A topological index is a real number obtained from the chemical graph structure. It can predict the physicochemical and biological properties of many anticancer medicines like blood, breast and skin cancer. This can be done through degree-based topological indices.. In this article, the drugs, azacitidine, buslfan, mercaptopurine, tioguanine, nelarabine, etc. which are used in order to cure blood cancer are discussed and the purpose of the QSPR study is to determine the mathematical relation between the properties under investigation (eg, boiling point, flash point etc.) and different descriptors related to molecular structure of the drugs. It is found that topological indices (TIs) applied on said drugs have a good correlation with physicochemical properties in this context.
Citation: Sumiya Nasir, Nadeem ul Hassan Awan, Fozia Bashir Farooq, Saima Parveen. Topological indices of novel drugs used in blood cancer treatment and its QSPR modeling[J]. AIMS Mathematics, 2022, 7(7): 11829-11850. doi: 10.3934/math.2022660
Related Papers:
[1]
Mohamed S. Eliwa, Essam A. Ahmed .
Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms. AIMS Mathematics, 2023, 8(1): 29-60.
doi: 10.3934/math.2023002
[2]
Naif Alotaibi, A. S. Al-Moisheer, Ibrahim Elbatal, Salem A. Alyami, Ahmed M. Gemeay, Ehab M. Almetwally .
Bivariate step-stress accelerated life test for a new three-parameter model under progressive censored schemes with application in medical. AIMS Mathematics, 2024, 9(2): 3521-3558.
doi: 10.3934/math.2024173
[3]
Mashail M. Al Sobhi .
The modified Kies-Fréchet distribution: Properties, inference and application. AIMS Mathematics, 2021, 6(5): 4691-4714.
doi: 10.3934/math.2021276
[4]
Tahani A. Abushal, Alaa H. Abdel-Hamid .
Inference on a new distribution under progressive-stress accelerated life tests and progressive type-II censoring based on a series-parallel system. AIMS Mathematics, 2022, 7(1): 425-454.
doi: 10.3934/math.2022028
[5]
Hanan Haj Ahmad, Ehab M. Almetwally, Dina A. Ramadan .
A comparative inference on reliability estimation for a multi-component stress-strength model under power Lomax distribution with applications. AIMS Mathematics, 2022, 7(10): 18050-18079.
doi: 10.3934/math.2022994
[6]
Bing Long, Zaifu Jiang .
Estimation and prediction for two-parameter Pareto distribution based on progressively double Type-II hybrid censored data. AIMS Mathematics, 2023, 8(7): 15332-15351.
doi: 10.3934/math.2023784
[7]
Neama Salah Youssef Temraz .
Analysis of stress-strength reliability with m-step strength levels under type I censoring and Gompertz distribution. AIMS Mathematics, 2024, 9(11): 30728-30744.
doi: 10.3934/math.20241484
[8]
Najwan Alsadat, Mahmoud Abu-Moussa, Ali Sharawy .
On the study of the recurrence relations and characterizations based on progressive first-failure censoring. AIMS Mathematics, 2024, 9(1): 481-494.
doi: 10.3934/math.2024026
[9]
Abdulhakim A. Al-Babtain, Rehan A. K. Sherwani, Ahmed Z. Afify, Khaoula Aidi, M. Arslan Nasir, Farrukh Jamal, Abdus Saboor .
The extended Burr-R class: properties, applications and modified test for censored data. AIMS Mathematics, 2021, 6(3): 2912-2931.
doi: 10.3934/math.2021176
[10]
Hatim Solayman Migdadi, Nesreen M. Al-Olaimat, Maryam Mohiuddin, Omar Meqdadi .
Statistical inference for the Power Rayleigh distribution based on adaptive progressive Type-II censored data. AIMS Mathematics, 2023, 8(10): 22553-22576.
doi: 10.3934/math.20231149
Abstract
A topological index is a real number obtained from the chemical graph structure. It can predict the physicochemical and biological properties of many anticancer medicines like blood, breast and skin cancer. This can be done through degree-based topological indices.. In this article, the drugs, azacitidine, buslfan, mercaptopurine, tioguanine, nelarabine, etc. which are used in order to cure blood cancer are discussed and the purpose of the QSPR study is to determine the mathematical relation between the properties under investigation (eg, boiling point, flash point etc.) and different descriptors related to molecular structure of the drugs. It is found that topological indices (TIs) applied on said drugs have a good correlation with physicochemical properties in this context.
1.
Introduction
The purpose of a life testing is to analyze failure times of test units that obtained under normal operating conditions. As it is well known, the modern products are designed to last longer, so for these products, collecting failure data under ordinary circumstances is entirely difficult or even impractical. In such situations, items should be exposed to stress higher than the manufacture levels of stress, in order to obtain data about their failure times. We can call this kind of life test under tough conditions accelerated life testing (ALT), and the failure times of the test units from such ALT can be used to determine life characteristics under normal usage conditions.
There are many models to apply acceleration in the experiments such as Arrhenius model and inverse power model. Choosing the appropriate model of acceleration depends on the type of stress (voltage, pressure, or temperature) that the researcher wants to apply. Arrhenius model and inverse power model are usually used for the thermal and non-thermal stresses, respectively. The common kinds of ALTs are constant-stress ALT (CSALT), step-stress ALT (SSALT) and progressive-stress ALT (PSALT). In the CSALT the stress is constant during the whole time of the experiment, but the stress is increased gradually at specific times, in the SSALT. On the Contrast, the stress is increasing function of the time in the PSALT. ALTs can be divided based on the number of stress levels into two types: Simple and multiple level ALT. Simple ALT only has two levels of stress while, multiple ALT contains more than wo levels of stress.
