J. Nig. Soc. Phys. Sci. 1 (2019) 12–19 Journal of the Nigerian Society of Physical Sciences Original Research Application of the Exponentiated Log-Logistic Weibull Distribution to Censored Data Adeniyi F. Fagbamigbea,∗, Gomolemo K. Baseleb, Boikanyo Makubateb, Broderick O. Oluyedec aDepartment of Epidemiology and Medical Statistics, College of Medicine, Faculty of Public Health, University of Ibadan, Nigeria bDepartment of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Palapye, BW cDepartment of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, USA Abstract In a recent paper, a new model called the Exponentiated Log-Logistic Weibull (ELLoGW) distribution with applications to reliability, survival analysis and income data was proposed. In this study, we applied the recently developed ELLoGW model to a wide range of censored data. We found that the ELLoGW distribution is a very competitive model for describing censored observations in life-time reliability problems such as survival analysis. This work shows that in certain cases, the ELLoGW distribution performs better than other parametric model such as the Log-Logistic Weibull, Exponentiated Log-Logistic Exponential, Log-Logistic Exponential distributions and the non-nested Gamma-Dagum (GD). Keywords: Generalized Distribution, Exponentiated log-logistic Distribution, Weibull Distribution, Survival Analysis. Article History : Received: 31-03-2019 Received in revised form: 23-04-2019 Accepted for publication: 23-04-2019 Published: 26-04-2019 c©2019 Journal of the Nigerian Society of Physical Sciences. All Rights Reserved. Communicated by: B. J. Falaye 1. Introduction Oluyede et al. had recently proposed a new model called the Exponentiated Log-Logistic Weibull (ELLoGW) distribu- tion with applications to reliability, survival analysis and in- come data [1]. In this paper, we are primarily concerned with the application of the new model (ELLoGW) to censored data. The details of the model and all its associated properties includ- ing the hazard function, reverse hazard function, quantile func- tion, moments, conditional moments, mean deviations, Bonfer- roni and Lorenz curves, entropy and order statistics have been reported earlier [1]. Censored data arise when a study comes to an end before every subject been studied experience the event of interest [2, ∗Corresponding Author Tel. No: +2348061348165 Email address: fadeniyi@cartafrica.org (Adeniyi F. Fagbamigbe ) 3]. In this study we applied data collected in situations where an experimentalist involved n samples in a life-testing experi- ment. The samples were kept under observation until “failure” occur. The samples could be electrical or mechanical compo- nents, system, individuals, or perhaps computer chips in a reli- ability experiment. It could also be patients, subjects or individuals that are re- cruited for drug or clinical trials. In most cases, such experi- ments are terminated before all the samples could have “failed”. The same could be said of drug or clinical trials wherein the re- cruited subjects might have been lost to follow up, dropped out of the study, withdrew from the study or the study may come to an end due to unforeseen circumstances or economic conditions or exhausted time frame. Also, samples could break accidentally, malfunction in an industrial experiment and thus referred to as “failure”. Gen- erally, the data obtained under these or similar scenarios are 12 Fagbamigbe et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 12–19 13 called censored data. Literature is replete on application of sta- tistical distribution to different types of data [1, 4, 5, 6]. A brief review of the model, the different types of censored data and the application of the model to different types of censored data are discussed in the subsequent sections. 