==== Front PLoS One PLoS One plos plosone PLoS ONE 1932-6203 Public Library of Science San Francisco, CA USA 10.1371/journal.pone.0241970 PONE-D-20-26265 Research Article Physical Sciences Mathematics Statistics Statistical Data Physical Sciences Mathematics Probability Theory Probability Distribution Skewness Computer and Information Sciences Data Management Data Visualization Infographics Graphs Physical Sciences Physics Thermodynamics Entropy Engineering and Technology Industrial Engineering Reliability Engineering Reliability Physical Sciences Mathematics Probability Theory Statistical Distributions Physical Sciences Mathematics Probability Theory Random Variables Engineering and Technology Kumaraswamy inverse Gompertz distribution: Properties and engineering applications to complete, type-II right censored and upper record data Kumaraswamy inverse Gompertz distributionhttps://orcid.org/0000-0002-7585-5519El-Morshedy M. Data curationFormal analysisMethodologyProject administrationResourcesSoftwareSupervisionValidationVisualization12* El-Faheem Adel A. Formal analysisFunding acquisitionMethodologyValidationWriting – original draft3 El-Dawoody M. ConceptualizationProject administrationResourcesSoftwareSupervisionVisualizationWriting – review & editing13 1 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt 3 Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt Chen Feng Editor Tongii University, CHINA Competing Interests: The authors have declared that no competing interests exist. * E-mail: m.elmorshedy@psau.edu.sa 3 12 2020 2020 15 12 e024197021 8 2020 25 10 2020 © 2020 El-Morshedy et al2020El-Morshedy et alThis is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.This article proposes and studies a new three-parameter generalized model of the inverse Gompertz distribution, in the so-called Kumaraswamy inverse Gompertz distribution. The main advantage of the new model is that it has "an upside down bathtub-shaped curve hazard rate function" depending upon the shape parameters. Several of its statistical and mathematical properties including quantiles, median, mode, moments, probability weighted moment, entropy function, skewness and kurtosis are derived. Moreover, the reliability and hazard rate functions, mean time to failure, mean residual and inactive lifetimes are also concluded. The maximum likelihood approach is done here to estimate the new model parameters. A simulation study is conducted to examine the performance of the estimators of this model. Finally, the usefulness of the proposed distribution is illustrated with different engineering applications to complete, type-II right censored, and upper record data and it is found that this model is more flexible when it is compared to well-known models in the statistical literature. The author(s) received no specific funding for this work. Data AvailabilityAll relevant data are within the paper.Data Availability All relevant data are within the paper. ==== Body 1. Introduction The two-parameter Gompertz (G) distribution was offered by [1] and it can be displayed as an extension of the exponential distribution. It has an influential role in survival analysis for forming adequate actuarial and human mortality tables. Also, it is a beneficial model for survival distributions characterized by increasing hazard rate and also to describe the distribution of adult life spans by demographers and actuaries; see [2]. Several authors have contributed to the studies that accentuate the statistical characterization and methodology of the G distribution; like [3–10]. The inverse distributions were introduced in the modeling literature in demography, biological and actuarial surveys; look in [10–15]. The inverse Gompertz (IG) distribution was proposed by [16] and it was introduced as a lifetime model. Suppose that X is a random variable that has an IG distribution whose shape and scale parameters are λ > 0 and β > 0, respectively. Then, the cumulative distribution function (CDF) of X takes the formula G(x)=e-λβ(eβx-1);x>0;λ,β>0.(1) Also, the probability density function (PDF) of X takes the formula g(x)=λx2eβxe-λβ(eβx-1);x>0;λ,β>0.