Several authors have investigated the CSALT, Mohie El-Din et al. [32] considered the estimation problem of the CSALT for the extension of the exponential distribution under progressive type-II censoring. Mohie El-Din et al. [33] discussed the problem of obtaining the optimal CSALT designs for the Lindley distribution. The concern of estimation for CSALT for generalized half normal distribution under complete sampling was addressed by Wang and Shi [39]. Abd El-Raheem [6] developed CSALT's optimal designs for extension of the exponential distribution. The problem of the optimum configuration of CSALT to extension of the exponential distribution under censoring was considered by Abd El-Raheem [7]. For further references on the CSALT, see Abd El-Raheem [8,9,10].
The SSALTs are discussed by several authors see for example, Balakrishnan and Han [17], Ismail [24], Mohie El-Din et al. [30,31], Chandra and Khan [37] and Hafez et al. [22]. For more reading about PSALTs, we can refer to Abdel-Hamid and Al-Hussaini [3,2,4], AL-Hussaini et al. [12], Abdel-Hamid and Abushul [5], Mohie El-Din et al. [35,34] and Abd El-Raheem [11].
Censoring appears normally in reliability examinations. It happens when lifetimes are known only for a portion of the units under examination, the residual lifetimes being known only to exceed determined values. There two main types of censoring schemes which are type-I and type-II censoring schemes (CSs). The fundamental difference between type-I and type-II CSs is that the first depends on the time of ending the experiment and the second depends on the number of failures. The test units in these two CSs cannot be excluded from the test until the end the test. A progressive type-II (PT-II) censoring is an extension of type-II censoring. It encourages the experimenter to remove experimented units at different times throughout the test. For more reading about PT-II CS, we can refer to Balakrishnan and Aggarwala [16], Almetwally et al. [13], Alshenawy et al. [14] and El-Sherpieny et al. [21].
Due to the great importance of PT-II censoring in reliability experiments, many researchers addressed the issue of statistical inference of ALTs under PT-II censored data, we can refer to Abdel-Hamid [1], Abdel-Hamid and AL-Hussaini [2,4], Jaheen et al. [25] and Mohie El-Din et al. [29].
The motivation and contribution in this research is that we apply CSALT for the lifetime of units that follow the MKEx distribution under PT-II censored sample with binomial removal, also we estimate the parameters using Bayesian and classical methods. Moreover, we use a real life censored data of transforms insulation to illustrate the proposed methods.
The paper is organized as follows: The MKEx distribution and test assumptions for CSALT are discussed in Section 2. The maximum likelihood estimation (MLE) for the model parameters are provided in Section 3. Bayes estimates (BEs) under different loss functions are discussed in Section 4. In Section 5, asymptotic and bootstrap confidence intervals (CIs) are presented. Section 6 contains the simulation numerical results. Section 7 contains the transformers turn insulation application. Section 8 contains the conclusion of the paper and the major findings in this research.
2.
Lifetime distribution and test assumptions
2.1. MKEx distribution
In 2013, Kumar and Dharmaja [26] introduced and studied the reduced Kies distribution. In many references, the reduced Kies distribution is known as modified Kies (MK) distribution. Kumar and Dharmaja [26] find out that the MK distribution can perform better than common lifetime distributions such as Weibull distribution and some of its extensions in modelling lifetime data. Kumar and Dharmaja [27] presented the exponentiated MK distribution and introduced its statistical properties. Dey et al. [20] estimated the distribution parameters of MK distribution under progressive type-II censoring and introduced the recurrence relations for the moments of the MK distribution. Based on MK distribution and the T−X family, Al-Babtain et al. [15] introduced a new lifetime distribution, they called it modified Kies exponential (MKEx) distribution. Aljohani et al. [40] discussed the parameter estimation of the MKEx distribution using the MLE method, based on ranked set sampling. It has bathtub shape, increasing and decreasing failure rate. Furthermore, It has the ability to model negatively and positively skewed data. Moreover, it has a closed form cumulative distribution function (CDF) and very easy to handle which make the distribution is candidate to use in different fields such as life testing, reliability, biomedical studies and survival analysis. In this context, Al-Babtain et al.[15] used two different types of real data to show that this distribution may be a good alternative to many popular distributions such as Weibull, Marshall-Olkin exponential, Kumaraswamy exponential, beta exponential, gamma, and exponentiated exponential distribution. The CDF of the MKEx distribution is
F(x;α,σ)=1−e−(eσx−1)α,x>0,α,σ>0.
(2.1)
The corresponding probability density function (PDF) of (2.1) is given by
f(x;α,σ)=ασeασx(1−e−σx)α−1e−(eσx−1)α,x>0,α,σ>0.
(2.2)
2.2. Multiple CSALT's assumptions
In this subsection, we introduce the assumption of CSALT under PT-II CS with binomial removal. Suppose an ALT contains number of stress levels L≥2 such that the stress is arranged ascendingly where ϕ1<ϕ2<...<ϕL, within the level l, l=1,2,...,L, identical nl units are exposed to an accelerated condition, so that the number of units under the lifetime experiment are ∑Ll=1nl=n, where n is the whole number of tested items in the test. In each stress level, ϕl, l=1,2,...,L, at the time of the first failure xl1:ml:nl, Rl1 of the nl−1 remaining units are randomly excluded from the test. In the same manner Rl2 of the surviving units, nl−Rl1−1, are randomly excluded from the test after the second failure xl2:ml:nl is detected. This mechanism continues until the failure of mthl occurs. The remaining surviving units Rlml=nl−ml−∑ml−1j=1Rlj are excluded from the test after the mthl occurs, and the test is terminated. Suppose that the elimination of an individual unit from the test is independent of the others but with the same probability of removal P. Then, the number of units withdrawn at each failure time has a binomial distribution. That is R1∼binomial(nl−ml,P), Rlj∼binomial(nl−ml−∑ml−1l=1Rlj,P),l=2,...,ml and Rlml=nl−ml−∑ml−1j=1Rlj. In this context, the assumptions of multiple CSALT are as follows:
1.In each stress level ϕl, the lifetime of the experimental units follow MKEx(α,σl) distribution.
