Abstract
A new continuous version of the inverse flexible Weibull model is proposed and studied. Some of its properties such as quantile function, moments and generating functions, incomplete moments, mean deviation, Lorenz and Bonferroni curves, the mean residual life function, the mean inactivity time, and the strong mean inactivity time are derived. The failure rate of the new model can be “increasing-constant,” “bathtub-constant,” “bathtub,” “constant,” “J-HRF,” “upside down bathtub,” “increasing,” “upside down-increasing-constant,” and “upside down.” Different copulas are used for deriving many bivariate and multivariate type extensions. Different non-Bayesian well-known estimation methods under uncensored scheme are considered and discussed such as the maximum likelihood estimation, Anderson Darling estimation, ordinary least square estimation, Cramér-von-Mises estimation, weighted least square estimation, and right tail Anderson Darling estimation methods. Simulation studies are performed for comparing these estimation methods. Finally, two real datasets are analyzed to illustrate the importance of the new model.
1. Introduction
The Weibull model [1] is a very useful distribution in modeling real data exhibiting monotonic hazard rate function (HRF). But it cannot be used in modeling and studying data which have nonmonotonic HRF such as the “bathtub shape (U-HRF).” For avoiding this drawback, Bebbington et al. [2] have defined a new two-parameter distribution which is an extension of the Weibull distribution referred to as a flexible Weibull (FW) extension distribution; it has a failure function that can be “decreasing,” “increasing,” or “bathtub-shaped.” Analogously, El-Goharyet et al. [3] derived the two-parameter inverse flexible Weibull (IFW) model which is the reciprocal of a random variable (RV) which has FW model. Several mathematical properties of this distribution such as the mode, moments, and moment generating function (MGF) have been discussed. El-Gohary et al. [3] proved that the hazard rate function (RRF) of the IFW model can be “upside down constant,” the cumulative distribution function (CDF) of IFW distribution is given bywhere the two parameters and control the shape of the distribution. The corresponding probability density function (PDF) is
Many authors introduced new families of continuous distributions that extend well-known existing ones and, at the same time, provide great flexibility in modeling data in practice, for example, the Marshall-Olkin-generated family [4], the Kumaraswamy-G [5], the Burr X generator of distributions [6], the Topp-Leone odd log-logistic family [7], the transmuted Topp-Leone family [8], the Burr XII system of densities [9], and the odd log-logistic Topp-Leone family [10], among others. Recently, Elgarhy et al. [11] represented a new flexible class of continuous distributions with an extra positive parameter called the type II Topp-Leone generated (TIITL-G) family of distributions. Due to [11], the CDF of the TIITL-G family is given by
The PDF is defined bywhere is a shape parameter, is the baseline PDF, and is a baseline CDF.
The remainder of the paper is organized as follows: in Section 2, we define the CDF, PDF, and HRF of TIITLIFW model and provide a simple expansion of the PDF. Simple type copula is derived in Section 3. Various mathematical properties are discussed in Section 4. Non-Bayesian estimation methods under uncensored schemes are given in Section 5. Section 6 presents a comparison under the non-Bayesian estimation methods using uncensored schemes via a simulation study. Section 7 presents a comparison under uncensorship with some competitive models. Concluding remarks are contained in Section 8.