(2) [17] presented a new lifetime model having only one parameter and it is called A distribution which is characterized by an upside-down bathtub shaped hazard function. The random variable Z has an A distribution with scale parameter β > 0, if the CDF of Z takes the form F(z)=e-1β(eβz-1);z>0;β>0.(3) The PDF of Z takes the form f(z)=1z2eβze-1β(eβz-1);z>0;β>0.(4) Setting λ = 1 in Eqs (1) and (2), we deduce the CDF and PDF of A distribution with the parameter β. That is, the A distribution is a particular case of IG distribution. In many workable circumstances, the classical distributions don’t give a sufficient fit to actual data. Therefore, various generators are proposed to produce a new models; see [18–28]. The Kumaraswamy-G (K-G) family is one of the essential generators that have an increased interest after the persuasive debate on the pitfalls of the beta-G family suggested by [29]. Cordeiro and de Castro (2011) clarified the CDF of the two-parameter K-G that takes the formula F(x)=1-{1-G(x)θ}γ.(5) The corresponding PDF to Eq (5) will be f(x)=γθg(x)G(x)θ-1{1-G(x)θ}γ-1,(6) where, g(x) and G(x) are the PDF and CDF of a baseline random variable X. Also, θ > 0 and γ > 0 are two extra shape parameters. The principal suggest in our paper is a generalization of the IG distribution called the Kumaraswamy inverse Gompertz distribution, abbreviated KuIG, depending on the Eqs (5) and (6). The failure rate function of the new model takes the form of "an upside down bathtub-shaped". Another important characteristic of the KuIG is that it suitable for testing the goodness of fit of some special sub-models, such as the IG and A distributions. The article is distributed as follows: Section 2 introduces the CDF and the corresponding PDF of the KuIG. Section 3 presents several fundamental statistical properties. Some essential functions used in reliability analysis are introduced in Section 4. The maximum likelihood approach is mentioned in Section 5 to appreciate the parameters of the KuIG. In Sections 6 and 7, we will obtain the maximum likelihood estimators for type-II right censored and upper record data, respectively. The performance of the KuIG estimators is appreciated in Section 8 using a simulation study. We will analyze five actual data sets are three complete, one type-II right censored, and one upper record data in Section 9 and the results are compared with different known distributions. Finally, in Section 10, a conclusion for the obtained results is presented. 2. Kumaraswamy inverse Gompertz distribution 2.1. Specifications of KuIG The non-negative random variable X is said to have the KuIG with the vector of parameters Ω = (α, β, γ), say X ∼ KuIG(Ω), if its CDF is given by the formula F(x)=1-{1-e-αβ(eβx-1)}γ;x>0;α,β,γ>0,(7) where, α = λθ. We can easily see that, if we substitute the CDF of A distribution, instead of the CDF of IG in Eq (5), we will obtain the same result. So, the proposed model can be named KuIG or KuA. The PDF of the KuIG will be f(x)=αγx2eβxe-αβ(eβx-1){1-e-αβ(eβx-1)}γ-1;x>0;α,β,γ>0.(8) The two parameters α and γ are the shape parameters and β is the scale parameter. Setting γ = 1 in the Eqs (7) and (8), we will get the CDF and PDF of the IG with the two parameters α and β, respectively. Moreover, if we put α = γ = 1 in the above two equations, we will obtain the CDF and PDF of A distribution with the parameter β. This confirms the fact that the IG and A distributions are particular cases of our proposed KuIG. Fig 1 shows the graphical behavior of the PDF for KuIG with different values of α, β and γ. 10.1371/journal.pone.0241970.g001Fig 1 The graphs of the PDF for KuIG. 3. Statistical characteristics of KuIG 3.1. The quantiles and median An explicit formula for the quantile and the median of KuIG are derived in this subsection. The quantile xq of the KuIG is given as follows xq=βln{1-βαln(1-(1-q)1γ)};00;α,β,γ>0.(23) The failure (hazard) rate function (HRF) of X is found by the formula h(x)=αγx2eβx(eαβ(eβx-1)-1)-1.(24) The graphic behavior of the HRF of KuIG with various choices of α, β and γ is offered in Fig 3. 10.1371/journal.pone.0241970.g003Fig 3 The graphs of the HRF for KuIG. If X ∼ KuIG(Ω), then the reversed failure rate function of X is r(x)=αγx2eβxe-αβ(eβx-1)(1-e-αβ(eβx-1))γ-1{1-(1-e-αβ(eβx-1))γ}-1.