2.The scale parameter in each stress level σl and the stress level ϕl is linked by the following relation.
log(σl)=ζ+βηl,l=0,1,...,L,
(2.3)
where ζ∈(−∞,∞) and β>0 are the unknown model parameters and ηl=η(ϕl) is an increasing function of ϕ.
(a) If η(ϕl)=log(ϕl), the model in (2.3) becomes the inverse power model.
(b) If η(ϕl)=1−ϕl, the model in (2.3) becomes Arrhenius model.
(c) If η(ϕl)=ϕl, the model in (2.3) becomes exponential model.
For more extensive reading about acceleration and its different models we can refer to the book of Nelson [36], specifically Chapter 2.
From Eq (2.3), we have
σl=σ0exp{β(ηl−η0)}=σ0θzl>0,l=0,1,...,L,
(2.4)
where θ=exp{β(η1−η0)}=σ1σ0>1, σ0>0 is the scale parameter of the MKEx distribution under usage conditions ϕ0, and
The transformation from the parameters (α,σl)=(α,ζ,β) to the new parameters (α,σ0,θ) is an one-to-one mapping. Since the Jacobian determinant from (α,ζ,β) to (α,σ0,θ) does not equal zero. Thus, the unknown parameters should be estimated are α, σ0 and θ.
3.
Estimation via maximum likelihood method
In this section, the classical estimates of the parameters of MKEx distribution under PT-II CS with binomial removal are obtained. As we mentioned later that the Rlj has a binomial distribution, then the PDF of Rlj, l=1,2,...,L is given as follows:
where 0≤rlj≤nl−ml−∑l−1j=1rlj. Furthermore, we suppose that Rlj is independent of Xlj:ml:nl for all l, l=1,2,...,L. Therefore, the likelihood function α, σ0, θ under PT-II censoring with binomial removal is given by
since Xlj:ml:nl and Rlj for all l=1,2,...,L are independent, then the MLE of P can be derived by maximizing Pr(Rlj=rlj) directly. {Hence the MLE of P is given by
where ϵlj=σ0θzlτlj. The MLE of (α,σ0,θ) is (ˆα,^σ0,ˆθ), which may be derived simultaneously by solving Eqs (3.4)-(3.6). Regrettably, solving these equations will be very hard, so we have to use numerical techniques like the Newton-Raphson method.
4.
Bayesian estimation
This section includes the BEs of α, σ0 and θ. We assume that α, σ0 and θ are independent and have gamma priors. Gamma prior for the acceleration factor θ>1 was first considered by DeGroot and Goel [19]. They stated that in most problems of accelerated life testing the accelerating factor θ will be greater than 1. However, in order to not restrict the applicability of the acceleration model we shall consider prior distributions for θ that assign positive density to all positive values of θ. If the experimenter is almost certain that θ>1, then he can choose a gamma prior distribution that assigns a suitably small probability to the interval 0<θ<1. For more details about this point, see Section 3 of DeGroot and Goel [19].} The gamma priors for distribution parameters are as follows:
To determine suitable and superior values of the hyper-parameters of the independent joint prior, we can use estimate and variance-covariance matrix of MLE method. By equating mean and variance of gamma priors as the following equations, the estimated hyper-parameters can be computed as
The BEs of u(Θ)=u(α,σ0,θ) using squared error (SE) and LINEX loss functions are as follows
˜uSE(Θ)=E(u(Θ)),
(4.8)
and
˜uLINEX(Θ)=−1clog[E(e−cu(Θ))],c≠0.
(4.9)
It is obvious that both BEs of u(α,σ0,θ) in (4.8) and (4.9) are considered as the division of more than one integration over each other. As we know multiple integrals is very tough to be solved analytically or even mathematically by hand. Therefore, we have to use the Markov Chain Monte Carlo (MCMC) technique to find an approximate value of integrals. An important methods of the MCMC technique, is the Metropolis-Hastings (MH) algorithm, some times they call it the random walk algorithm. It's similar to acceptance-rejection sampling, the MH algorithm consider for each iteration of the algorithm, a candidate value can be generated from normal proposal distribution}.
The MH algorithm produces a series of draws from MKEx distribution as follows:
1. initiate with α(0)=ˆα,σ(0)0=^σ0,θ(0)=ˆθ.
2. Set i=1.
3. Simulate α∗ from proposal distribution N(α(i−1),var(α(i−1))).
4. Evaluate the acceptance probability A(α(i−1)|α∗)=min[1,C∗(α∗|σ(i−1)0,θ(i−1))C∗(α(i−1)|σ(i−1)0,θ(i−1))].
5. Draw U∼U(0,1).
6. If U≤A(α(i−1)|α∗), put α(i)=α∗, else put α(i)=α(i−1).
7. Do the Steps 2-6 for σ0 and θ.
8. Put i=i+1.
9. Repeat Steps 3-8, N times to obtain (α(1),σ(1)0,θ(1)), ..., (α(N),σ(N)0,θ(N)).
Then, the BEs of u(α,σ0,θ) using MCMC under SE, and LINEX loss functions are respectively
The Bayesian estimates have CIs which are called the credible intervals or some times we call it the highest posterior density (HPD) intervals, for more information see, Chen and Shao [18]. They performed a technique that was used extensively to generate the HPD intervals of unknown parameters of the distribution. In this method, samples are drawn with the proposed MH algorithm that are used to generate estimates, for the HPD algorithm see Chen and Shao [18].
5.
Confidence intervals
In this section, the asymptotic, percentile Bootstrap and Bootstrap-t confidence intervals (CIs) for the unknown distribution parameters α,σ0,θ are obtained.
