Abstract
Modelling data in applied areas particularly in reliability engineering is a prominent research topic. Statistical models play a vital role in modelling reliability data and are useful for further decision-making policies. In this paper, we study a new class of distributions with one additional shape parameter, called a new generalized exponential-X family. Some of its properties are taken into account. The maximum likelihood approach is adopted to obtain the estimates of the model parameters. For assessing the performance of these estimators, a comprehensive Monte Carlo simulation study is carried out. The usefulness of the proposed family is demonstrated by means of a real-life application representing the failure times of electronic components. The fitted results show that the new generalized exponential-X family provides a close fit to data. Finally, considering the failure times data, the Bayesian analysis and performance of Gibbs sampling are discussed. The diagnostics measures such as the Raftery–Lewis, Geweke, and Gelman–Rubin are applied to check the convergence of the algorithm.
1. Introduction
Generally speaking, lifetime distributions have been frequently applied to model lifetime data in many fields especially in reliability engineering and biomedical sciences. Due to the variability of the data, the selection of the statistical models greatly affects the quality of the modelling to provide the best description of the phenomena under consideration, for instance, data modelling with the exponential and Rayleigh models when the data experience the decreasing failure rate, or the utilization of the Rayleigh model when the data has a constant failure behaviour. Henceforth, it is always of interest to provide the best fit to data under consideration. In such situations, the utilization of the Weibull model may be a suitable choice to analyze data having increasing, decreasing, or constant failure rates, for example, modified Weibull distribution [1], beta modified Weibull distribution [2], and a new modified Weibull distribution [3].
However, in a number of situations, where data behaves nonmonotonically such as unimodal, modified unimodal, or bathtub shaped failure rates, then the Weibull model is not a good candidate model to use. So, for accurate and precise data modelling, new extensions and modifications of the existing models are required. Therefore, many statistical methods are designed to find new extensions of the existing models to provide a better fit to the data of interest.
In the literature, most of the modifications of the Weibull model have been derived by introducing new families of distributions; see the beta extended Weibull family [4], the Weibull-G family [5], a new Weibull-X family [6]. For a more brief review, we refer to Ahamd et al. [7].
We are also continuing this research area and proposing a new statistical distribution family, namely, a new generalized exponential-X (NGE-X) family of distributions.
Genesis: let be the probability density function (pdf) of a random variable, say T, where , , and let be a function of cumulative distribution function (cdf) of a random variable, say X, satisfying the conditions given below:(1)(2) is differentiable and monotonically increasing(3) as and as
The cdf of the T-X family of distributions [8] is defined bywhere satisfies the conditions stated above. The pdf corresponding to (1) is
Using the approach of the T-X method, one can introduce new members of the survival family via the cdf given bywhere is the survival function of the baseline distribution.
Taking inspiration from (1), we introduce a new flexible class of distributions, namely, a new generalized exponential-X (NGE-X) family of distributions. Let T exp (1); then, its cdf is given by
The density function corresponding to (4) is
If follows (5) and setting in (1), we define the cdf of the NGE-X family given by
The corresponding pdf is
Based on the proposed procedure defined in (6), a special case is being studied, namely, a new generalized exponential-Weibull (NGE-Weibull), in the belief that it will be most effective in all areas where the Weibull model is applicable. The new distribution is a flexible model that is able to play an important role in reliability analysis as it can take on a variety of shapes of the failure rate function.
Furthermore, we consider the maximum likelihood and Bayesian approaches in order to estimate the parameters of the model. In the Bayesian discussion, we consider different types of symmetric and asymmetric loss functions including weighted squared error, squared error loss, precautionary, -loss, and modified squared error loss function to estimate the unknown parameters of the NGE-Weibull model. Since all the parameters are positive, we use gamma prior distributions. Bayesian credible and highest posterior density (HPD) intervals [9] are given for each of the proposed model parameters. The posterior samples were extracted via the Gibbs sampling process. From the graphical point of view, we sketch the posterior summary plots. Next, to explore the MCMC process in Bayesian analysis, we used the Gelman–Rubin, Geweke, and Raftery–Lewis diagnostic methods for testing the convergence of the algorithm.
