Inverse Exponentiated Lomax Power Series Distribution: Model, Estimation, and Application
In this paper, we introduce the inverse exponentiated Lomax power series (IELoPS) class of distributions, obtained by compounding the inverse exponentiated Lomax and power series distributions. The IELoPS class contains some significant new flexible lifetime distributions that possess powerful physical explications applied in areas like industrial and biological studies. The IELoPS class comprises the inverse Lomax power series as a new subclass as well as several new flexible compounded lifetime distributions. For the proposed class, some characteristics and properties are derived such as hazard rate function, limiting behavior, quantile function, Lorenz and Bonferroni curves, mean residual life, mean inactivity time, and some measures of information. The methods of maximum likelihood and Bayesian estimations are used to estimate the model parameters of one optional model. The Bayesian estimators of parameters are discussed under squared error and linear exponential loss functions. The asymptotic confidence intervals, as well as Bayesian credible intervals, of parameters, are constructed. Simulations for a one-selective model, say inverse exponentiated Lomax Poisson (IELoP) distribution, are designed to assess and compare different estimates. Results of the study emphasized the merit of produced estimates. In addition, they appeared the superiority of Bayesian estimate under regarded priors compared to the corresponding maximum likelihood estimate. Finally, we examine medical and reliability data to demonstrate the applicability, flexibility, and usefulness of IELoP distribution. For the suggested two real data sets, the IELoP distribution fits better than Kumaraswamy–Weibull, Poisson–Lomax, Poisson inverse Lomax, Weibull–Lomax, Gumbel–Lomax, odd Burr–Weibull–Poisson, and power Lomax–Poisson distributions.
In recent years, many families of distributions were proposed by combining certain beneficial continuous and power series (PS) distributions by several researchers. This procedure is used extensively in engineering applications including risk measurement, reliability, and survival analysis. A discrete random variable, W, of PS distributions (truncated at zero) has probability mass function (pmf) given bywhere depends only on , is the scale parameter, , and and denote the first and second derivatives of , respectively. Important quantities of some PS distributions (truncated at zero) such as Poisson, logarithmic, geometric, and binomial are provided in Table 1.
The principal idea of introducing these models is that a lifetime of a system with W (discrete random variable) components and the positive continuous random variable, say Xi (the lifetime of ith component), can be denoted by the nonnegative random variable or based on whether the components are series or parallel. In the last few decades, several papers have discussed the derivation of new probabilistic families by compounding different distributions with the PS model. Some notable compound classes proposed by several authors are as follows: exponential-PS family , Weibull-PS family , generalized exponential PS family , Burr XII-PS family , complementary Poisson Lindley-PS family , exponentiated extended Weibull family , complementary exponentiated inverted Weibull-PS family , Gompertz PS family , generalized modified Weibull-PS family , generalized inverse Weibull-PS family , exponential Pareto-PS family , exponentiated power Lindley-PS family , Burr–Weibull PS family , odd log-logistic PS family , generalized inverse Lindley PS family , exponentiated generalized PS family , exponentiated power generalized Weibull-PS family , new Lindley–Burr XII-PS , power function-PS family , inverse gamma PS family , and power quasi-Lindley PS family , among others. Recently, more generalized forms were provided by the compounding G-classes together with discrete distributions (see, for example, [22, 23]).
Several univariate continuous distributions have been extensively used in environmental, engineering, financial, and biomedical sciences, among other areas for modeling lifetime data. However, there is still a strong need for significant improvement of the classical distributions through different techniques for modeling a variety of data lifetime. In this regard, the inverted (or inverse) distribution is one of the procedures that explore extra properties of the phenomenon which cannot be created from noninverted distributions. The inverted distributions can be applied in several areas such as econometrics, engineering sciences, biological sciences, survey sampling, and medical research. In the literature, several studies related to inverted distributions have been handled by several researchers (see, for example, [24–29]).
Our interest here is with the recently inverted exponetaited Lomax (IELo) distribution (see ). The IELo distribution is the reciprocal of the exponentiated Lomax with the following probability density function (pdf).where is the scale parameter and and are the shape parameters. The cumulative distribution function (cdf) related to (2) is given by
The IELo distribution has several desirable properties: (i) it includes the inverse Lomax (ILo) distribution as a special model (for ); (ii) its hazard rate function (hrf) has flexible characteristics as decreasing, increasing, upside-down bathtub, and reversed J-shaped; and (iii) it is practical applications show that the IELo model often gives better fits than the other well-established models as mentioned in .