5.1. Asymptotic confidence intervals
Asymptotic CI is the most popular approach to establish the approximate confidence limits for parameters, we use the MLE to obtain the observed Fisher information matrix I(ˆΩ), which consists of of the negative second derivative of the natural logarithm of the likelihood function evaluated at ˆΩ=(ˆα,^σ0,ˆθ), where
I(ˆΩ)=[IˆαˆαI^σ0ˆαI^σ0^σ0IˆθˆαIˆθ^σ0Iˆθˆθ]
and by inverseing this matrix we can find the asymptotic variance-covariance matrix. Now we can find the asymptotic variance-covariance matrix of the parameter vector Ω is V(ˆΩ)=I−1(ˆΩ).
So the 100(1−γ)% asymptotic confidence intervals for parameters α, σ0 and θ can be established as follows:
(ˆϑl,ˆϑu)=ˆϑ±Z1−γ/2√V(ˆϑ),
(5.1)
where ϑ is α,σ0 or θ, and Zq is the 100q−th percentile of a standard normal distribution.
5.2. Bootstrap confidence interval
In this subsection, two parametric bootstrap methods: Percentile bootstrap (B-P) and the bootstrap-t (B-T) [23]are considered to obtain CIs for α,σ0, and θ.
The percentile bootstrap CIs can be obtained in such a way.
1. Find the values of the MLE of MKEx distribution.
2. To find the estimates of the bootstrap, we must generate a bootstrap samples, (αb,σb0,θb), using MLEs of (α,σ0,θ).
3. Repeat step number (2) B times to have (αb(1),αb(2),…,αb(B)),(σb(1)0,σb(2)0,…σb(B)0) and (θb(1),θb(2),…,θb(B)).
4.Arrange (αb(1),αb(2),…,αb(B)),(σb(1)0,σb(2)0,…σb(B)0) and (θb(1),θb(2),…,θb(B)) from smallest to the biggest, (αb[1],αb[2],…,αb([B])),(σb[1]0,σb[2]0,…σb[B]0) and (θb[1],θb[2],…,θb[B]).
5. The two side 100(1−γ)% percentile bootstrap confidence interval for α,σ0 and θ are evaluated by [αb([Bγ2]),αb([B(1−γ2)])],[σb([Bγ2])0,σb([B(1−γ2)])0] and [θb([Bγ2]),θb([B(1−γ2)])].
The bootstrap-t CIs can be obtained in such a way.
1. Repeat the first two steps in the percentile bootstrap algorithm.
2. Evaluate T=ˆϑb−ˆϑ√V(ˆϑb), where ϑ is α,σ0 or θ, and V(ˆϑb) is asymptotic variances of ˆϑb.
3. Repeat the above two steps B times and rearrange (T(1),T(2),…,T(B)) in ascending order.
4. A two side 100(1−γ)% bootstrap-t CI for α,σ0 and θ are given by [α−Tb([Bγ2])α√V(αb),α+√V(αb)Tb([B(1−γ2)])α], [σ0−Tb([Bγ2])σ0√V(σb0),σ0+√V(σb0)Tb([B(1−γ2)])σ0] and [θ−Tb([Bγ2])θ√V(θb),θ+√V(θb)Tb([B(1−γ2)])θ]
6.
Simulation study
In this section, we conduct a Monte Carlo simulation to find the estimates of the distribution parameters with different sample sizes nl and different censoring schemes Rl with different probability of binomial removal P. We compare the performance of the MLEs and the BEs under different loss functions in terms of their relative absolute bias (RABias), and the mean square error (MSE). Furthermore, we estimate the length of asymptotic CI (L.CI), length of B-P CI (L.BP), length of B-T CI (L.BT) and for Bayesian estimation method we estimate length of credible interval (L.Cr). In the simulation, we consider different sample sizes, nl, different number of failures, ml, and different ratio of failures, rl, where rl=ml/nl. In addition, probability of binomial removal P is considered to be 0.35, and 0.8 for each stress level l, l=1,2,...,L. The true values of the parameters used in the simulation study are (α=2, σ0=1.5, θ=2) and (α=0.8, σ0=1.5, θ=2). The simulation study is done using 10,000 iterations, and the average of the results of these iterations are tabulated in Tables 1-9.
Table 1.
RABias and MSE of the MLE for binomial removal parameter P based on CSALT.
The MCMC iterations and the kernel histograms of the posterior samples of the parameters α, σ0, and θ are plotted in Figure 1.
Figure 1.
MCMC iterations and the kernel histograms of the posterior samples for each parameter with true values: α=2,σ0=1.5,θ=2,n1=60;n2=40;n3=20;n4=10,m1=54; m2=36;m3=18;m4=9,andP=0.8.
Figure 1 shows that the MCMC samples are well mixed and stationary achieved. Also, it indicates that posterior distributions of the three parameters are symmetric.
Lengths of asymptotic, percentile bootstrap, bootstrap-t, and credible CIs when L=2, α=2, σ0=1.5 and θ=2.
Lengths of asymptotic, percentile bootstrap, bootstrap-t, and credible CIs when L=4, α=0.8, σ0=1.5 and θ=2.
The following observations are conducted from the simulation study:
1. For fixed ml, P, ηl and true values of the parameters, the RABias, MSE and length of CIs decrease as nl increases.
2. The best method of estimation is the Bayesian estimation according to the the values of the MSE.
3. The BEs under SE loss function are better than the corresponding estimates under LINEX loss function with c=−0.5 for all parameters.
4. BEs under LINEX loss function for α and σ0 with c=0.5 are better than the corresponding estimates under SE loss function for all cases.
5. BEs under LINEX loss function for θ, with c=0.5 is better than SE loss function when α is less than 1, while, when α is greater than 1, BEs under SE loss function are better.
6. The BEs under LINEX loss function with parameter c=0.5 have MSE less than corresponding estimates with c=−0.5, for all parameters.