The remaining sections of the article are organized as follows: Section 2 offers the proposed model with its graphical illustrations. In Section 3, the statistical properties are obtained. Section 4 is devoted to the parameter estimation by maximum likelihood estimation (MLE) and Monte Carlo simulation study. In Section 5, the proposed distribution is illustrated by analyzing the failure time data. The Bayesian analysis is provided in Section 6. Finally, this research is concluded in the last section.
2. Submodel Description
The two-parameter Weibull distribution has pdf, cdf, survival function (sf), hazard rate function (hrf), and cumulative hazard rate function (chrf) given by , , , and , respectively. Then, the cdf and pdf of the NGE-Weibull distribution are given by (for )respectively.
The NGE-Weibull density and hazard rate plots for chosen parameter values are provided, respectively, in Figures 1 and 2.
3. Mathematical Properties
This section is devoted to deriving the mathematical properties of the NGE-X distributions including the quantile function, expansions of cdf and pdf, reliability, random number generation, noncentral moments, and entropy with numerical illustrations. Furthermore, a characterization theorem extending the NGE-X class of distributions is also derived.
3.1. Quantile Function
Letting , we must solve the following equation for :
Letting , then solving the following equation for (using software like MATHEMATICA)it can be shown thatwhere gives the principal solution for in . Thus,where is the quantile of the baseline distribution with cdf .
Remark 1. Throughout, we assume the Weibull distribution has quantile:where and . Whenever the baseline distribution is Weibull, we refer to the submodel as .
Some numerical values of the quantile measure are provided in Table 1.
The quantile function is used to measure the effect of the shape parameters on the skewness and kurtosis. Henceforth, using the quantile function of the NGE-Weibull distribution, we obtained the expressions for skewness and kurtosis. The formulas for Bowley’s skewness and Moor’s kurtosis are givenreceptively.
For and different values of and , graphs for the skewness and kurtosis of the NGE-Weibull are sketched in Figures 3 and 4.
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3.2. Expansion for Distribution Function
Using the binomial theorem, we have
Using the power series representation for the exponential function (EP), we have
Put
It now follows that the cdf can be expressed as
3.3. Expansion for Density Function
From the binomial theorem, we have
By the power series representation for the exponential function, we can write
Put
It follows that we can write
Finally, the expansion for the pdf is given as
3.4. Reliability
In the concept of reliability theory, we know that the life of a component has a random strength with random stress. Random strength can be modelled by a random variable, say , and the random stress can be modelled by a random variable, say . The probability that the component works satisfactorily is , which is a known measure of component reliability for many applications. In particular,where is a pdf and is a cdf.
Let be a vector of parameters associated with the NGE-X distributions, where is a vector of parameters associated with the distribution of the random variable . If is distributed as NGE-X with parameter vector and is distributed as NGE-X with parameter vector , then from the expansion of the pdf and cdf, we havewhere is given bywithand is given bywith
3.5. Random Number Generation
Random numbers from the NGE-X distributions can be obtained fromwhere is the quantile of the baseline distribution with cdf , gives the principal solution for in , , and .
3.6. The Noncentral Moments
We know the random variable:where is the quantile of the baseline distribution with cdf , gives the principal solution for in , , and , following the NGE-X family of distributions. According to Nasiru et al. [10], we can writewhere the coefficients are suitably chosen real numbers that depend on the parameters of the distribution. For a power series raised to a positive integer , we havewhere are obtained from with for ; see Gradshteyn and Ryzhik [11]. Thus, we have the following:where is an expectation. By the Binomial series, we can writeas
By integer powers of the Lambert W function, we can writeas
By the binomial series, we can writeas
Put
Thus, the noncentral moment is given by
Some numerical descriptions of the ordinary moments are presented in Table 2.
3.7. Renyi Entropy
Using the binomial series, we can write , as
Using the power series representation for the exponential function, we can write , as
By the binomial theorem, we can write as
Put
Thus, the Renyi entropy for , , can be expressed as
3.8. Characterization Theorem
It is known that the failure rate function, , of a twice differentiable function, , satisfies the first-order differential equation:
In this section, we present a Weibull-NGE-X distribution. The result here is inspired by Alizadeh et al. [12]. First, let us introduce the following.
Definition 1. We say a random variable follows a Weibull- model if its cdf is given bywhere is some baseline distribution, , and is a vector of parameters in the baseline distribution whose support depends on , , and .
