A New Three-Parameter Exponential Distribution with Variable Shapes for the Hazard Rate: Estimation and Applications

In this paper, we study a new flexible three-parameter exponential distribution called the extended odd Weibull exponential distribution, which can have constant, decreasing, increasing, bathtub, upside-down bathtub and reversed-J shaped hazard rates, and right-skewed, left-skewed, symmetrical, and reversed-J shaped densities. Some mathematical properties of the proposed distribution are derived. The model parameters are estimated via eight frequentist estimation methods called, the maximum likelihood estimators, least squares and weighted least-squares estimators, maximum product of spacing estimators, Cramér-von Mises estimators, percentiles estimators, and Anderson-Darling and right-tail Anderson-Darling estimators. Extensive simulations are conducted to compare the performance of these estimation methods for small and large samples. Four practical data sets from the fields of medicine, engineering, and reliability are analyzed, proving the usefulness and flexibility of the proposed distribution.


Introduction
The exponential distribution has been extensively used in analyzing lifetime data due to its lack of memory property and its simple form. However, the exponential distribution with only a constant hazard rate shape is not able to fit data sets with different hazard shapes as increasing, decreasing, bathtub, or unimodal (upside down bathtub) shaped failure rates, often encountered in engineering and reliability, among others. very important in the reliability engineering context. The interesting point is that the EOWEx distribution, with three parameters, can have the bathtub and modified bathtub failure rates as, in general, most distributions used to model such data are complicated, and usually may include four or five parameters to obtain these failure rates.
• It can be considered as a suitable distribution for fitting skewed data that may not be properly fitted by other extensions of the exponential distribution and can also be used in many problems in applied areas, such as medicine, engineering, survival analysis, and industrial reliability. • Four applications to real data from the medicine, engineering and reliability fields prove that the EOWEx model performs better than four other competing lifetime distributions, motivating its use in applied areas. • Its cumulative distribution function (CDF) and hazard rate function (HRF) have simple closed forms, therefore it can be utilized to analyze censored data sets.
Furthermore, we focus on eight different estimation procedures and study how these estimators of the EOWEx unknown parameters behave for several sample sizes and for several parameter combinations. We also develop a guideline for choosing the best estimation method to estimate the EOWEx parameters, which we think would be of interest to applied statisticians and reliability engineers. We consider different estimators called, the maximum likelihood estimators, least-squares and weighted least-squares estimators, percentiles estimators, Cramér-von-Mises estimators, maximum product of spacings estimators, Anderson-Darling estimators, and right-tail Anderson-Darling estimators. We conduct an extensive simulation study to assess and compare the performance of these estimators.
The EOWEx distribution is constructed based on the extended odd Weibull-G (ExOW-G) family proposed by Alizadeh et al. [10]. Let G(x; ξ) = 1 − G(x; ξ) and g (x; ξ) = dG(x; ξ)/dx denote the survival function (SF) and probability density function (PDF) of a baseline model with parameter vector ξ, then the CDF of the EOW-G family has the form F(x; α, β, ξ) = 1 − 1 + β G(x; ξ) G(x; ξ) The corresponding PDF of (1) is defined by f (x; α, β, ξ) = α g(x; ξ) G(x; ξ) α−1 G(x; ξ) α+1 1 + β G(x; ξ) G(x; ξ) where α and β are positive shape parameters. The random variable with PDF (2) is denoted by X ∼ExOW-G(α, β, ξ). When β → 0 + , we have the Weibull-G family. The HRF of the EOW-G family takes the form where τ(x; ξ) is the baseline HRF. By inverting (1), we obtain the quantile function (QF) of the ExOW-G family where Q G (u) = G −1 (u) is the QF of the baseline G distribution and u ∈ (0, 1). The rest of this article is organized as follows. In Section 2, we define the proposed EOWEx distribution. In Section 3, we derive a linear representation for the EOWEx density function and obtain some of its properties. Eight estimation methods to estimate the EOWEx parameters are presented in Section 4. In Section 5, we perform a simulation study to compare the performance of these estimation methods. Four real data applications are used to prove the usefulness of the EOWEx distribution in Section 6. Finally, we conclude the paper by some remarks in Section 7.