7. Lengths of credible CIs are always shorter than the corresponding lengths of asymptotic CIs.
8. The percentile bootstrap CI has the shortest length among all considered CIs.
7.
Application of transformer insulation
In this section, a real data set is analyzed to illustrate the proposed methods in the previous sections. Furthermore, this data set is used to show that the MKEx distribution can be a possible alternative to widely known distributions such as exponential distribution, generalized exponential distribution and Weibull distribution.
Nelson[36] presented in Chapter three of his book the results of a constant-stress accelerated life test of a transformer insulation. The test consisted of three levels of constant voltage, which are respectively 35.4kv, 42.4kv and 46.7kv with normal voltage is 14.4kv. The results of such test are presented in Table 10. In this table, the sign "+" refers to the censored data.
Table 10.
Failure times of transformers insulation.
For each level of this test, we suggested using the following progressives CSs as follows:
1. For ϕ1=35.4kv: n1=10, m1=8 and R1j=0,j=1,...,7, R18=2, with P=0.0277.
2.For ϕ2=42.4kv: n2=10, m2=9 and R2j=0,j=1,...,7, R28=1, R29=0, with P=0.0123.
3. For ϕ3=46.7kv: n3=10, m3=9 and R3j=0, j=1,...,8, R39=1, with P=0.0123.
In the following subsection, we explain how to perform a goodness of fit test for the data in Table 10 and the proposed MKEx distribution.
7.1. Modified KS algorithm for fitting progressive censored data
When the data is PT-II censored data, we have to use modified Kolmogorov-Smirnov (KS) goodness of fit test. The modified Kolmogorov-Smirnov statistic for PT-II censored data was originally introduced by Pakyari and Balakrishnan [38]. This algorithm is based on several steps, first, find the estimates of the parameters for the proposed distribution and next transforming the data to normality, then testing the goodness of fit of the transformed data to normality. Let τ1:m:n<τ2:m:n<...<τm:m:n be a PT-II censored sample with CS (R1,R2,...,Rm) from a distribution function F(t;θ), then the modified KS statistic for PT-II censored data is
Dm:n=max{D+m:n,D−m:n},
(7.1)
where
D+m:n=maxi=1,2,...,m{νi:m:n−ui:m:n},
and
D−m:n=maxi=1,2,...,m{ui:m:n−νi−1:m:n},
where νi:m:n=E(Ui:m:n) is the expected value of the ith PT-II censored order statistic from the U(0,1) distribution, given by
The following algorithm was proposed by Pakyari and Balakrishnan [38] to apply the KS test for PT-II censored data.
1.Find the MLE of the parameter θ, and calculate αi:m:n=F(ti:m:n;ˆθ) for i=1,2,...,m.
2.Evaluate yi:m:n=Φ−1(αi:m:n) for i=1,2,...,m.
3.Considering y1:m:n,y2:m:n,...,ym:m:n as a PT-II censored data from a normal distribution with mean μ and standard deviation σ, calculate the MLEs ˆμ and ˆσ.
4.Evaluate ui:m:n=Φ{(yi:m:n−ˆμ)/ˆσ} for i=1,2,...,m.
5.Evaluate Dm:n according to (7.1).
6.Evaluate the P-value of the test, and reject the null hypothesis H0 at significance level 0.05 if P-value less than 0.05.
Table 11 contains the MLEs and BEs using SE and LINEX loss functions of α, σ0 and θ for the real data set.
Table 11.
Different estimates of unknown parameters for the real data set.
Table 13 contains the value of the life characteristics σl, l=0,1,2,3 for each stress level, also it contains the mean time to failure (MTTF) for each stress level.
Table 13.
The values of life characteristic and MTTF for each stress level for the MKEx distribution.
From Figures 2-5, we can note that with the increase in the stress value, the reliability function tends to zero faster.
The remainder of this section deals with the comparison of the proposed model, MKEx, generalized exponential (GE), Weibull, and exponential distributions. The comparison is performed using the real data shown in Table 10 and the PT-II CSs previously presented in this section. To clarify which of these distributions is more suitable for the data in Table 10, the parameters for the four distributions are estimated and the Akaike information criteria (AIC) is calculated for each distribution. The results of these calculations are summarized in Table 14. Furthermore, the results of modified KS test for the GE, Weibull, and exponential distributions are presented in Table 15.
Table 14.
MLEs of the parameters of MKEx, GE, Weibull, and exponential distributions with AIC.
From Tables 14 and 15, we can see that the MKEx distribution provides a better fit to the given data compared to exponential, GE, and Weibull regarding AIC.
8.
Conclusions
This paper discussed the statistical inference of CSALT under PT-II censoring with binomial removal when the lifetimes of test units follow the MKEx distribution. In this context, we obtained the point and interval estimates for the unknown parameters using both classical and Bayesian methods. We concluded that the Bayesian method was better than the classical method according to MSE and relative absolute bias of the estimates. Regarding the interval estimates, we noted that the percentile bootstrap interval was the best one according to the shortness of the interval length. Furthermore, An application about the insulation of transformers was discussed and used to illustrate the theoretical results. Moreover, the data of insulation of transformers was used to show that the suggested model, MKEx, can be a possible alternative to some well known distributions.
Acknowledgments
The authors are thankful for the Taif University researchers supporting project number TURSP-2020/160, Taif University, Taif, Saudi Arabia.
Conflict of interest
The authors declare no conflict of interest.