In the present work, we introduce a new class of lifetime distributions called the inverse exponentiated Lomax power series (IELoPS) distribution. This class is formed by considering a system with series components and by compounding the IELo distribution with the PS distributions. This class of distributions exhibits a variety of hazard rate shapes, comprises a new class, and contains some new inverse ELo types of distributions compounded with discrete distributions (truncated at zero). We provide several distributional properties including quantile function, expansion of its pdf, moment measures, Lorenz and Bonferroni curves, mean residual life, mean inactivity time, and some uncertainty measures. The maximum likelihood (ML) and Bayesian procedures are used to estimate the parameters of one selective model of the class. A numerical simulation experiment is conducted to examine the precision of the obtained estimators. The potentiality of the one selective model is studied using medical and reliability data. We provide three motivations of the IELoPS class of distribution, which can be applied in some interesting situation:(i)It can arise in many industrial application and biological organisms due to the stochastic reorientation of (ii)It can be used to model approximately the time to the first failure of a system of identical components that are connected in a series system(iii)The nonmonotonic failure rates in the IELoPS family of distributions, such as bathtub, inverted bathtub, and increasing-decreasing failure rates, display certain interesting features that are more likely to be observed in real-life situations
We organize this paper as follows. In Section 2, we introduce the IELoPS distribution and its particular models. We derive some structural properties of the IELoPS distribution in Section 3. In Section 4, we discuss the ML estimator for one selective model, specifically IELo Poisson (IELoP) distribution, and provide expressions for the approximate confidence intervals (ACIs). The Bayesian estimators under squared error (SELF) and linear exponential (LINEX) loss functions are provided in Section 5. In Section 6, a simulation study is designed to assess and compare the performance of the ML and Bayesian estimates. Two real data examples are regarded in Section 7 to reveal the flexibility and potentiality of the IELoP distribution. Conclusions of the paper are given in Section 8.
2. The Class of IELoPS Distribution
In this section, we present the IELoPS class and introduce some of its special models. An important physical explanation of the class of distributions, particularly for use in survival and reliability studies, is as follows. Assume that the failure of a device, system, or component is caused by the presence of an unknown number of initial defects of the same kind, say W, which are only detectable after failure and are perfectly corrected. If the Xi’s are independent and identically distributed (iid) IELo random variables independent of W, a truncated PS random variable, where Xi indicates the time until the device fails owing to the ith defect, for . Then, the time to the first failure, that is , may be described by a distribution in the class of IELoPS distributions.
Definition 1. Let be independent and identically distributed (iid) IELo random variables with pdf (2) and cdf (3). Suppose that W is a discrete random variable following a PS distribution (truncated at zero) with pmf (1). The pdf of IELoPS distributions is derived as follows:
Given , the conditional cdf of is given byHence, is the IELo distribution with parameters , and , so we obtainSo, the marginal cdf of X is given bywhere and . A random variable with cdf (6) following IELoPS distribution with parameters , and will be denoted by .
The pdf, survival function, and hrf of the IELoPS class corresponding to (6) are given, respectively, byandBased on cdf (6), some new compound distributions are listed as follows:(i)For , then the IELoPS distribution gives the inverse Lomax power series class (new)(ii)For , then the IELoPS class gives the IELoP distribution (new)(iii)For , then the IELoPS class gives the ILo Poisson distribution (new)(iv)For , then the IELoPS class gives the IELo logarithmic distribution (new)(v)For , then the IELoPS class gives the ILo logarithmic distribution (new)(vi)For , then the IELoPS class gives the IELo geometric distribution (new)(vii)For , then the IELoPS class gives the ILo geometric distribution (new)(viii)For , then the IELoPS class gives the IELo binomial distribution (new)(ix)For , then the IELoPS class gives the ILo binomial distribution (new)The cdf of the IELoP distribution is obtained, using , from cdf (6) as follows:The pdf of the IELoP distribution corresponding to (10) is as follows:In addition, the hrf takes the following form:The pdf and hrf plots of the IELoP distribution are represented in Figures 1 and 2. The pdf plots are right-skewed, reversed J-shaped, s-shaped, unimodal, and increasing and decreasing for some selected values of the parameters giving the shapes obtained in the below plots. The hrf plots of the IELoP distribution are decreasing, increasing, reversed J-shaped, s-shaped, upside-down shaped for the selected values of parameters. These observations can be seen as significant evidence considered indicating the great flexibility of the IELoP distribution in fitting several data.