The pdf of the Weibull- model is given bywhere is the pdf of the baseline distribution. Clearly, the hazard rate function of the Weibull- distribution is given by
Theorem 1. Let be a continuous random variable. The pdf of isfor some baseline distribution with pdf , cdf , , and , if and only if its hazard rate function satisfies the differential equation given bywith , with the initial condition for .
Proof. If has pdf as stated in the theorem, then the differential equation as stated in the theorem holds. Now if the stated differential equation holds, thenorwhich is the hazard function of Weibull-
Clearly, a characterization of the Weibull-NGE-X distribution is obtained from the above theorem by letting the baseline pdf and cdf be given as in Section 1.
4. Estimation of the Parameters and Monte Carlo Simulation
The section deals with the derivation of the MLEs (maximum likelihood estimators) of distribution of the NGE-textit X and then conducts a simulation study using the Monte Carlo approach to evaluate the MLEs.
4.1. Maximum Likelihood Estimation
Consider a set of observed values, say , observed from the NGE-X distributions with parameters and . The total LLF (log-likelihood function) for is
The partial derivatives of the LLF are given byand
Setting these equations to zero and solving them simultaneously yields the MLEs of and .
4.2. Monte Carlo Simulation Study
In this subsection, the MLEs of the NGE-Weibull distribution are evaluated via the Monte Carlo simulation approach. Measures such as mean square error (MSE), biases, and absolute biases are used for evaluation purposes. We generate of samples size from the proposed NGE-Weibull model using the inverse transformed technique. For each generated sample, MLEs of the NGE-Weibull are obtained. The estimated biases and MSEs are calculated via the formulas given byrespectively.
Figures 5–10 illustrate the simulation results for the above measures. These plots show that increasing sample size n results in decreasing the estimated biases. Also, increasing sample size n results in decreasing the estimated MSEs decay toward zero as n increases. These results reveal the efficiency as well as the consistency property of the MLEs.
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5. Data Analysis
This section is devoted to illustrating the NGE-Weibull distribution by analyzing an application taken from reliability engineering. The data set representing the failure times of 50 electronic components (per 1000h) is given by 0.036, 0.058, 0.061, 0.074, 0.078, 0.086, 0.102, 0.103, 0.114, 0.116, 0.148, 0.183, 0.192, 0.254, 0.262, 0.379, 0.381, 0.538, 0.570, 0.574, 0.590, 0.618, 0.645, 0.961, 1.228, 1.600, 2.006, 2.054, 2.804, 3.058, 3.076, 3.147, 3.625, 3.704, 3.931, 4.073, 4.393, 4.534, 4.893, 6.274, 6.816, 7.896, 7.904, 8.022, 9.337, 10.940, 11.020, 13.880, 14.730, 15.080. For more details about this data, see Aryal and Elbatal [13].
The MLEs of the NGE-Weibull and other competing distributions are determined, and seven analytical measures including three goodness-of-fit statistics such as Cramer–Von Mises (CM) test statistic, Anderson Darling (AD) test statistic, and Kolmogorov–Smirnov (KS) statistic along with p-value and four discrimination measures such as Akaike information criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian information criterion (BIC), and Hannan–Quinn information criterion (HQIC) are considered. For more details about these measures, we refer to Ahmad et al. [7]. All the computations have been carried via the optim()R-function with the argument method = “BFGS”; see Appendix.
In general, a model with the smaller values of these statistics represents the better fit to data. We fit the proposed model and other models to the failure time data set. The other fitted models are: the two-parameter Weibull and three-parameter exponentiated Weibull (EW) of Modhalkar and Sarivasta [14] and Marshall–Olkin Weibull (MOW) of the Marshall and Olkin [15]. The pdfs of the competing models are as follows:(i)Weibull distribution:(ii)EW distribution:(iii)MOW distribution:
The MLEs and their corresponding standard errors (in parentheses) of the model parameters are presented in Table 3. The discrimination measures of all the competing models are presented in Table 4 whereas the goodness-of-fit measures are reported in Table 5.
The results of the NGE-Weibull are compared with the Weibull, EW, and MOW models in Tables 4 and 5. From these results, we see that NGE-Weibull has the lowest values for the considered measures. Therefore, we conclude that the NGE-Weibull model can be selected as the best competitive model for data related to the failure times of the electronic devices.