The EOWEx Distribution
In this section, we define the three-parameter EOWEx model. The PDF and CDF of the Ex distribution are g (x; λ) = λ exp (−λx) and G (x; λ) = 1 − exp (−λx), x > 0, λ > 0. By inserting the CDF of the Ex model in (1), we obtain the CDF of the EOWEx distribution The corresponding PDF follows, by inserting the PDF and CDF of the Ex distribution in (2), as Thereforeforth, a random variable with PDF (6) is denoted by X ∼EOWEx(α, β, λ). The EOWEx model reduces to the two-parameter Weibull Ex distribution for β → 0 + .
The HRF and QF of the EOWEx distribution are given, respectively, by Figures 1 and 2 display some possible shapes of the PDF and HRF of the EOWEx distribution. These figures indicate that the PDF of the EOWEx distribution can be left-skewed, right-skewed, reversed-J shaped, and symmetric. Further, the HRF of the EOWEx distribution has some important shapes, including constant, increasing, decreasing, upside down bathtub, reversed bathtub, reversed-J shaped, which are desirable characteristics for a lifetime distribution. It can be seen, from the application section, that the EOWEx distribution allows greater flexibility and can be used to model skewed data and can be widely applied in different areas such as reliability, biomedical studies, biology, engineering, and survival analysis.

Some Properties
In this section, we obtain some properties of the EOWEx distribution including the linear representation, moments, moment generating function (MGF), mean residual life, mean inactivity time, and order statistics.

Linear Representation
We provide a useful linear representation for the EOWEx density. Alizadeh et al. [10] derived a mixture representation of the EOW-G density as follows, where a k,j = −β k Γ (αk + j) (−1/β) k /k!j!Γ (αk) and h αk+j (x) = (αk + j) g(x)G(x) αk+j−1 is the Exp-G density with positive power parameter αk + j. Using the PDF and CDF of the Ex distribution, the last equation can be rewritten as Applying the binomial expansion to [1 − exp (−λx)] αk+j−1 , the above equation reduces to Equation (7) can be expressed as where and g m+1 (x) = (m + 1)λ exp (−(m + 1)λx) denotes the Ex density with scale parameter (m + 1)λ. Then, the EOWEx PDF can be expressed as a single linear combination of Ex densities. Let Z be a random variable having the Ex distribution with PDF g (x; λ) = λ exp (−λx), x > 0, λ > 0. Then, the rth ordinary and incomplete moments, and MGF of Z are respectively, where Γ (a + 1) = ∞ 0 w s e −w dw is the gamma function and γ (a + 1, λt) = λt 0 w a e −w dw is the lower incomplete gamma function.

Moments and MGF
The rth moment of X follows simply from Equation (8) as Table 1 displays the numerical values of the mean (µ), variance (σ 2 ), skewness (γ 1 ), and kurtosis (γ 2 ) of the EOWEx distribution for λ = 1 and some selected values of α and β. The values in Table 1 illustrate that the skewness of the EOWEx distribution is ranging in the interval (−0.24265, 2.67091), whereas the spread of its kurtosis is much larger ranging from 3.34256 to 14.4083. Furthermore, the EOWEx distribution can be left skewed or right skewed, and it can be leptokurtic (γ 2 > 3). Therefore, the EOWEx distribution can be used to model the skewed data due to its flexibility.
The rth incomplete moment of X can be obtained from (8) as The first incomplete moment of X follows from the last equation as Based on Equation (8), the MGF of the EOWEx distribution takes the form

Mean Residual Life and Mean Inactivity Time
The mean residual life (MRL) (also known as the life expectancy at age t) represents the expected additional life length for a unit, which is alive at age t and is defined by where ϕ 1 (t) is given by (10) and S(t) is the SF of the EOWEx distribution. Inserting Equation (10) in (11), we have The mean inactivity time (MIT) is defined by M X (t) = E (t − X | X ≤ t) (for t > 0) and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, t).
The MIT of X is Combining Equations (10) and (12), the MIT of X is as follows,