3. Statistical Properties
Here, some structural properties of the IELoPS class including, expansion, quantile function, rth moment, incomplete moments, and some entropy measures are obtained. Then, these measures are obtained for IELoP distribution.
Here, we provide two expansions. In the first expansion, we show that the IELo distribution is the limiting distribution for IELoPS class. Secondly, we show that the pdf of IELoPS is expressed as an infinite mixture of IELo distribution.
Firstly; the limiting distribution of the IELoPS class is obtained for and setting in cdf (6) as follows:
Using L’Hospital’s rule in (13),
Hence,which is the cdf of the IELo distribution. Secondly, we show that the pdf of IELoPS class can be represented as a linear combination of the pdf of , using , in pdf (7) as follows:where is the pdf of , given by
3.2. Quantile Function
The quantile function, denoted by defined by, is obtained by inverting cdf (6) as follows:
Using , and in (18), we obtain the quantile function for IELoP distribution as follows:
In the setting, u = 0.5, in (19), we obtain the median of IELoP distribution. Based on quantiles, Bowley’s skewness and Moor’s kurtosis are given by
Plots of skewness and kurtosis of IELoP distribution have appeared in Figure 3. Figure 3 shows different values of shapes with increasing, decreasing, and constant. The kurtosis has range from 0 to 7. The skewness has range from 0 to 1.
3.3. Moment Measures
The rth moment of X has the IELoPS class which is obtained from pdf (16) as follows:
Let , then (21) takes the form,
Using binomial expansion in (22) leads towhere B (.,.) is the beta function. Furthermore, as a particular case, the rth moment of the IELoP distribution is obtained by setting in (23). Some numerical measures including mean , variance (), coefficient of variation (CV), skewness (Sk), and kurtosis (Ku) of the IELoP distribution for sets (i) (ii) (iii) (iv) and (v) are recorded in Table 2.
We conclude from Table 2 the following:(i)Based on (i) and (ii) for the same values of and as the value of increases, we observe that , , and CV measures are increasing, while the skewness and kurtosis measures are decreasing(ii)Based on sets (iii) and (iv), as the value of increases and decreases for the same values of we observe that , , and CV measures are increasing, while the skewness and kurtosis measures are decreasing(iii)Based on sets (iv) and (v), as the value of increases, we observe that , , and CV measures are increasing, while the skewness and kurtosis measures are decreasing(iv)In general, the distribution is skewed to right and leptokurtic
3.4. Incomplete Moments
The incomplete moments have numerous applications in lifetime models. The incomplete moments are used to calculate the mean deviations, Bonferroni and Lorenz curves, mean residual life (MRL), and the mean waiting time (MWT). Here, we obtain the expression for the incomplete moments of the IELoPS class. The rth incomplete moment of the IELoPS class, based on (16), is given by
Let then (24) takes the formwhere B (.,.) is the incomplete beta function. In particular, the Lorenz and Bonferroni curves can be expressed viz , respectively, as follows:which are useful in economics, demography, insurance, engineering, and medicine. Furthermore, the MRL and the MWT can be expressed viz as, respectively,
3.5. Some Information Measures
Entropy has been used in various positions in science and engineering. The entropy measures the uncertainty of the data, that is, the larger value of entropy leads to larger uncertainty in the data. Numerous entropy measures have been proposed and studied in the literature. This section is devoted to obtain the expression for different entropy measures of the IELoPS class. The Rényi entropy (RE) is defined bywhere using (7) takes the form
From , we have the following relation
The Havrda and Charvat entropy (HCE) is defined by
The Arimoto entropy (AE) is defined by
Tsallis entropy (TE) is defined by
From Table 3 we observe the following:(i)The HCE takes the largest values, for all sets (i)−(v) at , compared to other entropies measures which lead to less information(ii)The RE takes the largest values, for set (i) at , compared to other entropy measures, i.e., we have less information(iii)The HCE takes the smallest values, for sets (v) and (vi) at , compared to other entropies measures which lead to more information(iv)As the value of shape parameter increases, for the same value of parameter, all the entropies measures decrease for all values (v)All the entropies measures increase as both shape parameters increase for all values of (vi)According to the above observations, we conclude that the parameter values have a strong effect on the entropy measure with different level of randomness
4. Maximum Likelihood Estimation
This section deals with the ML estimators of the IELoP distribution parameters. Moreover, the ACIs of the parameters are also obtained.