2. The New Model
Using (1) in (3), the CDF of the TIITLIFW distribution can be written as
The corresponding PDF is given by
The HRF can be expressed as
Figure 1 shows some plots of the PDF of the TIITLIFW distribution for some different values of the parameters. Figure 2 shows some plots of the HRF of the TIITLIFW distribution for some different parameter values. Based on Figure 1, we conclude that the new PDF can have many right skewed heavy tail shapes. Based on Figure 2, it is observed that the new HRF can be “increasing-constant,” “bathtub-constant,” “bathtub,” “constant,” “J-HRF,” “upside down bathtub,” “increasing,” “upside down-increasing-constant,” and “upside down”
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The PDF of the TIITLIFW distribution can be written as
Proof. Supposing is a real noninteger, we have the power series expansionusing the power series (9) in equation (8), and the fact that , we getexpanding using Taylor seriesagain, using a series expansion of , and after some algebras, the PDF can be written aswhere
3. Copula
3.1. Bivariate TIITLIFW (BTIITLIFW) via Morgenstern Family
The CDF of the Morgenstern family of two random RVs can be derived aswheresettingthen we have
3.2. Via Clayton Copula
The BTIITLIFW type extension: the weighted version of the Clayton copula can be expressed as
Let us assume that ∼ TIITLIFW and ∼ TIITLIFW . Then, settingthe associated CDF of the BTIITLIFW type distribution can be written as
A straightforward -dimensional extension from the above will be
3.3. Via Modified Farlie-Gumbel-Morgenstern (FGM) Copula
The joint CDF of the modified FGM copula is given as or , where , and , where and are being two continuous functions on the interval where . Let
Then, where Type I: Recalling the following functional form for both and . Then, the BTIITLIFW-FGM (Type-I) can be derived from where Therefore, Type II: Let and be two functional forms, satisfying all the conditions mentioned above, where Then, the corresponding BTIITLIFW-FGM (Type-II) can be derived from Then, Type-III:
Let and be two functional forms, satisfying all the abovementioned conditions where
In this case, one can also derive a closed form expression for the associated CDF of the BTIITLIFW-FGM (Type-III) fromas follows
3.4. Via Renyi’s Entropy
Let and Then, the Renyi’s entropy copula can be expressed as
Then, the associated BTIITLIFW can be directly derived from as
4. Statistical and Reliability Measures
4.1. Quantile Function
For RV has CDF of the TIITLIFW distribution, the quantile function of the TIITLIFW distribution is given by the following equation:where
4.2. Moments and Generating Functions
The moment of the TIITLIFW distribution is obtained using the formulahence using equation (7) we obtain
Setting , it follows thatwhereis a gamma function. In particular, if and , we obtain the mean and variance of the TIITLIFW distribution. The MGF of the TIITLIFW is given by
The mathematical form of the Galton skewness and Moors kurtosis of TIITLIFW distribution can be computed using the quantile function and well-known relationships. The first four moments, skewness, and kurtosis of the TIITLIFW distribution for different values of parameters are represented in Table 1. Table 1 shows that the skewness is always positive, and kurtosis is always greater than three.
Figure 3 shows two plots of the skewness of the TIITLIFW distribution. Figure 4 presents two plots of the kurtosis of the TIITLIFW distribution. Based on Figures 3 and 4, we conclude that the TIITLIFW model can have many useful skewness and kurtosis shapes.
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4.3. Incomplete Moments
The lower and upper incomplete moments of are defined byrespectively, for any real = . The lower incomplete moment of the TIITLIFW distribution iswhereis the lower incomplete gamma function. Similarly, the upper incomplete moment of the TIITLIFW distribution iswhereis the upper incomplete gamma function.
4.4. Mean Deviation, Lorenz, and Bonferroni Curves
For RV with PDF, , distribution function , mean , and , the mean deviation about the mean and median, respectively, is given bywhere
The Lorenz curve for a positive RV is defined as
Then, we havewhere . Also, Bonferroni curve is defined by
Then,
4.5. The Mean and Strong Mean Inactivity Times Functions
The mean inactivity time (MIT) can be derived from
Then, the MIT can be derived as
The strong mean inactivity time (SMIT) is a new reliability function given by
Therefore, the SMIT can be expressed as
5. Non-Bayesian Estimation Methods under Uncensored Schemes
5.1. The MLE Method
Let be a random sample of size from TIITLIFW. The log-likelihood function for the vector of parameters can be written as
The associated score function is given by
The can be maximized by solving the following nonlinear likelihood equations and . Then,
The maximum likelihood estimation (MLE) of , say , is obtained by solving the system of nonlinear equations
5.2. The CVME Method
The CVMEs of the parameters , and [12] are obtained by minimizing the following expression with respect to the parameters , and , respectively, wherewhererefers to the empirical estimate of the CDF at computed from a certain sample and
Then, CVMEs of the parameters , and are obtained by solving the following nonlinear system:where
5.3. The OLSE and WLSE Method
Let denote the CDF of the TIITLIFW model and be the ordered RS. The OLSEs [13] are obtained by minimizingwhere
Then,
The OLSEs are obtained by solvingwhere , , and are as defined above. The WLSE is obtained by minimizing the function WLSE with respect to , and . Then,where
The WLSEs are obtained by solving
5.4. The ADE Method
The ADEs are obtained by minimizing the functionwhere
Then, the parameter estimates are derived by solving the nonlinear equations
5.5. The ADE (R-T) Method
The ADEs (R-T) are obtained by minimizing
Then, the estimates follow by solving the nonlinear equations
6. Comparing the Non-Bayesian Estimation Methods under Uncensored Schemes via a Simulation Study
A numerical simulation study is conducted to compare the non-Bayesian estimation methods. The simulation study is based on which generated datasets from the TIITLIFW distribution where , and and , and . The comparison is performed based on the bias and root mean - standard error .