Order Statistics
Order statistics are important in many areas of statistical theory and practice. According to Alizadeh et al. [10], the PDF of ith order statistic of the EOW-G class, X (i) (for i = 1, . . . , n), can be expressed as Here, h α(k+1)+s is the exponentiated exponential density with power parameter α (k + 1) + s and Let X 1 , · · · , X n be a random sample from the EOWEx model and let X (1) , · · · , X (n) be the associated order statistics. The PDF of ith order statistic reduces to Applying the binomial series, the last equation becomes where Equation (14) means that the PDF of EOWEx order statistics is a mixture of Ex densities with scale parameter (r + 1) λ. Therefore, some of their mathematical properties are obtained from those of the Ex distribution. For example, the qth moments of

Maximum Likelihood Method
Let x 1 , · · · , x n be a random sample from the EOWEx distribution with parameters α, β, and λ. The log-likelihood function has the form = n log (α) + n log (λ) + αλ The MLEs of α, β and λ can be obtained by maximizing the last equation with respect to α, β and λ, or by solving the following nonlinear equations, The R (optim function), Ox program (sub-routine MaxBFGS), SAS (PROCNLMIXED), Mathcad program, or Newton-Rapshon method can be used to maximize the log-likelihood function to obtain the MLEs. The log-likelihood is maximized using a wide range of starting values. The starting values were taken to correspond to all combinations of the model parameters, where α = 0.1, 0.5, . . . , 10, β = 0.1, 0.5, . . . , 10 and λ = 0.1, 0.5, . . . , 10. The call to optim converged about 98 percent of the time. The maximum likelihood solution was unique, when the calls to optim did converge. The elements of the observed information matrix are given in explicit expressions as follows,

Least Squares and Weighted Least Squares Methods
The least squares (LS) and weighted least square (WLS) methods are used to estimate the parameters of the beta distribution (Swain et al. [11]). Let x (1) < x (2) < · · · < x (n) be the sample order statistics of size n from the EOWEx distribution; therefore, the LS estimators (LSEs) and WLS estimators (WLSEs) of the EOWEx parameters α, β and λ can be obtained by minimizing with respect to α, β, and λ, where v i = 1 in case of LSEs and v i = (n + 1) 2 (n + 2)/ [i(n − i + 1)] in case of WLSEs. Furthermore, the LSEs and WLSEs follow by solving the nonlinear equations

Percentile Method
Here, we use the percentile method (Kao,[14]) to estimate the unknown parameters of the EOWEx distribution by equating the sample percentile points with the population percentile points.
Let u i = i/ (n + 1) be an unbiased estimator of F x (i) |a, b, λ . Then, the percentile estimators (PCEs) of the EOWEx parameters are obtained by minimizing the following function with respect to α, β, and λ,

Anderson-Darling and Right-Tail Anderson-Darling Methods
The Anderson-Darling estimators (ADEs) are another type of minimum distance estimators. The ADEs of the EOWEx parameters are obtained by minimizing with respect to α, β, and λ. The ADEs can also be obtained by solving the nonlinear equations where ∆ s x (i) |α, β, λ = 0, (for s = 1, 2, 3) is defined by (15). The right-tail Anderson-Darling estimators (RTADEs) of the EOWEx parameters α, β and λ are obtained by minimizing the following function with respect to α, β and λ,

Simulation Results
In this section, the performance of eight different estimators of the EOWEx parameters is assessed by a simulation study. We consider different sample sizes n = {20, 50, 100} for different parameters values α = (3.5, 0.75), β = (3, 1.5, 0.25) and λ = (1, 0.5). We generate N = 1000 random samples from EOWEx distribution. For each estimate, we obtain the average values of the estimates (AEs) and their corresponding mean squares error (MSEs).
The performance of different estimators are evaluated in terms of MSEs, i.e., the most efficient estimation method will be the one whose MSEs values are closer to zero. The simulation results are obtained via the R software. Tables 2-4 show the AEs and MSEs (in parentheses) of the MLEs, LSEs, WLSEs, MPSEs, PCEs, CVMEs, ADEs, and RTADEs. Further, the AEs based on all estimation methods tend to the true parameter values, as the sample size increase in all cases, which indicates that all estimators are asymptotically unbiased. The figures in these tables means that MLEs, LSEs, WLSEs, MPSEs, PCEs, CVMEs, ADEs, and RTADEs perform very well, in terms of MSEs, for estimating the EOWEx parameters.