Let denote the observation obtained from a IELoP sample with a set of parameters . The likelihood function of the IELoP distribution can be expressed as follows:where Based on equation (42), the natural logarithm of the likelihood function, denoted by , is given by
The ML estimators, denoted by and are obtained by maximizing directly. So, the ML estimators of are derived by solving the following nonlinear equations:
Furthermore, ACIs of the IELoP distribution parameters are obtained. So, ACI can be approximated by numerically inverting Fisher’s information matrix. Thus, the approximate and the two-sided ACIs for and can be, respectively, easily obtained bywhere is the standard normal percentile and is the standard deviation for ML estimates.
5. Bayesian Estimation
The Bayesian approach deals with the parameters as random variables having a probability distribution. The ability to incorporate prior knowledge into research makes the Bayesian method very useful in reliability analysis. We assume that the prior of and has a gamma distribution with the following pdfs.
The independent joint pdf of and can be written as follows:
Reference  discussed how to elicit the hyperparameters of the informative priors. These informative priors will be obtained from the ML estimates for and by equating the estimate and variance by the inverse of Fisher information matrix of , and Equating mean and variance of , and for gamma priors, we get
Hence, the estimated hyper-parameters can be written as
The Bayesian estimators are obtained based on the most commonly loss functions, specifically SELF and LINEX. The Bayesian estimators of are defined as posterior mean and obtained as follows:
The Bayesian estimator of based on the LINEX loss function is obtained as follows:
Integrals (54)-(55) are complicated to be solved analytically, so the Markov chain Monte Carlo (MCMC) approach will be used. An important subclass of the MCMC techniques is Gibb’s sampling and more general Metropolis within Gibbs samplers. The Metropolis–Hastings (MH) algorithm together with the Gibbs sampling are the two most popular examples of an MCMC method. The MH algorithm considers that, for each iteration, of the process, a candidate value can be produced from a proposal distribution, similar to acceptance-rejection sampling. The MH algorithm generates a sequence of draws from this distribution as follows:(1)Start with any initial values satisfying (2)Using the initial value, sample a candidate point from proposal (3)Given the candidate point , calculate the acceptance probability (4)Draw a value of u from the uniform distribution, if , accept as (5)Otherwise, reject and do (6)Repeat steps 2–5 times until we get J draws.
According to reference , we obtain Bayes credible intervals by using highest probability density of the parameters as follows:(1)Arrange as , , and , where M is length of simulation generated(2)The symmetric credible intervals of become , , , and
6. Simulation Study
In this section, a Monte-Carlo simulation is done to examine and compare the performance of proposed estimates of the IELoP distribution. The Bayesian estimators are obtained using gamma priors under SELF and LINEX loss functions. The main difficulty in the Bayesian procedure is that of obtaining the posterior distribution. The MH algorithm together with the Gibbs sampling is used to simulate the deviates from the posterior pdf. The following steps are outlined as follows:(i)Generate 10000 random samples of size n = 40, 80, and 150 from the IELoP distribution(ii)Using the quantile function in Equation Error! of the IELoP distribution(iii)Four different cases of IELoP parameters values are selected as follows:(i)Case I: (ii)Case II: (iii)Case III: (iv)Case IV: (iv)The ML estimates (MLEs) and associated ACI at are calculated; also, Bayesian estimates (BEs) and associated credible intervals at are also computed(v)Evaluating the performance of the estimates through accuracy measures, including bias, mean squared errors (MSE), and lengths of CI (L. CI)
The simulation outcomes are recorded in Tables 5–7, and the following remakes are noticed.(i)As n increases, the bias, MSE, and the L. CI associated with the parameter estimates decrease for both methods of estimation.(ii)In case I, the bias, MSE, and the L. CI for estimates of and decrease when increases. The bias, MSE, and the L. CI are associated with the parameter estimates of and increase for both methods of estimation as increases value in approximately most of the situations.(iii)In case II, the bias, MSE, and the L .CI for estimates of and increase as the value of increases.(iv)In case III, the bias, MSE, and the L. CI for MLE of and increase as the value of increases.(v)In case IV, the bias, MSE, and the L.CI for MLE of and decrease as the value of increases in majority of situations.(vi)In generality of situations, we notice that the measures (bias, MSE, and L. CI) of BEs are preferable than the corresponding measures of MLE. Also, the measures of BEs based on LINEX loss function are preferable than the measures of BEs based on SELF.(vii)In the generality of situations, the BEs based on LINEX (c = 2) loss function are preferable than BEs based on LINEX (c = −2) loss function.(viii)The length of Bayes credible intervals is shorter than the length of ACI estimates, in most of the situations.