From Table 2, we note the following: 1) The tends to as n increases and tends to which means that all estimators are nonbiased. 2) The tends to as n increases and tends to which means incidence of consistency property.
7. Comparing Models under Uncensorship
We illustrate the flexibility and the performance of the TIITLIFW distribution as compared to some alternative models using two real data applications. The goodness-of-fit statistics for this distribution are compared with other competitive distributions. The MLEs of the distribution parameters are determined numerically. To compare the distributions, we consider the measures of goodness-of-fit, such as Akaike information criterion (), consistent Akaike information criterion (), and Bayesian information criterion () statistic. The better distribution to fit the data corresponds to smaller values of these statistics.
We consider two uncensored datasets for comparing competitive models. The first data present the remission times (in months) of a random sample of 128 bladder cancer patients [14]: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.69, 4.23, 5.41, 0.90, 2.69, 4.18, 5.34, 7.59, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 5.71, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 10.66, 15.96, 36.66, 1.05, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, and 22.69. The second data present the lifetimes of 38 devices provided by [14]: 0.1, 1, 1, 1, 2, 3, 6, 40, 45, 46, 47, 50, 55, 0.2, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 1, 82, 83, 84, 84, 84, 85, 85, 1, 60, 63, and 86, 86. According to [15], the total time in test (TTT) plots, box plots, quantile-quantile (QQ) plots, and kernel density estimation (KDE) plots are shown in Figure 5 and Figures 6(a) and 6(c) for bladder cancer data and lifetimes data, respectively. Based on Figures 6(a) and 6(c), the HRF of the bladder data is “upside down” or “reversed U-shape.” Based on Figures 6(a) and 6(c), the HRF of the lifetimes data is “U-shape.” The box plots (middle panels) are presented along with its corresponding normal quantile-quantile plot (right panels) in Figures 5 and 6 for discovering the outliers and normality.
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The following competitive models are considered in the comparison: the exponentiated IFW (Exp-IFW) [12], IFW [3], exponentiated generalized IW (ExpG-IFW) [16], generalized IW (G-IW) [17], IW [18], and [2].
Tables 3 and 4 present the MLEs for the bladder cancer data and lifetimes data. Tables 5 and 6 show the statistics criteria for the bladder cancer data and lifetime data. From Tables 5 and 6, it is clear that the TIITLIFW distribution provides the best fits for the two datasets. Figures 7 and 8 show the estimated PDFs (EPDFs) (left panel) and the estimated HRF (EHRFs) (right panel) for bladder cancer data and lifetimes data, respectively. Figures 9 and 10 show the profile of the log-likelihood function for bladder cancer data and lifetimes data, respectively. From Figures 7 and 8, we conclude that the new model can achieve a good fit.
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8. Concluding Remarks
A three-parameter lifetime distribution, so-called the TIITLIFW distribution, is introduced as an extension of the inverse flexible Weibull distribution. Some explicit expressions for mathematical quantities of the TIITLIFW distribution are derived. The hazard rate function allows constant, decreasing, increasing, upside down bathtub, or bathtub-shaped forms. We consider six different estimation methods to estimate the parameters of the TIITLIFW distribution. The performance of these proposed estimation methods is conducted via some simulations. A real data application proves that the TIITLIFW model provides consistently better fits compared to some other well-known competitive models.
Data Availability
The data used to support the findings in this study are included within the paper.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Acknowledgments
The author acknowledges the College of Science at Imam Mohammad Ibn Saud Islamic University, Saudi Arabia, for supporting this project.