(25) 4.2. The mean time to failure If X ∼ KuIG(Ω), then the mean time to failure (MTTF) of X is found by MTTF=∫0∞xf(x;Ω)dx=μ(1). Substituting from Eq (13), when r = 1, we find that MTTF=∑k=0γ-1∑i=0∞∑j=0i∑m=0∞(γ-1k)(-1)i+j+k+1(k+1)i(1+i)mαi+1β-ij-mγΓ(m)j!m!(i-j)!.(26) 4.3. The mean residual lifetime In life testing situations and reliability theory, the mean residual lifetime (MRL), say Mr(t), is known as the anticipated remaining lifetime T − t, provided that the component has been survived until a time t. The Mr(t) is found using the formula Mr(t)=E(T-t|T>t). If T ∼ KuIG(Ω), then the MRL of T is Mr(t)=1R(t)∫t∞R(x)dx=11−F(t)[μ(1)−∑k=0γ∑i=0∞∑j=0i(γk)(−1)i+j+kαikiβ−ij!(i−j)!∫0te(i−j)βxdx]=11−F(t)[μ(1)−∑k=0γ∑i=0∞∑j=0i∑m=0∞(γk)(−1)i+j+kαikiβm−i(i−j)mt1−mj!m!(i−j)!(1−m)].(27) 4.4. The mean inactive lifetime The mean waiting (inactive) lifetime (MWT), say Mw(t), measures the time elapsed since the component fails with lifetime T, provided that it has failed some time before t, t > 0. It is defined as Mw(t)=E(t-T|T≤t). If T ∼ KuIG(Ω), then the MWT of T is Mw(t)=1F(t)∫0tF(x)dx=1F(t)[t−∑k=0γ∑i=0∞∑j=0i∑m=0∞(γk)(−1)i+j+kαikiβm−i(i−j)mt1−mj!m!(i−j)!(1−m)].(28) 5. Maximum likelihood estimators for complete data In this section, we will discuss and study how to use the maximum likelihood approach to appreciate the unknown parameters (α, β, γ) of the KuIG. Suppose that x1, x2, …, xn be a randomly selected sample with size n from the KuIG(Ω), thus the log-likelihood function L(Ω) for it is given by L(Ω)=nln(αγ)+β∑i=1n1xi-2∑i=1nln(xi)-αβ∑i=1n(eβxi-1)+(γ-1)∑i=1nln(1-e-αβ(eβxi-1)).(29) By deriving the first partial derivatives of L(Ω) with regard to α, β and γ and put it equal to zero, the normal equations of L(Ω) will take the forms nγ^+∑i=1nln(1-e-α^β^(eβ^xi-1))=0,(30) nα^-1β^∑i=1n(eβ^xi-1)+(γ^-1)β^∑i=1neβ^xi-1eα^β^(eβ^xi-1)-1=0(31) and ∑i=1n1xi-α^β^∑i=1n1xieβ^xi+α^β^2∑i=1n(eβ^xi-1)+α^(γ^-1)β^2∑i=1n(β^xi-1)eβ^xi+1eα^β^(eβ^xi-1)-1=0.(32) Eqs (30), (31) and (32) don’t have an explicit solutions to α^,β^ and γ^. Therefore, we will solve the previous system of equations numerically to obtain the maximum likelihood estimators (MLEs) (α^,β^,γ^). 6. Maximum likelihood estimators for type-II right censored data If a life testing experiment stopped over when a limited number of items are observed to be failed, then the remaining items are indicated to be a type-II right censored. The inference associated with this type of data is less efficient than the inference associated with the complete data because some information about the parameters under study will be missed with the type-II right censored data; see [34]. If x(1), x(2), ……, x(k), k ≤ n denote the ordered values of a random sample x1, x2, ……, xn (failure times) and observations terminate after the kth failure occurs, then the likelihood function take the form ([35]) lcen.=n!(n-k)!(R(xk))n-k∏i=1kf(xi).(33) If x1, x2, …, xn be a random sample taken from KuIG(Ω), then the L(Ω) of x(1), x(2), ……, x(k), k ≤ n is found by the formula L(Ω)=kln(αγ)+ln(n!(n−k)!)+(n−k)ln(1−e−αβ(eβxk−1))γ+β∑i=1k1xi−2∑i=1kln(xi)−αβ∑i=1k(eβxi−1)+(γ−1)∑i=1kln(1−e−αβ(eβxi−1)).(34) The first partial derivatives of L(Ω) are obtained by differentiating Eq (34) for α, β and γ as ∂L∂γ=kγ+∑i=1kln(1-e-αβ(eβxi-1)),(35) ∂L∂α=kα+γ(n-k)(eβxk-1)β(eαβ(eβxk-1)-1)-1β∑i=1k(eβxi-1)+(γ-1)β∑i=1keβxi-1eαβ(eβxi-1)-1(36) and ∂L∂β=αγ(n−k)((βxk−1)eβxk+1)β2(eαβ(eβxk−1)−1)+∑i=1k1xi−αβ∑i=1k1xieβxi+αβ2∑i=1k(eβxi−1)+α(γ−1)β2∑i=1k(βxi−1)eβxi+1eαβ(eβxi−1)−1.(37) Equating Eqs (35), (36) and (37) to zero, we will get the normal equations of L which don’t have an explicit solutions to α^,β^, and γ^ and must be solved numerically to find the maximum likelihood estimators (MLEs) (α^,β^,γ^). 7. Maximum likelihood estimators for upper record data The study of record values has extensive applications to real world situations such as sporting events, meteorological and seismological sciences and life testing studies. The upper record value is that one which is larger than all watched values so far. Suppose that X = {XU(1), XU(2), …, XU(n)} is an upper record values taken from a random sample x1, x2, …, xn that follow the KuIG, then the likelihood function of X can be expressed by [36] lreco.=f(xU(n);Ω)∏i=1n-1f(xU(i),Ω)R(xU(i),Ω),0≤xU(1)