References
[1]
B. Figuerola, C. Avila, The phylum bryozoa as a promising source of anticancer drugs, Mar. Drugs, 17 (2019), 477. https://doi.org/10.3390/md17080477 doi: 10.3390/md17080477
[2]
G. Genovese, A. K. Kähler, R. E. Handsaker, J. Lindberg, S. A. Rose, S. F. Bakhoum, et al., Clonal hematopoiesis and blood-cancer risk inferred from blood DNA sequence, New Eng. J. Med., 371 (2014), 2477-2487. https://doi.org/10.1056/NEJMoa1409405 doi: 10.1056/NEJMoa1409405
[3]
T. Terwilliger, M. J. B. C. J. Abdul-Hay, Acute lymphoblastic leukemia: A comprehensive review and 2017 update, Blood Cancer J., 7 (2017), e577-e577. https://doi.org/10.1038/bcj.2017.53 doi: 10.1038/bcj.2017.53
[4]
A. Aslam, Y. Bashir, S. Ahmad, W. Gao, On topological indices of certain dendrimer structures, Z. Naturforsch., 72 (2017), 559-566. https://doi.org/10.1515/zna-2017-0081 doi: 10.1515/zna-2017-0081
[5]
S. M. Hosamani, D. Perigidad, S. Jamagoud, Y. Maled, S. Gavade, QSPR anlysis of certain degree based topological indices, J. Statis. Appl. Prob., 6 (2017), 1-11. https://doi.org/10.18576/jsap/060211 doi: 10.18576/jsap/060211
[6]
M. Randic, Comparative structure-property studies: regressions using a single descriptor, Croat. Chem. Acta, 66 (1993), 289-312.
[7]
M. Randic, Quantitative structure-propert relationship: boiling points and planar benzenoids, New J. Chem., 20 (1996) 1001-1009.
[8]
M. C. Shanmukha, N. S. Basavarajappa, K. N. Anilkumar, Predicting physico-chemical properties of octane isomers using QSPR approach, Malaya J. Math., 8 (2020) 104-116. https://doi.org/10.26637/MJM0801/0018 doi: 10.26637/MJM0801/0018
[9]
S. Hayat, M. Imran, J. Liu, Correlation between the Estrada index and Q-electronic energies for benzenoid hydrocarbons with applica-tions to boron nanotubes, Int. J. Quant. Chem., 2019. https://doi.org/10.1002/qua.26016 doi: 10.1002/qua.26016
[10]
A. Aslam, S. Ahmad, W. Gao, On topological indices of boron triangular nanotubes, Z. Naturforsch., 72 (2017), 711-716. https://doi.org/10.1515/zna-2017-0135 doi: 10.1515/zna-2017-0135
[11]
S. Hayat, S. Wang, J. Liu, Valency-based topological descrip-tors of chemical networks and their applications, Appl. Math. Model., 2018. https://doi.org/10.1016/j.apm.2018.03.016 doi: 10.1016/j.apm.2018.03.016
[12]
S. Hayat, M. Imran, J. Liu, An efficient computational technique for degree and distance based topological descriptors with applications, 2019. https://doi.org/10.1109/ACCESS.2019.2900500
[13]
E. Estrada, L. Torres, L. Rodriguez, I. Gutman, An atom-bond connectivity index: Modeling the enthalpy of formation of alkanes, Indian J. Chem., 37A (1998), 849-855.
[14]
T. Barbui, J. Thiele, H. Gisslinger, H. M. Kvasnicka, A. M. Vannucchi, P. Guglielmelli, et al., The 2016 WHO classification and diagnostic criteria for myeloproliferative neoplasms: document summary and in-depth discussion, Blood Cancer J., 8 (2018), 1-11. https://doi.org/10.1038/s41408-018-0054-y doi: 10.1038/s41408-018-0054-y
[15]
M. Randic, On Characterization of molecular branching, J. Am. Chem. Soc., 97 (1975), 6609-6615. https://doi.org/10.1021/ja00856a001 doi: 10.1021/ja00856a001
[16]
B. Zhou, N. Trinajstic, On general sum-connectivity index, J. Math. Chem., 47 (2010), 210-218. https://doi.org/10.1007/s10910-009-9542-4 doi: 10.1007/s10910-009-9542-4
[17]
D. Vukicevic, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem., 46 (2009), 1369-1376. https://doi.org/10.1007/s10910-009-9520-x doi: 10.1007/s10910-009-9520-x
[18]
M., Adnan, S. A. U. H. Bokhary, G. Abbas, T. Iqbal, Degree-based topological indices and QSPR analysis of antituberculosis drugs, J. Chem., 2022, Article ID 5748626. https://doi.org/10.1155/2022/5748626 doi: 10.1155/2022/5748626
[19]
I. Gutman, Degree based topological indices, Croat.Chem. Acta, 86 (2013), 351-361. https://doi.org/10.5562/cca2294 doi: 10.5562/cca2294
[20]
S. Fajtlowicz, On conjectures of grafitti Ⅱ, Congr. Numerantium, 60 (1987), 189-197.
[21]
G. H. Shirdel, H. RezaPour, A. M. Sayadi, The hyper-zagreb index of graph operations, Iran. J. Math. Chem., 4 (2013), 213-220.
[22]
M. Imran, M. K. Siddiqui, A. Q. Baig, W. Khalid, H. Shaker, Topological properties of cellular neural networks, J. Intell. Fuzzy Syst., 37 (2019), 3605-3614. https://doi.org/10.3233/JIFS-181813 doi: 10.3233/JIFS-181813
[23]
B. Furtula, I. Gutman, A forgotton topological index, J. Math. Chem., 53 (2015), 213-220. https://doi.org/10.1007/s10910-015-0480-z doi: 10.1007/s10910-015-0480-z
[24]
W. Gao, W. Wang, M. K. Jamil, M. R. FArhani, Electron energy studing of moleculer structurevia forgotten topological index computation, J. Chem., 2016. https://doi.org/10.1155/2016/1053183 doi: 10.1155/2016/1053183
[25]