7. Data Analysis
Dataset I: reference  used the data set in their analysis of the generalized Lindley distribution. The data depict the time it took for twenty patients to feel better after taken an analgesic.
Dataset II: we look at a set of strength data that originally published in . The data are for single carbon fibers and impregnated 1000-carbon fiber tows and are measured in GPA. Tension was applied to single fibers with gauge lengths of 10 mm.
We compare the fits of IELoP distribution with some chosen distributions including Kumaraswamy–Weibull (KW) , Poisson–Lomax (PL) , Poisson inverse Lomax (PIL) , Weibull–Lomax (WL) , Gumbel–Lomax (GL) , odd Burr–Weibull–Poisson (OBWP) , and power Lomax–Poisson (PLP)  distributions. We use the following accuracy measures for model comparison: Akaike information criterion (AIC), Bayesian information criterion (BIC), corrected AIC (CAIC), Hannan–Quinn information criterion (HQIC), and Kolmogorov–Smirnov statistics (KSS) with value. Tables 8 and 9 give suggested criteria values as well as the MLEs of the suggested models associated with their standard errors (SE). Comparing the likelihood values, values are based on the KSS, AIC, BIC, CAIC, and HQIC in Tables 8 and 9 for both data sets. We see from these tables that the IELoP distribution is a good alternative model comparing with other fitted models. Also, the value for KSS has its highest value when the lifetime is IELoP distribution. Estimated cdf and pdf for different distributions are shown for dataset I and II in Figures 4 and 5. Q-Q and P-P plots are shown for dataset I and II in Figures 6 and 7, indicating that our distribution is a good choice for modeling the above real data.
Furthermore, the MLE and BE together with their SE of the IELoP model for both data sets are displayed in Tables 10 and 11. Plots of MCMC estimates for and using MCMC sampler performance are represented for data set I and II in Figures 8 and 9, respectively.
As seen in Figures 8 and 9, the algorithm works well with this initial condition and proposal distribution. The samples are correlated, but the Markov chain mixes well. The trace within any iteration would not look much different. This indicates that the convergence in distribution takes place rapidly. The posterior distributions of parameters have a normal distribution. Also, it indicates that averaged four estimators are convergence after 3000 iterations.
8. Summary and Conclusion
In this study, we introduce and define a new class of compound distributions called the IELoPS distribution. The IELoPS class comprises some new flexible lifetime distributions applied in many areas. We obtain several useful structure forms, including quantile function, moments, and incomplete moments, some measures of uncertainty. Plots of density and hazard rate functions for the optional distribution have great flexibility with various shapes. Numerical values of uncertainty measures for IELoP distribution showed that the parameters values have a great influence on the level of uncertainty. The maximum likelihood and Bayesian procedures are employed for estimating the population parameters for IELoP distribution of the class. Bayesian estimator is assessed using symmetric and asymmetric loss functions. Furthermore, approximate confidence intervals and Bayesian credible intervals are obtained. We implement Monte Carlo simulation investigation for the IELPoP distribution. Two real-life data examples are provided from the perspective of practical applications showed that the superiority of the IELoP distribution compared to some other recent models. For further research, Neutrosophic statistics could be used, which is an extension of classical statistics that is used when data come from a complex process or from an uncertain environment [44–46].
All data are available in the paper, and all references for all data and all links are included in the paper.
Conflicts of Interest
The authors declare no conflicts of interest.
This work was funded by the Researchers Supporting Project (no. RSP-2021/363), King Saud University, Riyadh, Saudi Arabia.
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