In addition to the numerical results provided in Tables 4 and 5, the plots of the fitted density and distribution function of the NGE-Weibull model are displayed in Figure 11. The Kaplan–Meier survival and the probability-probability (PP) are shown in Figure 12. From the results provided in Tables 4 and 5 and displayed graphically in Figures 11 and 12, we see that the NGE-Weibull provides a close fit to the reliability engineering data.
In addition, for the failure times of electronic components, we calculate the KS statistical values of the NGE-Weibull distribution and other competitors. Subsequently, we applied the parametric bootstrap method [16] and bootstrapped the value for all distributions. The KS statistic and the corresponding bootstrapped value are reported in Table 6. Based on the results presented in Table 6, we see that NGE-Weibull is a good competitor among the competing models for modeling the failure times of the electronic component's data.
6. Bayesian Estimation
The Bayesian inference has been taken into consideration by a number of researchers. In the Bayesian analysis, we do not know the exact value of the model parameters, which can be negatively affected by the loss when selecting an estimator. These losses can be measured by the function of the parameter and the corresponding estimator. Here, in the Bayesian analysis, we consider different types of symmetric and asymmetric loss functions; see Table 7.
For further details, we refer to Kharazmi et al. [17] and Ahmad et al. [7].
Next, we provide a Bayesian discussion for estimating the parameters of NGE-Weibull distribution via analyzing complete sample data.
6.1. Joint Posterior and Marginal Posterior Distributions
Assume that the parameters , , and of the NGE-Weibull have independent prior distributions defined bywhere , , , , , > 0. Consequently, the joint prior density function can be formulated as follows:
For simplicity, let us define the function as
The joint posterior distribution defined from equation (64) and the likelihood function is
Therefore, the joint posterior pdf can be expressed bywhere, and is given as
Moreover, the marginal posterior pdf of , and assuming that , can be given aswhere , and is the th member of a vector .
6.2. Bayesian Point Estimation
Under the marginal posterior pdf as in (70) and the loss functions which are given in Table 7, the Bayesian point estimation for the parameter vector is obtained via minimizing the expectation of loss function under the marginal posterior pdf as follows:
However, in practice, because of the intractable integral in relation (71), using the well-known Gibbs sampler [18] or Metropolis Hastings algorithms [19, 20] is suggested to generate posterior samples. We will argue this issue more precisely in subsection 6.5.
6.3. Credibility Interval
In the Bayesian framework, interval estimation is done via credibility interval conception. Consider the parameter vector , which is associated with the NGE-Weibull distribution and denote the marginal posterior pdf of the parameter as in (70). For a given value of , the credibility interval is defined as
By considering the relation (72), it is very difficult to obtain the marginal pdf from the joint posterior pdf. We use the Gibbs sampler to generate posterior samples. Let (where ) be a posterior random sample of size , which is extracted from the joint posterior pdf as in (67). Using these generated posterior samples, the marginal posteriors pdfs of given can be given bywhere shows the vector of posterior samples when the component is removed. Using (73) in (72), one can be able to compute the credibility intervals for as follows:
6.4. The Highest Posterior Density Interval
The highest posterior density interval is a kind of credibility interval which imposed a specific restriction. A HPD interval for , is the simultaneous solution of the following integral equations:
6.5. Generating Posterior Samples
It is clear from (67) and (70) that there are no explicit expressions for the Bayesian point estimators under the loss functions; see Table 7. Due to the intractable integrals associated with joint posterior and marginal posterior distributions, therefore, we require numerical software to solve numerically the integral equations via MCMC methods such as Gibbs sampling and the Metropolis–Hastings algorithm.
Suppose that the general model is associated with parameter vector and observed data . Thus, the joint posterior distribution is . We also assume that is the initial values vector to start the Gibbs sampler. The Gibbs sampling approach draws the values for each iteration in steps by drawing a new value for each parameter from its full conditional given the most recently drawn values of all other parameters. The steps for any iteration, say iteration , are as follows:(i)Starting with an initial estimate (ii)Draw from (iii)Draw from and so on down to(iv)Draw from
In the case of the NGE-Weibull distribution, by considering the parameter vector and initial parameter vector , the posterior samples are extracted by the above Gibbs sampler where the full conditional distributions are given as
Here, since there is not any prior information about hyperparameters in (57), we implement the idea of Congdon [21] and the hyperparameters values are set as . So, we can use the MCMC procedure to extract posterior samples of (70) by means of the Gibbs sampling process in OpenBUGS software.