I. B. M. Corp, Released. IBM SPSS Statistics for Windows, Version 24.0 (Armonk, NY: IBM Corp., 2016).
[26]
J. Liu, M. Arockiaraj, M. Arulperumjothi, S. Prabhu, Distance based and Bond additive topological indices of certain repurposed antiviral drug compounds tested for treating COVID- 19, Int. J. Quantum Chem., 121 (2021), e26617. https://doi.org/10.1002/qua.26617 doi: 10.1002/qua.26617
[27]
S. Prabhu, G. Murugan, M. Arockiaraj, M. Arulperumjothi, V. Manimozhi, Molecular topological characterization of three classes of polycyclic aromatic hydrocarbons, J. Mol. Struct., 1229 (2021), 129501. https://doi.org/10.1016/j.molstruc.2020.129501 doi: 10.1016/j.molstruc.2020.129501
[28]
S. Prabhu, Y. S. Nisha, M. Arulperumjothi, D. Sagaya Rani Jeba, V. Manimozhi, On detour index of Cycloparaphenylene and polyphenylene molecular structures, Sci. Rep-UK, 2021. https://doi.org/10.1038/s41598-021-94765-6 doi: 10.1038/s41598-021-94765-6
[29]
Y. Chu, K. Julietraja, P. Venugopal, M. K. Siddiqui, S. Prabhu, Degree- and irregularity-based molecular descriptors for benzenoid systems, Eur. Phys. J. Plus, 136 (2021), 78. https://doi.org/10.1140/epjp/s13360-020-01033-z doi: 10.1140/epjp/s13360-020-01033-z
This article has been cited by:
1.
Mazen Nassar, Farouq Mohammad A. Alam,
Analysis of Modified Kies Exponential Distribution with Constant Stress Partially Accelerated Life Tests under Type-II Censoring,
2022,
10,
2227-7390,
819,
10.3390/math10050819
2.
Gamal M. Ibrahim, Amal S. Hassan, Ehab M. Almetwally, Hisham M. Almongy,
Parameter Estimation of Alpha Power Inverted Topp-Leone Distribution with Applications,
2021,
29,
1079-8587,
353,
10.32604/iasc.2021.017586
3.
Omar Alzeley, Ehab M. Almetwally, Ahmed M. Gemeay, Huda M. Alshanbari, E. H. Hafez, M. H. Abu-Moussa, Ahmed Mostafa Khalil,
Statistical Inference under Censored Data for the New Exponential-X Fréchet Distribution: Simulation and Application to Leukemia Data,
2021,
2021,
1687-5273,
1,
10.1155/2021/2167670
4.
I. Elbatal,
A new lifetime family of distributions: Theoretical developments and analysis of COVID 19 data,
2021,
31,
22113797,
104979,
10.1016/j.rinp.2021.104979
5.
Eslam Hussam, Randa Alharbi, Ehab M. Almetwally, Bader Alruwaili, Ahmed M. Gemeay, Fathy H. Riad, Dost Muhammad Khan,
Single and Multiple Ramp Progressive Stress with Binomial Removal: Practical Application for Industry,
2022,
2022,
1563-5147,
1,
10.1155/2022/9558650
6.
Hisham M. Almongy, Fatma Y. Alshenawy, Ehab M. Almetwally, Doaa A. Abdo,
Applying Transformer Insulation Using Weibull Extended Distribution Based on Progressive Censoring Scheme,
2021,
10,
2075-1680,
100,
10.3390/axioms10020100
7.
Yusra A. Tashkandy, Ehab M. Almetwally, Randa Ragab, Ahmed M. Gemeay, M.M. Abd El-Raouf, Saima Khan Khosa, Eslam Hussam, M.E. Bakr,
Statistical inferences for the extended inverse Weibull distribution under progressive type-II censored sample with applications,
2023,
65,
11100168,
493,
10.1016/j.aej.2022.09.023
8.
Ahlam Tolba, Dina Ramadan, Ehab Almetwally, Taghreed Jawa, Neveen Sayed-Ahmed,
Statistical inference for stress-strength reliability using inverse Lomax lifetime distribution with mechanical engineering applications,
2022,
26,
0354-9836,
303,
10.2298/TSCI22S1303T
9.
Dina A. Ramadan, Ehab M. Almetwally, Ahlam H. Tolba,
Statistical Inference to the Parameter of the Akshaya Distribution under Competing Risks Data with Application HIV Infection to AIDS,
2022,
2198-5804,
10.1007/s40745-022-00382-z
10.
Refah Alotaibi, H. Rezk, Sanku Dey,
MCMC Method for Exponentiated Lomax Distribution based on Accelerated Life Testing with Type I Censoring,
2021,
20,
2224-2880,
319,
10.37394/23206.2021.20.33
11.
Refah Alotaibi, Faten S. Alamri, Ehab M. Almetwally, Min Wang, Hoda Rezk,
Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions,
2022,
10,
2227-7390,
1602,
10.3390/math10091602
12.
Amal S. Hassan, Ehab M. Almetwally, Samia C. Gamoura, Ahmed S. M. Metwally, Ljubisa Kocinac,
Inverse Exponentiated Lomax Power Series Distribution: Model, Estimation, and Application,
2022,
2022,
2314-4785,
1,
10.1155/2022/1998653
13.
Ahlam Tolba, Ehab Almetwally, Neveen Sayed-Ahmed, Taghreed Jawa, Nagla Yehia, Dina Ramadan,
Bayesian and non-Bayesian estimation methods to independent competing risks models with type II half logistic weibull sub-distributions with application to an automatic life test,
2022,
26,
0354-9836,
285,
10.2298/TSCI22S1285T
14.
Ramy Aldallal, Ahmed M. Gemeay, Eslam Hussam, Mutua Kilai, Anoop Kumar,
Statistical modeling for COVID 19 infected patient’s data in Kingdom of Saudi Arabia,
2022,
17,
1932-6203,
e0276688,
10.1371/journal.pone.0276688
15.