2. The ELLoGW Distribution In this section, the generalized or exponentiated log-logistic Weibull (ELLoGW) distribution is presented. The generalized or exponentiated log-logistic Weibull (ELLoGW) (see Oluyede et al. [1] for details) distribution function is given by GELLoGW (x) = [ 1 − ( 1 + ( x s )c)−1 exp(−αxβ) ]δ , (1) for s, c, α, β, δ, and x ≥ 0. The corresponding probability den- sity function (pdf) is given by gELLoGW (x) = δ [ 1 − ( 1 + ( x s )c)−1 exp(−αxβ) ]δ−1 e−αx β × [ 1 + ( x s )c]−1{ αβxβ−1 + cxc−1 (sc + xc) } , (2) for s, c, α, β, δ > 0, and x ≥ 0. The corresponding hazard function is given by hELLoGW (x) = δ [ 1 − ( 1 + ( x s )c)−1 e−αx β ]δ−1 × e−αx β [ 1 + ( x s )c]−1{ αβxβ−1 + cxc−1 (sc + xc) } × 1 − {1 − (1 + ( x s )c)−1 exp(−αxβ) }δ−1 . 3. Censoring In the following section we construct log-likelihood func- tions of the ELLoGW distribution to deal with type I right cen- soring and type II doubly censored observations. 3.1. Type I Right Censoring This is the most common form of incomplete data often en- countered in survival analysis. Type I censoring arises when the study is conducted over a specified time period that can ter- minate before all the units have failed. Each individual has a fixed censoring time Ci, which would be the time between the date of entry and the end of the study, so that the complete fail- ure time of an individual will be known only if it is less than or equal to the censoring time Ti ≤ Ci; otherwise, only a lower bound of the individual lifetime is available Ti > Ci. The data for this design are conveniently indicated by pairs of random variables (Ti, εi), i = 1, ..., n. Consider a sample of size n of independent positive random variables T1, ..., Tn, such that Ti is associated with an indicator variable εi=0 if Ti is a censoring time. Let Θ=(s,c,α,β,δ)T , then the likelihood function, L(Θ), of a type I right censored sample (t1,ε1), ..., (tn,εn) from the EL- LoGW distribution with pdf gELLoGW (.) and survival function S ELLoGW (.) can be written as L(Θ) = n∏ i=1 gELLoGW (ti) �i S ELLoGW (ti) 1−�i, (3) where S ELLoGW (ti)= 1 - GELLoGW (ti). The log-likelihood func- tion, l(Θ), based on data, from the ELLoGW distribution is, l(Θ) = n∑ i=1 { �i ln [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αti β )δ−1 × e−αti β ( 1 + ( ti s )c)−1( αβti β−1 + ctic−1 sc + tic )] (4) + (1 − �i) ln [ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αti β )δ]} . Elements of the score vector are given by( ∂l(Θ) ∂s , ∂l(Θ) ∂c , ∂l(Θ) ∂α , ∂l(Θ) ∂β , ∂l(Θ) ∂δ )T . The MLEs Θ̂ = (ŝ, ĉ, α̂, β̂, δ̂)T are obtained from the nu- merical maximization of equation (4). Let Θ = (s, c,α,β,δ)T be the parameter vector and Θ̂ = (ŝ, ĉ, α̂, β̂, δ̂)T be the max- imum likelihood estimate of Θ = (s, c,α,β,δ)T . Under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, [12] we have: √ n(Θ̂ − Θ) d −→ N5(0, I−1(Θ)), where I(Θ) is the ex- pected Fisher Information Matrix. The asymptotic behavior is still valid if I(Θ) is replaced by the observed information matrix evaluated at Θ̂, that is J(Θ̂). The multivariate normal distribu- tion N5(0, J(Θ̂)−1), where the mean vector 0 = (0, 0, 0, 0, 0)T , can be used to construct confidence intervals and confidence re- gions for the individual model parameters and for the survival and hazard functions. 3.2. Type II Censoring For type II censoring, the data consists of the rth smallest lifetimes X(1) ≤ X(2) ≤ ... ≤ X(r) out of a random sample of n lifetimes X1, X2, ..., Xn from the ELLoGW distribution. Assum- ing X1 ≤ X2 ≤ ... ≤ Xn are independent and identically dis- tributed and have a continuous distribution with pdf gELLoGW (.) and survival function S ELLoGW (.), it follows that the joint pdf of X(1), ..., X(r) is L(Θ) = n! (n − r)! [ r∏ i=1 gELLoGW (x(i)) ][ S ELLoGW (x(r)) ]n−r ,(5) 13 Fagbamigbe et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 12–19 14 Table 1: Times to Infection Data Estimates Statistics Model s c α β δ −2 log L AIC AICC BIC W∗ A∗ KS pvalue S S ELLoGW 2.1674 0.8092 0.0090 1.7083 4.1106 98.0372 108.0372 114.7039 111.5775 0.0225 0.1556 0.999 0.0196 (2.5612) (0.7057) (0.0450) (1.4104) (2.9263) LLoGW 20.4723 1.5446 0.0071 