Next, we provide Bayesian estimation results. It is evident from equation (70) that there are no closed-form expressions for Bayesian estimators, which are extracted based on the loss functions in Table 7. Therefore, a MCMC procedure via the Gibbs sampler process is designed using the expressions (72), (73), and (74), with 10,000 replicates to obtain the Bayesian estimators. In Table 8, we provide the corresponding point and posterior risk estimations. Furtherer, credible and HPD intervals are provided in Table 9. In order to provide a visual inspection, we provide posterior summary plots in Figures 13–15. These plots verify that the convergence of the Gibbs sampling process has occurred.
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Next, for evaluation of the MCMC procedure in Bayesian analysis, we report some diagnostics measures such as Gelman–Rubin (GR), Geweke (G), and Raftery–Lewis (RL) for checking the convergence of the Gibbs algorithm in Table 10. For more details about these indexes, see Lee et al. [22]. The GR diagnostic for parameters , , and is equal to 1. Hence, based on the GR diagnostic measure, the chains are acceptable. Figure 16 shows that the estimates come from state spaces of the corresponding parameters. From Table 10, Geweke’s test statistics for parameters , , and are 0.846, 0.080, and 1.727, respectively. Hence, the G diagnostic measure also confirms the acceptance of chains as shown in Figures 17 and 18. Moreover, the reported diagnostic statistics for parameters , , and based on the RL method do not show a significant degree of dependence between estimates.
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7. Concluding Remarks
We have introduced a new generalized exponential-X family. A special submodel of this family named a new generalized exponential-Weibull distribution is discussed. The density of the NGE-Weibull model can take different shapes of density and failure rate functions. Parameters of the NGE-Weibull distribution are estimated using the method of maximum likelihood estimation. A simulation study was conducted to evaluate the behavior of the estimators. Statistical properties of the NGE-X distributions are also obtained. A real application related to the failure times data is considered and it is observed that the NGE-Weibull model provides the best fit to data than other well-known competitors. Finally, the Bayesian estimation method is used to estimate the model parameters and conduct the Bayesian analysis under five different loss functions. Furthermore, the diagnostics measures such as the Gelman–Rubin, Geweke, and Raftery Lewis are also discussed to evaluate the MCMC procedure in the Bayesian analysis.
Appendix
A. R Code for Analysis
Note: in the following R-code, pm is used for the proposed model. Data = c(0.036, 0.058, 0.061, 0.074, 0.078, 0.086, 0.102, 0.103, 0.114, 0.116, 0.148, 0.183, 0.192, 0.254, 0.262, 0.379, 0.381, 0.538, 0.570, 0.574, 0.590, 0.618, 0.645, 0.961, 1.228, 1.600, 2.006, 2.054, 2.804, 3.058, 3.076, 3.147, 3.625, 3.704, 3.931, 4.073, 4.393, 4.534, 4.893, 6.274, 6.816, 7.896, 7.904, 8.022, 9.337, 10.940, 11.020, 13.880, 14.730, 15.080) ################################################################# ############### pdf of the proposed model ################################################################# pdf_pm < - function(par,x) { alpha = par[1] theta = par[2] gamma = par[3] 2thetaalphagamma(x^(alpha − 1))exp(-gammax^alpha) (1 − exp(-gammax^alpha)) ((1 − ((1 − exp(-gammax^alpha))^2))^(theta − 1)) (2 − ((1 − exp(-gammax^alpha))^2)) (1/(exp (theta((1 − exp(−gammax^alpha))^2)))) } ################################################################# ############### cdf of the proposed model ################################################################# cdf_pm < - function(par,x) { alpha = par[1] theta = par[2] gamma = par[3] 1 − ((((1 − ((1-exp(-gammax^alpha))^2))^(theta))) /((exp (theta((1 − exp(-gammax^alpha))^2))))) } set.seed(0) goodness.fit(pdf = pdf_pm, cdf = cdf_pm, starts = c(0.5,0.5,0.5), data = data, method = “BFGS”, domain = c(0,Inf), mle = NULL)
Data Availability
The data used to support the findings of this study are provided within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by Yazd University, Iran.