually identically distributed and independent ϵi∼N0,σ2 are assumed. However, these assumptions have no bearing on the estimation problem but are essential for statistical hypothesis testing of parameters. Nevertheless, without checking these assumptions, the variance of this estimator can be obtained by the bootstrap resampling method.


### 3.2. MLE

The log-likelihood function is(14)la,b,θ;x=nlna+∑i=1nlnbxi+θ+b−2∑i=1nlnxi−∑i=1nθxi−a∑i=1nxibe−θ/xi,Then, the likelihood equations are(15)∂∂ala,b,θ;x=a−n∑i=1nxibexp−θ/xi=0,∂∂bla,b,θ;x=∑i=1nlnxi+1b+θ/xiexp−θ/xi−a∑i=1nxiblnxiexp−θxi=0,(16)∂∂θla,b,θ;x=−∑i=1n1xi+∑i=1n1bxi+θ+a∑i=1nxib−1exp−θxi=0.The log-likelihood expression can be maximized with respect to the parameters to calculate the MLE. In another approach, the likelihood equations can be solved to obtain the estimates.The Fisher information matrix is(17)K=E−∂2l∂a2E−∂2l∂a∂bE−∂2l∂a∂θE−∂2l∂b∂aE−∂2l∂b2E−∂2l∂b∂θE−∂2l∂θ∂aE−∂2l∂θ∂bE−∂2l∂θ2,where l=lnfX represents the log-likelihood function. The variance-covariance matrix of the MLE is approximated by n−1K−1, where K−1 is the inverse of the information matrix. Moreover, the standardized estimators nAa^−a,b^−b,θ^−θ converge asymptotically to a standard normal random vector. Note that the coefficient matrix A satisfies the condition K−1=AAT.


### 3.3. Right Censored Data

The lifetimesx1,x2,...,xn can be subjected to right censorship by a random censorship variable C. Then, xi is said to be censored when xi>ci and observed if xi≤ci, where c1,c2,…,cn follows from C. Thus, the observations consist of pairs vi,δii=1,2,…,n where vi=minxi,ci and δi=1 if the observation is not censored, xi≤ci, and δi=0 if vi has a censored item, xi>ci.For right-censored data, we should replace the empirical reliability estimator used in the WPP method with the Kaplan-Meier survival estimator, which is as follows:(18)R^x=∏s<x1−dN¯sY¯s,where dN¯s and Y¯s are the number of items failed at time s and the number of items at risk at s respectively, see Fleming and Harrington [17] for more details about Kaplan-Meier estimator.Given the censored datavi,δii=1,2,…,n, we can specify the log-likelihood function as follows:(19)la,b,θ;x=∑i=1nδilnfvi+∑i=1n1−δilnRvi,where f and R are the density and reliability functions of the IMW model, respectively. Thus, the MLE for censored data can be calculated by maximizing this function with respect to the parameters.


## 3.1. WPP Method

As explained in the previous section, takeyi=ln−lnR^xi and ti=lnxi, i=1,2,...,n where R^x=n−1∑j=1nlxi>x is the empirical reliability function. To estimate the parameters, we fit the following regression model to these data:(13)yi=α+bti−θe−ti+εi,i=1,2,...,n,and usually identically distributed and independent ϵi∼N0,σ2 are assumed. However, these assumptions have no bearing on the estimation problem but are essential for statistical hypothesis testing of parameters. Nevertheless, without checking these assumptions, the variance of this estimator can be obtained by the bootstrap resampling method.


## 3.2. MLE

The log-likelihood function is(14)la,b,θ;x=nlna+∑i=1nlnbxi+θ+b−2∑i=1nlnxi−∑i=1nθxi−a∑i=1nxibe−θ/xi,Then, the likelihood equations are(15)∂∂ala,b,θ;x=a−n∑i=1nxibexp−θ/xi=0,∂∂bla,b,θ;x=∑i=1nlnxi+1b+θ/xiexp−θ/xi−a∑i=1nxiblnxiexp−θxi=0,(16)∂∂θla,b,θ;x=−∑i=1n1xi+∑i=1n1bxi+θ+a∑i=1nxib−1exp−θxi=0.The log-likelihood expression can be maximized with respect to the parameters to calculate the MLE. In another approach, the likelihood equations can be solved to obtain the estimates.The Fisher information matrix is(17)K=E−∂2l∂a2E−∂2l∂a∂bE−∂2l∂a∂θE−∂2l∂b∂aE−∂2l∂b2E−∂2l∂b∂θE−∂2l∂θ∂aE−∂2l∂θ∂bE−∂2l∂θ2,where l=lnfX represents the log-likelihood function. The variance-covariance matrix of the MLE is approximated by n−1K−1, where K−1 is the inverse of the information matrix. Moreover, the standardized estimators nAa^−a,b^−b,θ^−θ converge asymptotically to a standard normal random vector. Note that the coefficient matrix A satisfies the condition K−1=AAT.