Applications in Medicine, Engineering, and Reliability
In this section, the EOWEx distribution is fitted to four data sets from fields of medicine, engineering, and reliability. The EOWEx model is compared with other some competitive models called, the exponentiated exponential (EEx) (Gupta and Kundu, [2]), beta exponential (BEx) (Nadarajah and Kotz, [18]), alpha power exponential (APEx) (Mahdavi and Kundu, [19]), and exponential (Ex) distributions. The densities of these models are given by The fit of these distributions is evaluated using some measures including Cramér-von Mises (W * ), Anderson-Darling (A * ), and Kolmogorov Smirnov (KS) statistics with its p-value.
The first set of data was studied by Lee and Wang [20], and it represents the remission times (in months) of a random sample of 128 bladder cancer patients. These data were analyzed by Sen et al. [21], Afify et al. [22], and Mansour et al. [23]. The second set of data was studied by Kundu and Raqab [24], and it represents the gauge lengths of 20 mm of a sample of 74 observations. This data set was analyzed by Afify et al. [25] and Afify et al. [26]. The third set of data consists of the failure times of 20 mechanical components (Murthy et al. [27]). The fourth set of data refers to breaking stress of carbon fibres (in Gba) and it consists of 100 observations (Nichols and Padgett, [28]). These data were analyzed by Afify et al. [29].
Tables 5-8 provide the values of W * , A * , and KS as well as the p-value for the models fitted to the four data sets, respectively. Further, Tables 5-8 display the MLEs and standard errors (SEs) (appear in parentheses) of the parameters of the EOWEx, EEx, BEx, APEx, and Ex models. In Tables 5-8, we compare the fits of the EOWEx model with the EEx, BEx, APEx, and Ex models. The figures in these tables indicate that the EOWEx distribution has the lowest values of W * , A * , KS and largest p-value, among all fitted models. The fitted EOWEx PDF, CDF, SF, and P-P plots of the four data sets are displayed in Figures 3 and 4, respectively.   Furthermore, we use the eight estimation methods discussed in Section 4 to estimate the EOWEx parameters. Tables 9-12 display the estimates of the EOWEx parameters using these estimation methods and the numerical values of KS and its p-value for the four data sets, respectively. Based on the values of KS and p-value, in Tables 9-12, the LSEs is recommended to estimate the EOWEx parameters for cancer data, failure times data, and breaking stress of carbon fibers data, whereas the MLEs is recommended to estimate the EOWEx parameters for gauge lengths data. However, all estimation methods perform very well for the four data sets. The P-P plots of the EOWEx distribution using the four best estimation methods are displayed in Figures 5 and 6, for the four data sets, respectively.

Concluding Remarks
In this paper, we propose the three-parameter extended odd Weibull exponential (EOWEx) distribution. The EOWEx density is a linear combination of exponential densities. Some of its mathematical properties are obtained. The EOWEx parameters are estimated by eight different estimation methods called, MLEs, LSEs, WLSEs, MPSEs, PCEs, CVMES, ADEs, and RTADEs. An extensive simulation study is conducted to compare the performance of these different estimators to identify the best performing estimators. The simulation results reveal that all estimators perform very well in terms of their mean square errors. Four real data applications are used to prove the EOWEx flexibility and potentiality. These applications show that the EOWEx model can yield better fits than some other existing extensions of the exponential distribution. We expect the utility of the newly proposed model in several fields such as reliability, medicine, engineering, and life testing.