1.7720 1 98.2032 106.2032 112.8698 109.0354 0.0244 0.1655 0.9988 0.0213 (0.0309) (1.0376) (0.0193) (0.8868) - ELLoGE 0.0192 0.3982 0.1266 1 28.8554 98.4424 106.4424 113.1091 109.2746 0.0242 0.1697 0.9922 0.0208 (0.0991) (0.4326) (0.0633) - (0.0002) LLoGE 12.5296 2.5301 0.0295 1 1 99.4833 105.4833 112.1500 107.6075 0.0283 0.1963 0.9981 0.0193 (7.2980) (1.0222) (0.0482) - - λ β δ α θ GD 9.0005 1.9151 0.4786 3.9043 0.0192 98.7984 108.7984 115.4651 112.3386 0.0274 0.1932 0.9676 0.0254 (13.2476) (2.5511) (0.4362) (3.4708) (0.0658) where S ELLoGW (x) = 1 − GELLoGW (x). The log-likelihood func- tion, l(Θ), based on data from the ELLoGW distribution is, l(Θ) = ln [ n! (n − r)! ] + r∑ i=1 { ln δ + (δ− 1) × ln [ 1 − ( 1 + ( x(i) s )c)−1 e−αx β (i) ] −αxβ(i) (6) − ln [ 1 + ( x(i) s )c] + ln { αβxβ−1(i) + cxc−1(i) sc + xc(i) } +(n − r) ln [ 1 − ( 1 − ( 1 + ( x(r1 ) s )c) e−αx β (r1 ) )δ]} . Elements of the score vector and MLE’s are similar to those in the type I right censoring scheme. 3.3. Type II Double Censoring Type II double censoring is a censoring procedure wherein a fixed number of observations is missing at the two ends of a n sample size unlike in type I censoring, where the number of censored observations is totally random and the study time is predetermined. The data consist of the remaining ordered observations tr+1,...,tm when the r smallest observations and the n−m largest observations are out of a sample of size n from the ELLoGW distribution. The likelihood function, L(Θ), of the type II doubly censored sample t(r+1),...,t(m) from the ELLoGW distribution with pdf gELLoGW (.), cdf GELLoGW (.) and survival function S ELLoGW (.) is given by L(Θ) = n! r!(n − m)! {GELLoGW (t(r+1))} r ×{S ELLoGW (t(m))} n−m m∏ i=r+1 gELLoGW (t(i)). (7) The log-likelihood function, l(Θ), for a type II doubly censored sample t(r+1),...t(m) from the ELLoGW distribution is given by l(Θ) = ln ( n! r!(n − m)! ) +r ln [( 1 − ( 1 + ( t(r+1) s )c)−1 e−αt(r+1) β )δ] +(n − m) ln [ 1 − ( 1 − ( 1 + ( t(m) s )c)−1 e−αt(m) β )δ] + m∑ i=r+1 ln { δ ( 1 − ( 1 + ( t(i) s )c)−1 e−αt(i) β )δ−1 (8) ×e−αt(i) β ( 1 + ( t(i) s )c)−1( αβt(i) β−1 + ct(i)c−1 sc + t(i)c )} . Elements of the score vector and their MLE’s are similar to those in the type I right censoring scheme. 4. Applications In this section, we give some applications to real life data. Maximum likelihood estimates of the model parameters under type I right and type II double censored data are obtained and comparisons with the Log-Logistic Weibull (LLoGW), Expo- nentiated Log-Logistic Exponential (ELLoGE) and Log-Logistic Exponential (LLoGE) distributions as well as the non-nested Gamma-Dagum (GD) [11, 13] distribution are presented. The MLEs of the parameters (standard error in parenthe- sis), -2log-likelihood statistic (−2 ln(L)), Akaike Information Criterion (AIC = 2p − 2 ln(L)), Bayesian Information Crite- rion (BIC = p ln(n) − 2 ln(L)), and Corrected Akaike Informa- tion Criterion ( AICC = AIC + 2 p(p+1)n−p−1 ) , where L = L(Θ̂) is the value of the likelihood function evaluated at the parameter esti- mates, n is the number of observations, and p is the number of estimated parameters are presented. The goodness-of-fit statis- tics: Kolmogorov-Smirnov p-value (KS p-value), Cramer-von Mises (W∗) and Anderson-Darling (A∗) were also presented in the Tables. These statistics can be used to verify which distri- bution fits better to the data. In general, the smaller the values of W∗ and A∗, the better the fit. The ELLoGW distribution is fitted to the data sets and these fits are compared to the fits using exponentiated log-logistic exponential (ELLoGE), log-logistic Weibull (LLoGW) and log-logistic exponential (LLoGE) distri- butions. 