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.­e IELoPS class contains some signicant new exible lifetime distributions that possess powerful physical explications applied in areas like industrial and biological studies. ­e IELoPS class comprises the inverse Lomax power series as a new subclass as well as several new exible 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. ­e methods of maximum likelihood and Bayesian estimations are used to estimate the model parameters of one optional model. ­e Bayesian estimators of parameters are discussed under squared error and linear exponential loss functions. ­e asymptotic condence 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 dierent 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, exibility, and usefulness of IELoP distribution. For the suggested two real data sets, the IELoP distribution ts better than Kumaraswamy–Weibull, Poisson–Lomax, Poisson inverse Lomax, Weibull–Lomax, Gumbel–Lomax, odd Burr–Weibull–Poisson, and power Lomax–Poisson distributions.


Introduction
In recent years, many families of distributions were proposed by combining certain bene cial continuous and power series (PS) distributions by several researchers. is 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 by P(W w; θ) a w θ w C(θ) , w 1, 2, 3, . . . , where a w depends only on w, θ > 0 is the scale parameter, C(θ) ∞ w 1 a w θ w , and C ′ (·) and C ″ (·) denote the rst and second derivatives of C(θ), respectively. Important quantities of some PS distributions (truncated at zero) such as Poisson, logarithmic, geometric, and binomial are provided in Table 1.
e 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 X i (the lifetime of i th component), can be denoted by the nonnegative random variable X � min X i W i�1 or Y � max X i W i�1 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 [1], Weibull-PS family [2], generalized exponential PS family [3], Burr XII-PS family [4], complementary Poisson Lindley-PS family [5], exponentiated extended Weibull family [6], complementary exponentiated inverted Weibull-PS family [7], Gompertz PS family [8], generalized modified Weibull-PS family [9], generalized inverse Weibull-PS family [10], exponential Pareto-PS family [11], exponentiated power Lindley-PS family [12], Burr-Weibull PS family [13], odd log-logistic PS family [14], generalized inverse Lindley PS family [15], exponentiated generalized PS family [16], exponentiated power generalized Weibull-PS family [17], new Lindley-Burr XII-PS [18], power function-PS family [19], inverse gamma PS family [20], and power quasi-Lindley PS family [21], 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.
e 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][25][26][27][28][29]).
Our interest here is with the recently inverted exponetaited Lomax (IELo) distribution (see [29]). e IELo distribution is the reciprocal of the exponentiated Lomax with the following probability density function (pdf ).
g(x; δ, c, φ) � cφδ where δ is the scale parameter and c and φ are the shape parameters. e cumulative distribution function (cdf ) related to (2) is given by e IELo distribution has several desirable properties: (i) it includes the inverse Lomax (ILo) distribution as a special model (for φ � 1); (ii) its hazard rate function (hrf ) has flexible characteristics as decreasing, increasing, upsidedown 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 [29].
In the present work, we introduce a new class of lifetime distributions called the inverse exponentiated Lomax power series (IELoPS) distribution. is class is formed by considering a system with series components and by compounding the IELo distribution with the PS distributions. is 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. e 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.
e 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 W � min(X 1 ; X 2 ; . . . ; X W ) (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) e 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). e 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.