## 3.3. Right Censored Data

The lifetimesx1,x2,...,xn can be subjected to right censorship by a random censorship variable C. Then, xi is said to be censored when xi>ci and observed if xi≤ci, where c1,c2,…,cn follows from C. Thus, the observations consist of pairs vi,δii=1,2,…,n where vi=minxi,ci and δi=1 if the observation is not censored, xi≤ci, and δi=0 if vi has a censored item, xi>ci.For right-censored data, we should replace the empirical reliability estimator used in the WPP method with the Kaplan-Meier survival estimator, which is as follows:(18)R^x=∏s<x1−dN¯sY¯s,where dN¯s and Y¯s are the number of items failed at time s and the number of items at risk at s respectively, see Fleming and Harrington [17] for more details about Kaplan-Meier estimator.Given the censored datavi,δii=1,2,…,n, we can specify the log-likelihood function as follows:(19)la,b,θ;x=∑i=1nδilnfvi+∑i=1n1−δilnRvi,where f and R are the density and reliability functions of the IMW model, respectively. Thus, the MLE for censored data can be calculated by maximizing this function with respect to the parameters.


## 4. Simulations

Given parametersa, b and θ, a random instance is generated from IMWa,b,θ by applying the relation X=F−1U, where U follows from the standard uniform distribution and F−1 is the inverse of the CDF, defined in 4. In each run of the simulation study, for selected parameters, r=5000 replications of samples of sizes n=50, 150 and 250 are simulated. Then, for each sample the parameters are estimated by maximum likelihood or the WPP plot method. The results of the simulation study are abstracted in Table 1. For computing the MLE, the “optim” function with the default method “Nelder-Mead” in R is used. The initial values for optimizations are taken from the uniform model, e g., for a on the intervals a×0.85,1.25 and similarly for b and θ. For computing the WPP plot estimates, the built-in function “lm” of R is used. The estimates are used to compute the bias (B) and mean square error (MSE) for all parameters.Table 1
Simulation results for MLE of the parameters ofIMWa,b,θ.
nMethodParameters50150250BMSEBMSEBMSEMLE0.1, 0.8, 0.050.01340.00190.00480.00060.00340.0004−0.00880.0102−0.00440.0033−0.00370.00210.22620.25430.06010.01690.03360.00620.1, 0.8, 0.10.01290.00190.00420.00050.00250.0003−0.00760.0103−0.00340.0030−0.00140.00190.23910.25830.06790.02560.04000.01050.05, 1.2, 0.050.01960.00320.00940.00170.00600.0011−0.03630.0339−0.02500.0130−0.01690.00800.47170.88930.17560.12230.11290.05440.05, 1.2, 0.10.01700.00220.00670.00070.00390.0004−0.03110.0304−0.01820.0098−0.01180.00560.50791.05790.19000.15710.10970.06510.7, 0.5, 0.70.02650.04930.00690.01480.00450.00840.01880.01360.00720.00430.00360.00230.03720.06560.00960.02040.00630.0116WPP0.1, 0.8, 0.050.01470.00200.00360.00050.00170.0003−0.00920.01210.00050.00370.00150.00220.04380.1242−0.00420.0081−0.00740.00330.1, 0.8, 0.10.01330.00220.00230.00050.00040.0003−0.00380.01240.00460.00390.00550.00230.02330.1210-0.01810.0128−0.01750.00640.05, 1.2, 0.050.01450.00310.00330.00030.00150.0001−0.01800.0367−0.00290.01030.00090.00580.07000.6265-0.00230.0684−0.00890.03070.05, 1.2, 0.10.01270.00200.00290.00030.00130.0001-0.00650.0373−0.00030.01030.00290.00600.02940.6365−0.01830.0889−0.02050.04040.7, 0.5, 0.7−0.02360.0691−0.03360.0222−0.03570.01440.06760.02890.04110.00990.03460.0064−0.11200.0910−0.07080.0323−0.06250.0211Each cell consists ofBa and MSEa in the first row, Bb and MSEb in the second row, and Bθ and MSEθ in the third row.Fora, the B and MSE are(20)Ba=1r∑i=1ra^i−a(21)MSEa=1r∑i=1ra^i−a2.Below are some critical observations on the simulation results:(i)
The MLE and WPP plot estimators are consistent, i.e., B and MSE decrease withn(ii)
The convergence rates for the MLE and WPP plot estimators are approximately the same(iii)
The small values of B and MSE indicate that both methods are suitable for application to real-world problems