14 Fagbamigbe et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 12–19 15 Figure 1: Fitted Densities, Probability Plots for Times to Infection Data Table 2: Maintenance Data: Type I right censoring Estimates Statistics Model s c α β δ −2 log L AIC AICC BIC W∗ A∗ KS p − value S S ELLoGW 0.4681 1.2222 0.0023 3.9238 3.2765 115.3248 125.3248 127.2603 133.3794 0.0330 0.2262 0.8368 0.0360 (0.4880) (0.2612) (0.0033) (0.8702) (2.9931) LLoGW 1.7033 1.4847 0.0049 3.4126 1 118.9152 126.9152 128.8507 133.3589 0.0744 0.4677 0.5637 0.0569 (0.5094) (0.3395) (0.0301) (3.6607) - ELLoGE 0.0169 0.6744 0.4229 1 27.9948 120.4324 128.4324 130.3678 134.8760 0.0576 0.3904 0.7290 0.0594 (0.0207) (0.2535) (0.1848) - (0.0003) LLoGE 1.9005 1.8526 0.1235 1 1 124.1565 130.1565 132.0920 134.9893 0.0902 0.5814 0.6966 0.0641 (1.1114) (0.3484) (0.2375) - - λ β δ α θ GD 1.7891 2.6292 0.4189 3.1506 0.0332 121.6979 131.6979 133.6333 139.7525 0.0686 0.4541 0.6322 0.0724 (3.2287) (6.4012) (0.7477) (8.2541) (0.1681) Table 3: Maintenance Data: Type II double censoring Estimates Statistics Model s c α β δ −2 log L AIC AICC BIC W∗ A∗ KS p − value S S ELLoGW 0.5157 1.3934 0.0003 4.9765 5.0079 97.5309 107.5309 109.9309 114.7009 0.0450 0.3378 0.5716 0.0535 (0.1059) (0.2409) (0.0003) (0.7204) (0.0146) LLoGW 21.0083 1.6085 0.1990 1.6298 1 105.1903 113.1903 115.5903 118.9262 0.1017 0.6910 0.3093 0.0976 (0.0046) (2.4384) (0.0849) (0.2486) - ELLoGE 0.0221 0.9696 0.3636 1 98.3757 102.7356 110.7356 113.1356 116.4715 0.0777 0.5469 0.7063 0.0845 (0.0337) (0.4440) (0.2369) - (0.00007) LLoGE 5.1884 25.1414 0.3791 1 1 106.8501 112.8501 115.2501 117.1521 0.0782 0.5817 0.1414 0.1575 (0.3930) (26.2939) (0.0784) - - λ β δ α θ GD 1.7696 2.9249 0.5022 3.0936 0.0341 104.1335 114.1335 116.5335 121.3034 0.0853 0.5947 0.5902 0.0895 (3.0503) (3.8793) (0.4692) (4.9479) (0.0966) 4.1. Type I right censoring: Times to Infection Data The following example utilizes the times to infection (in months) of kidney dialysis patients with a surgically placed catheter. This data is given in Table 1.2 of Klein and Moeschberger [7]. The infection times are given in Table A.1 and the censored observations are given in Table A.2 in the appendix. The initial values for the ELLoGW distribution in R code are s = 2, c = 0.9, α = 0.01, β = 1.9 and δ = 4. The MLEs of the parameters of the ELLoGW distribution and its related sub- models, AIC, AICC, BIC, W∗, A∗, KS p-value and SS are given in Table 1, along with standard errors in parentheses. Plots of the fitted densities and the histogram, observed probability vs predicted probability ([6]) are given in Figure 1. The likelihood ratio (LR) test statistic of the hypothesis H0: LLoGW against Ha: ELLoGW and H0: ELLoGE against Ha: ELLoGW are 0.166 (p-value = 0.6837) and 0.4052 (p-value 15 Fagbamigbe et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 12–19 16 Fitted PDF x f( x) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ELLoGW LLoGW ELLoGE LLoGE GD 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 The Graph of Observed vs Expected Probability Observed Probability E xp e ct e d P ro b a b ili ty ELLoGW(SS=0.0360) LLoGW(SS=0.0569) ELLoGE(SS=0.0594) LLoGE(SS=0.0641) GD(SS=0.07479690) Figure 2: Fitted Densities, Probability Plots for Type I right censoring maintenance data Fitted PDF x f( x) 0 1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5 ELLoGW LLoGW ELLoGE LLoGE GD 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 The Graph of Observed vs Expected Probability Observed Probability E xp e ct e d P ro b a b ili ty ELLoGW(SS=0.0546) LLoGW(SS=0.0976) ELLoGE(SS=0.0845) LLoGE(SS=0.1575) GD(SS=0.0895) Figure 3: Fitted Densities, Probability Plots for Type II double censoring maintenance data = 0.5244). We can conclude that there are no significant dif- ferences between fits of the LLoGW and ELLoGW distribu- tions, as well as between fits of the ELLoGE and ELLoGW distributions. The values of the goodness-of-fit statistics A∗ and W∗ shows that the ELLoGW distribution is the better fit for the time to infection data. The Kolmogorov-Smirnov p-value shows that the ELLoGW distribution fits the data better than the other models. Also, the value of SS from the probability plot in Figure 1 is the smallest for ELLoGW model. Also, Figure 1 showed that the new model ELLoGW provided better fit for the density and probability plots than the LLOGW, ELLoGW, LLoGE and GD models. 