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 X i 's are independent and identically distributed (iid) IELo random variables independent of W, a truncated PS random variable, where X i indicates the time until the device fails owing to the i th defect, for i ≥ 1. en, the time to the first failure, that is X � min X i W i�1 , may be described by a distribution in the class of IELoPS distributions. Definition 1. Let X 1 , . . . , X W 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). e pdf of IELoPS distributions is derived as follows: Given X � min X i W i�1 , the conditional cdf of X|W is given by Hence, X|W � w is the IELo distribution with parameters δ, c, and wφ, so we obtain So, the marginal cdf of X is given by where . A random variable with cdf (6) following IELoPS distribution with parameters δ, c, φ, andθ will be denoted by X ∼ IELoPS(δ, c, φ, θ). e pdf, survival function, and hrf of the IELoPS class corresponding to (6) are given, respectively, by Journal of Mathematics and h(x; Θ) � δcφθ Based on cdf (6), some new compound distributions are listed as follows: (i) For φ � 1, then the IELoPS distribution gives the inverse Lomax power series class (new) (ii) For C(θ) � e θ − 1, θ > 0, then the IELoPS class gives the IELoP distribution (new) (iii) For C(θ) � e θ − 1, θ > 0, φ � 1, then the IELoPS class gives the ILo Poisson distribution (new) (iv) For C(θ) � − ln(1 − θ), 0 < θ < 1, then the IELoPS class gives the IELo logarithmic distribution (new) IELoPS class gives the ILo binomial distribution (new) e cdf of the IELoP distribution is obtained, using C(θ) � e θ − 1, from cdf (6) as follows: e pdf of the IELoP distribution corresponding to (10) is as follows: In addition, the hrf takes the following form: e pdf and hrf plots of the IELoP distribution are represented in Figures 1 and 2. e 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. e hrf plots of the IELoP distribution are decreasing, increasing, reversed Jshaped, s-shaped, upside-down shaped for the selected values of parameters. ese observations can be seen as significant evidence considered indicating the great flexibility of the IELoP distribution in fitting several data.

Statistical Properties
Here, some structural properties of the IELoPS class including, expansion, quantile function, r th moment, incomplete moments, and some entropy measures are obtained.
en, these measures are obtained for IELoP distribution.

Expansions.
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 θ ⟶ 0 + and setting C(θ) � ∞ w�1 a w θ w in cdf (6) as follows: Using L'Hospital's rule in (13),

Journal of Mathematics
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 X � min X i W i�1 , using C ′ (θ) � ∞ w�1 wa w θ w− 1 , in pdf (7) as follows: where 3.2. Quantile Function. e quantile function, denoted by Q(u) defined byQ(u) � u, is obtained by inverting cdf (6) as follows: (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. e kurtosis has range from 0 to 7. e skewness has range from 0 to 1.

Moment Measures.
e r th moment of X has the IELoPS class which is obtained from pdf (16) as follows: Let y � (1 + (δ/x)) − c , then (21) takes the form, Using binomial expansion in (22) leads to where B (.,.) is the beta function. Furthermore, as a particular case, the r th moment of the IELoP distribution is obtained by . Some numerical measures including mean (μ 1 ′ ), variance (σ 2 ), coefficient of variation (CV), skewness (Sk), and kurtosis (Ku) of the IELoP distribution for sets Table 2.
We conclude from Table 2     (iii) Based on sets (iv) and (v), as the value of φ increases, we observe that μ 1 ′ , σ 2 , and CV measures are increasing, while the skewness and kurtosis measures are decreasing (iv) In general, the distribution is skewed to right and leptokurtic

Incomplete Moments.
e incomplete moments have numerous applications in lifetime models. e 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. e r th incomplete moment of the IELoPS class, based on (16), is given by Let y � (1 + (δ/x)) − c , then (24) takes the form where B (.,.) is the incomplete beta function. In particular, the Lorenz and Bonferroni curves can be expressed viz Ξ 1 (t), respectively, as follows: which are useful in economics, demography, insurance, engineering, and medicine. Furthermore, the MRL and the MWT can be expressed viz Ξ 1 (t) as, respectively, 3.5. Some Information Measures. Entropy has been used in various positions in science and engineering. e 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. is section is devoted to obtain the expression for different entropy measures of the IELoPS class. e Rényi entropy (RE) is defined by where (f(x; Θ)) ς using (7) takes the form 8 Journal of Mathematics From [30], we have the following relation Using (31) in (30), then we have erefore, (30) will be written as follows: where for t > 1, d ς,t � t − 1 t n�1 [n(ς + 1) − t]ρ n d ς,t− n , and d ς,0 � 1. Using binomial expansion in (33), (29) takes the form Setting (34) in (28), RE of IELoPS class is obtained as follows: e Havrda and Charvat entropy (HCE) is defined by Journal of Mathematics e HCE of IELoPS class is obtained after using (34) in (36) as follows: e Arimoto entropy (AE) is defined by Hence the AE of the IELoP class is given by using (34) in (38) as follows: Tsallis entropy (TE) is defined by Hence, the TE of IELoPS class is obtained by using (34) in (40) as follows: In particular, the RE, HCE, AE, and TE of the IELoP distribution are obtained after putting C(θ) � e θ − 1, in (35), (37), (39), and (41), respectively.
Numerical values of RE, HCE, AE, and TE, of the IELoP distribution for the same chosen values of parameters given in Section 3.3, are recorded in Table 3.
From Table 3 we observe the following:

Maximum Likelihood Estimation
is section deals with the ML estimators of the IELoP distribution parameters. Moreover, the ACIs of the parameters are also obtained.
Let x 1 , . . . , x n denote the observation obtained from a IELoP sample with a set of parameters Θ � (δ, c, φ, θ). e likelihood function of the IELoP distribution can be expressed as follows: where D i � (1 + (δ/x i )) Based on equation (42), the natural logarithm of the likelihood function, denoted by l(Θ), is given by e ML estimators, denoted by δ, c, φ, and θ are obtained by maximizing l(Θ) directly. So, the ML estimators of Θ are derived by solving the following nonlinear equations: Equations (44)− (47) have no closed form solutions; therefore, an iterative technique can be employed to obtain the parameter estimators using optimization algorism as Newton-Raphson.
Furthermore, ACIs of the IELoP distribution parameters are obtained. So, ACI can be approximated by numerically inverting Fisher's information matrix.

Bayesian Estimation
e Bayesian approach deals with the parameters as random variables having a probability distribution. e ability to incorporate prior knowledge into research makes the Bayesian method very useful in reliability analysis. We assume that the prior of δ, c, φ, and θ has a gamma distribution with the following pdfs.
(49) e independent joint pdf of δ, c, φ, and θ can be written as follows: Reference [31] discussed how to elicit the hyperparameters of the informative priors.
ese informative priors will be obtained from the ML estimates for δ, c, φ, and θ by equating the estimate and variance (V Θ , Θ � (δ, c, φ, θ)) by the inverse of Fisher information matrix of δ, c, φ, and θ. Equating mean and variance of δ, c, φ, and θ for gamma priors, we get Hence, the estimated hyper-parameters can be written as For more examples, see [32,33]. From the likelihood function and joint prior function, the joint posterior density function of the IELoP distribution is e Bayesian estimators are obtained based on the most commonly loss functions, specifically SELF and LINEX. e Bayesian estimators of Θ are defined as posterior mean and obtained as follows: e Bayesian estimator of Θ 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.

Simulation Study
In this section, a Monte-Carlo simulation is done to examine and compare the performance of proposed estimates of the IELoP distribution.
e Bayesian estimators are obtained using gamma priors under SELF and LINEX loss functions. e main difficulty in the Bayesian procedure is that of obtaining the posterior distribution.
e MH algorithm together with the Gibbs sampling is used to simulate the deviates from the posterior pdf.
e 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: (iv) e ML estimates (MLEs) and associated ACI at α � 0.05 are calculated; also, Bayesian estimates (BEs) and associated credible intervals at α � 0.05 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) e simulation outcomes are recorded in Tables 5-7, and the following remakes are noticed.

Data Analysis
Dataset I: reference [35] used the data set in their analysis of the generalized Lindley distribution. e 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 [36]. e 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) [37], Poisson-Lomax (PL) [38], Poisson inverse Lomax (PIL) [39], Weibull-Lomax (WL) [40], Gumbel-Lomax (GL) [41], odd Burr-Weibull-Poisson (OBWP) [42], and power Lomax-Poisson (PLP) [43] 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 P 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, P 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 P value for KSS has its highest value                   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 δ, c, φ, and θ 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. e samples are correlated, but the Markov chain mixes well. e trace within any iteration would not look much different. is indicates that the convergence in distribution takes place rapidly. e posterior distributions of parameters have a normal distribution. Also, it indicates that averaged four estimators are convergence after 3000 iterations.

Summary and Conclusion
In this study, we introduce and define a new class of compound distributions called the IELoPS distribution. e 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. e 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][45][46].

Data Availability
All data are available in the paper, and all references for all data and all links are included in the paper.

Conflicts of Interest
e authors declare no conflicts of interest.