## 5. Applications

In this section, we analyze a maintenance dataset to demonstrate the usefulness of the proposed model. Table2 shows a maintenance data set reported by Chhikara and Folks [18] consisting repair times (in hours) of an airborne communication transceiver.Table 2
Repair times (in hours) of an airborne communication transceiver.
0.20.30.50.50.50.50.60.60.70.70.70.80.81.01.01.01.01.11.31.51.51.51.52.02.02.22.52.73.03.03.33.34.04.04.54.75.05.45.47.07.58.89.010.322.024.5The WPP diagram in Figure4 shows a concave shape and confirms that the proposed IMW model fits the data set. The IMW model and some alternative models have been fitted to the data, and the MLEs are computed. The selected alternatives are the Weibull distribution, the inverse Weibull (IW) distribution, the exponentiated Weibull (EW) distribution, the modified Weibull (MW) distribution offered by Lai and others. [12], the Marshal-Olkin Weibull (MOW) distribution, the modified Weibull extension (MWE) distribution offered by Xie and others. [11] and competing risk Weibull (CRW) distribution each with corresponding CDFs(22)Fx=1−exp−axb,a>0,b>0,x≥0,Fx=exp−ax−b,a>0,b>0,x≥0,Fx=1−exp−axbθ,a>0,b>0,θ>0,x≥0,Fx=1−exp−axbeθx,a>0,b>0,θ>0,x≥0,Fx=1−θexp−axb1−1−θexp−axb,a>0,b>0,θ>0,x≥0,Fx=1−expaθ1−expxbab,a>0,b>0,θ>0,x≥0,(23)Fx=1−exp−θxa−θxb,a>0,b>0,θ>0,x≥0.Figure 4
The WPP plot for data set of Table2. The empirical and estimated distribution functions of the selected models are shown in Figure 5, right side. The concave shape of this plot indicates that the proposed IMW distribution may be a good candidate for describing this data set.The AIC, Kolmogorov–Smirnov (K–S), Anderson–Darling (AD), and Cramer–Von Mises (CVM) statistics are shown for all models in Table3.Table 3
Results of fitting IMW and some alternative distributions to data set of Table2.
Modela^b^θ^AICK–Sp-valueADp-valueCVMp-valueIMW (MLE)0.75610.55940.7568105.06990.07090.21550.03180.97480.98560.9709IMW (WPP)0.71020.59420.5485—0.10190.38670.05320.72610.86140.8591Weibull0.33370.8985—108.46970.12040.88800.12050.51700.42110.4949IW1.13151.0125—105.69070.08060.35690.05090.92590.88960.8728EW6.60310.1469893.29105.72920.07440.28470.04710.96060.94890.8957MW0.37430.85970.000001019110.60470.14290.86330.09690.30440.43690.6019MOW0.01261.48630.0331106.85610.09230.41150.06140.82820.83650.8074MWE34637.20.89790.1148110.47520.12040.89010.12100.51660.41980.4929CRW0.89850.89870.1668110.46290.12040.88780.12040.517142120.4951The first and second rows are related to the IMW model estimated by ML and WPP methods respectively. Other models are estimated by ML method.To calculate the MLE, the “optim” function is used with the standard “Nelder-Mead” method inR. Table 3 summarizes the results showing that the IMW outperforms the other distributions in terms of AIC, K–S, AD and CVM statistics. The small values of the K–S, AD and CVM statistics (large values and close to 1 of the corresponding p-values) indicate a good fit of IMW. Figure 5, left side, shows the scaled TTT plot, which shows an upside-down, bathtub-shaped hazard rate function. The estimated parameters of the IMW model confirm this shape for the HRF, see Proposition 1.Figure 5
The TTT plot for the data set in Table2 (left side). The empirical distribution function along with the estimated distributions from IMW and some alternatives (right side).
(a)(b)


## 6. Conclusion

A new IMW model with increasing and unimodal HRF is introduced and studied. Its form is simple, but it proves to be sufficiently flexible to fit data sets with such HRF. The WPP representation can be used to investigate the applicability of this model. Indeed, the model shows a good fit when the WPP representation is a concave curve. The simulation results show that the WPP and MLE estimators are sufficiently efficient and powerful and can be applied in real examples. A maintenance data set is analyzed with this model and other alternative models to show its applicability. The results lead to new ideas and applications in reliability, survival analysis, medical statistics, and risk theory. It is expected that the new model will be valuable to the scientific community in the future and will be a better option than the baseline model. Other aspects and applications of the new model may be explored in the future of this study. In particular, the following topics are interesting and still remain as open problems: Bayesian and E-Bayesian estimation based on complete and different censoring schemes; Proposing a bivariate family of distributions to extend the univariate case of this model.



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