16 Fagbamigbe et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 12–19 17 4.2. Type I right censoring: Maintenance Data In the following example, we consider a maintenance data set. The set of data is the maintenance data with 46 observa- tions reported on active repair times (hours) for an airborne communication transceiver. The data was discussed by Alven [8], Chhikara and Folks [9], and Dimitrakopoulou et al. [10]. The maintenance data set is given in Table A.3 in the appendix. The data given in Table A.4 has eight observations removed to illustrate type I right censoring (see appendix for the data). Initial values for the ELLoGW model in R code are s = 0.3, c = 1, α = 0.002, β = 3 and δ = 3. The MLEs of the param- eters of the ELLoGW distribution and its related submodels, AIC, AICC, BIC, W*, A*, KS p-value and SS are given in Ta- ble 2. Plots of the fitted densities and the histogram, observed probability vs predicted probability are given in Figure 2. The LR test statistic of the hypothesis H0: LLoGW against Ha: ELLoGW and H0: ELLoGE against Ha: ELLoGW are 3.5904 (p-value = 0.0581) and 5.1076 (p-value = 0.0238). We can conclude that there is a significant difference between the fits of the ELLoGE and ELLoGW distributions. There is also a significant difference between fits of the LLoGW and ELLoGW distributions at the 10 % level of significance. The values of the goodness-of-fit statistics A∗ and W∗ shows that the ELLoGW distribution is by far the better fit for the maintenance data. The KS p-value shows that the ELLoGW distribution fits the main- tenance data better than the other models. Figure 2 showed that the new model ELLoGW provided better fit for the density and probability plots than the LLOGW, ELLoGW, LLoGE and GD models. 4.3. Type II double censoring: Maintenance Data In the following example we consider the maintenance data set with 15 observations removed to illustrate type II Double censoring. The data is given in Table A.5 in the appendix. Initial values for the ELLoGW model are s = 0.4, c = 1.4, α = 0.001, β = 5 and δ = 5. The MLEs of the parameters of the ELLoGW distribution and its related sub models, AIC, AICC, BIC, W*, A*, KS p-value and SS are given in Table 3, along with standard errors in parentheses. Plots of the fitted densities and the histogram, observed probability vs predicted probability are given in Figure 3. The LR test statistic of the hypothesis H0: LLoGW against Ha: ELLoGW and H0: ELLoGE against Ha: ELLoGW are 7.6594 (p-value = 0.0056) and 5.2047 (p-value = 0.0225). We can conclude that there is a significant difference between the fits of the LLoGW and the ELLoGW distributions as well as be- tween the fits of the ELLoGE and ELLoGW distributions. The values of the goodness-of-fit statistics A∗ and W∗ shows that the ELLoGW distribution is by far the better fit for the maintenance data. The KS p-value for the ELLoGW distribution is greater than that of the LLoGW and LLoGE models. Figure 3 showed that the new model ELLoGW provided better fit for the density and probability plots than the LLOGW, ELLoGW, LLoGE and GD models. 5. Concluding Remarks We have presented the ELLoGW distribution under differ- ent censoring mechanisms. Applications of the model, under type I right censoring and type II double censoring, to real data are presented to illustrate its usefulness and applicability. The findings suggest that the ELLoGW distribution performed bet- ter than all other distributions considered. Acknowledgment We thank the referees for the positive enlightening com- ments and suggestions, which have greatly helped us in making improvements to this paper. References [1] B. O. Oluyede, G. Basele, S. Huang & B. Makubate, “A New class of Generalized Log-Logistic Weibull Distribution: Theory, Properties and Applications”, Journal of Probability and Statistical Sciences, 14 (2016) 171. [2] D. 