O.E. Abo-Kasem, Ehab M. Almetwally, Wael S. Abu El Azm,
Reliability analysis of two Gompertz populations under joint progressive type-ii censoring scheme based on binomial removal,
2023,
0228-6203,
1,
10.1080/02286203.2023.2169570
16.
Mazen Nassar, Sanku Dey, Liang Wang, Ahmed Elshahhat,
Estimation of Lindley constant-stress model via product of spacing with Type-II censored accelerated life data,
2021,
0361-0918,
1,
10.1080/03610918.2021.2018460
17.
Fathy H. Riad, Bader Alruwaili, Ahmed M. Gemeay, Eslam Hussam,
Statistical modeling for COVID-19 virus spread in Kingdom of Saudi Arabia and Netherlands,
2022,
61,
11100168,
9849,
10.1016/j.aej.2022.03.015
18.
Naif Alotaibi, Ibrahim Elbatal, Ehab M. Almetwally, Salem A. Alyami, A. S. Al-Moisheer, Mohammed Elgarhy,
Truncated Cauchy Power Weibull-G Class of Distributions: Bayesian and Non-Bayesian Inference Modelling for COVID-19 and Carbon Fiber Data,
2022,
10,
2227-7390,
1565,
10.3390/math10091565
19.
Refah Alotaibi, Ehab M. Almetwally, Qiuchen Hai, Hoda Rezk,
Optimal Test Plan of Step Stress Partially Accelerated Life Testing for Alpha Power Inverse Weibull Distribution under Adaptive Progressive Hybrid Censored Data and Different Loss Functions,
2022,
10,
2227-7390,
4652,
10.3390/math10244652
20.
Abdulaziz S. Alghamdi, Mubarak H. Elhafian, Hassan M. Aljohani, G.A. Abd-Elmougod,
Estimations of accelerated Lomax lifetime distribution with a dependent competing risks model under type-I generalized hybrid censoring scheme,
2022,
61,
11100168,
6489,
10.1016/j.aej.2021.12.006
21.
Yuge Du, Chunmei Zhang, Wenhao Gui,
Accelerated life test for Pareto distribution under progressive type-II censored competing risks data with binomial removals and its application in electrode insulation system,
2023,
0361-0918,
1,
10.1080/03610918.2023.2175868
22.
Osama Abdulaziz Alamri, Abdulrahman H Alessa, Eslam Hussam, Marwan H. Alhelali, Mutua Kilai,
Statistical modelling for the Covid-19 mortality rate in the Kingdom of Saudi Arabia,
2023,
68,
11100168,
517,
10.1016/j.aej.2023.01.024
23.
Pramote Charongrattanasakul, Wimonmas Bamrungsetthapong, Poom Kumam,
Designing Adaptive Multiple Dependent State Sampling Plan for Accelerated Life Tests,
2023,
46,
0267-6192,
1631,
10.32604/csse.2023.036179
24.
Najwan Alsadat,
A new modified model with application to engineering data sets,
2023,
72,
11100168,
1,
10.1016/j.aej.2023.03.050
25.
Talal Kurdi, Mazen Nassar, Farouq Mohammad A. Alam,
Bayesian Estimation Using Product of Spacing for Modified Kies Exponential Progressively Censored Data,
2023,
12,
2075-1680,
917,
10.3390/axioms12100917
26.
Ronghua Wang, Beiqing Gu, Xiaoling Xu,
RELIABILITY STATISTICAL ANALYSIS OF TWO-PARAMETER EXPONENTIAL DISTRIBUTION UNDER CONSTANT STRESS ACCELERATED LIFE TEST WITH INVERSE POWER LAW MODEL,
2024,
14,
2156-907X,
2993,
10.11948/20240017
27.
Refah Alotaibi, Ehab M. Almetwally, Min Wang, Hoda Rezk,
Optimal scheme and estimation for a bivariate step‐stress accelerated life test with the inverse Weibull distribution under type‐I progressive censored samples,
2023,
39,
0748-8017,
3082,
10.1002/qre.3418
28.
Hassan M. Aljohani,
Estimation for the P(X > Y) of Lomax distribution under accelerated life tests,
2024,
10,
24058440,
e25802,
10.1016/j.heliyon.2024.e25802
29.
Mohamed Sief, Xinsheng Liu, Abd El-Raheem Mohamed Abd El-Raheem,
Inference for a constant-stress model under progressive type-II censored data from the truncated normal distribution,
2024,
39,
0943-4062,
2791,
10.1007/s00180-023-01407-8
30.
Amany E. Aly, Magdy E. El-Adll, Haroon M. Barakat, Ramy Abdelhamid Aldallal,
A new least squares method for estimation and prediction based on the cumulative Hazard function,
2023,
8,
2473-6988,
21968,
10.3934/math.20231120
31.
Eslam Hussam, Ehab M. ALMetwally,
Statistical inference on progressive-stress accelerated life testing for the Perk distribution under adaptive type-II hybrid censoring scheme,
2025,
2214-1766,
10.1007/s44199-025-00103-4
32.
Potluri S S Swetha, Vasili B V Nagarjuna,
The Kumaraswamy modified Kies-G family of distributions: properties and applications,
2025,
100,
0031-8949,
035024,
10.1088/1402-4896/adb10c
Sumiya Nasir, Nadeem ul Hassan Awan, Fozia Bashir Farooq, Saima Parveen. Topological indices of novel drugs used in blood cancer treatment and its QSPR modeling[J]. AIMS Mathematics, 2022, 7(7): 11829-11850. doi: 10.3934/math.2022660
Sumiya Nasir, Nadeem ul Hassan Awan, Fozia Bashir Farooq, Saima Parveen. Topological indices of novel drugs used in blood cancer treatment and its QSPR modeling[J]. AIMS Mathematics, 2022, 7(7): 11829-11850. doi: 10.3934/math.2022660