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Elements of the Score Vector: Type I Right Censoring Elements of the score vector for the Type I right censoring scheme are given by, ∂`(Θ) ∂s = n∑ i=1 { �i [ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2 (−1) ( 1 + ( ti s ))−3( ti s )c−1( cti s2 ) e−αt β i ( αβtβ−1i + ctc−1i sc + tci ) + [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ][( 1 + ( ti s )c)−2( ti s )c−1( cti s2 )( αβtβ−1i + ctc−1i sc + tci ) − ( 1 + ( ti s )c)−1 c2(sti)c−1 (sc + tci ) 2 ]] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 +(1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−2( ti s )c−1( cti s2 )][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} , ∂`(Θ) ∂c = n∑ i=1 { �i [ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2 (−1) ( 1 + ( ti s ))−3( ti s )c ln ( ti s ) e−αt β i ( αβtβ−1i + ctc−1i sc + tci ) + [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ][ (−1) ( 1 + ( ti s )c)−1( ti s )c ln ( ti s )( αβtβ−1i + ctc−1i sc + tci ) − ( 1 + ( ti s )c)−1 × t2(c−1)i ln(ti)(s c + tci ) − ct c−1 i (s c ln s + tci ln ti) (sc + tci ) 2 ]][ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 +(1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−2 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−2( ti s )c ln ( ti s )][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} , ∂`(Θ) ∂α = n∑ i=1 { �i [[ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2( 1 + ( ti s ))−1 tβi e −2αtβi −δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1( 1 + ( ti s ))−1 tβi e −αtβi ] × ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci ) + δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−1 βtβ−1i ] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 + (1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 ×e−αt β i ( 1 + ( ti s )c)−1 (−tβi ) ][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} , ∂`(Θ) ∂β = n∑ i=1 { �i [[ δ(δ− 1) ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−2( 1 + ( ti s ))−1 αtβi e −2αtβi ln(ti) −δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1( 1 + ( ti s ))−1 ×αtβi e −αtβi ln(ti) ]( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci ) + δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s )c)−1 (αtβ−1i + αβt β−1 i ln(ti)) ] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 + (1 − �i) [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 ×e−αt β i ( 1 + ( ti s )c)−1 (−αtβi ) ln(ti) ][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} and ∂`(Θ) ∂δ = n∑ i=1 { �i [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ln ( 1 − ( 1 + ( ti s )c)−1 e−αt β i ) + ( 1 − ( 1 + ( ti s )c)−1 e−αt β i ] × [ δ ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ−1 e−αt β i ( 1 + ( ti s ))−1( αβtβ−1i + ctc−1i sc + tci )]−1 + (1 − �i) [( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ × ln ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )][ 1 − ( 1 − ( 1 + ( ti s )c)−1 e−αt β i )δ]−1} . 18 Fagbamigbe et al. / J. Nig. Soc. Phys. Sci. 1 (2019) 12–19 19 Tables with Complete and Censored data The infection times are given in Table A.1 and the censored observations are given in Table A.2. The complete and censored maintenance data are given in Tables A.3, A.4 and A.5, respectively. Table A.1: Infection Times 1.5 3.5 4.5 5.5 8.5 8.5 9.5 10.5 11.5 15.5 16.5 18.5 23.5 26.5 Table A.2: Censored Infection Times 2.5 2.5 3.5 3.5 3.5 4.5 4.5 5.5 6.5 6.5 7.5 7.5 7.5 7.5 8.5 9.5 10.5 11.5 12.5 12.5 13.5 14.5 14.5 21.5 21.5 22.5 22.5 25.5 27.5 Table A.3: Complete Maintenance Data 0.2 0.3 0.5 0.5 0.5 0.5 0.6 0.6 0.7 0.7 0.7 0.8 0.8 1.0 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 7.0 7.5 8.8 9.0 10.3 22.0 24.5 Table A.4: Type I Right Censored Maintenance Data 0.2 0.3 0.5 0.5 0.5 0.5 0.6 0.6 0.7 0.7 0.7 0.8 0.8 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 Table A.5: Maintenance Data II 0.6 0.7 0.7 0.7 0.8 0.8 1.0 1.0 1.0 1.1 1.3 1.5 1.5 1.5 1.5 2.0 2.0 2.2 2.5 2.7 3.0 3.0 3.3 3.3 4.0 4.0 4.5 4.7 5.0 5.4 5.4 19