Loading [MathJax]/jax/output/SVG/jax.js
Research article

The extended Burr-R class: properties, applications and modified test for censored data

  • Received: 24 October 2020 Accepted: 29 December 2020 Published: 08 January 2021
  • MSC : 60E05, 62F10

  • This article introduces a new three-parameter Marshall-Olkin Burr-R (MOB-R) family which extends the generalize Burr-G class. Some of its general properties are discussed. One of its special models called the MOB-Lomax distribution is studied in detail for illustrative purpose. A modified chi-square test statistic is provided for right censored data from the MOB-L distribution. The model parameters are estimated via the maximum likelihood and simulation results are obtained to assess the behavior of the maximum likelihood approach. Applications to real data sets are provided to show the usefulness of the proposed MOB-Lomax distribution. The modified chi-square test statistic shows that the MOB-Lomax model can be used as a good candidate for analyzing real censored data.

    Citation: 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[J]. AIMS Mathematics, 2021, 6(3): 2912-2931. doi: 10.3934/math.2021176

    Related Papers:

    [1] A. M. Abd El-Raheem, Ehab M. Almetwally, M. S. Mohamed, E. H. Hafez . Accelerated life tests for modified Kies exponential lifetime distribution: binomial removal, transformers turn insulation application and numerical results. AIMS Mathematics, 2021, 6(5): 5222-5255. doi: 10.3934/math.2021310
    [2] 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
    [3] 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
    [4] Hassan Okasha, Mazen Nassar, Saeed A. Dobbah . E-Bayesian estimation of Burr Type XII model based on adaptive Type-Ⅱ progressive hybrid censored data. AIMS Mathematics, 2021, 6(4): 4173-4196. doi: 10.3934/math.2021247
    [5] 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
    [6] Ehab M. Almetwally, Ahlam H. Tolba, Dina A. Ramadan . Bayesian and non-Bayesian estimations for a flexible reduced logarithmic-inverse Lomax distribution under progressive hybrid type-Ⅰ censored data with a head and neck cancer application. AIMS Mathematics, 2025, 10(4): 9171-9201. doi: 10.3934/math.2025422
    [7] Samah M. Ahmed, Abdelfattah Mustafa . Estimation of the coefficients of variation for inverse power Lomax distribution. AIMS Mathematics, 2024, 9(12): 33423-33441. doi: 10.3934/math.20241595
    [8] 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
    [9] Refah Alotaibi, Hassan Okasha, Hoda Rezk, Abdullah M. Almarashi, Mazen Nassar . On a new flexible Lomax distribution: statistical properties and estimation procedures with applications to engineering and medical data. AIMS Mathematics, 2021, 6(12): 13976-13999. doi: 10.3934/math.2021808
    [10] 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. AIMS Mathematics, 2023, 8(9): 21968-21992. doi: 10.3934/math.20231120
  • This article introduces a new three-parameter Marshall-Olkin Burr-R (MOB-R) family which extends the generalize Burr-G class. Some of its general properties are discussed. One of its special models called the MOB-Lomax distribution is studied in detail for illustrative purpose. A modified chi-square test statistic is provided for right censored data from the MOB-L distribution. The model parameters are estimated via the maximum likelihood and simulation results are obtained to assess the behavior of the maximum likelihood approach. Applications to real data sets are provided to show the usefulness of the proposed MOB-Lomax distribution. The modified chi-square test statistic shows that the MOB-Lomax model can be used as a good candidate for analyzing real censored data.



    Marshall Olkin in 1997 gave a general method to extend any existing distribution by adding one additional parameter called tiled parameter for the seek of goodness of fit. Later on many researchers adopted this method and proposed many new extended distributions, available in literature some of them are the Marshall Olkin Pareto distribution [7], Marshall Olkin beta distribution [14], Marshall Olkin extended Weibull distribution [10], Marshall Olkin exponentiated Burr XII distribution [11], Marshall Olkin additive Weibull distribution [2], Marshall Olkin alpha power family [22], Marshall Olkin Burr III family [1], and Marshall Olkin power generalized Weibull distribution [3], among many more.

    In this study, we introduced a new family of distributions called Marshall Olkin Burr-R (MOB-R) family by inserting the generalized Burr-G (Arslan et al. [20]) in the Marshall Olkin family (Marshall and Olkin [16]), we obtain the following cdf and pdf of the MOB-R class as

    G(x;η,θ,k,ξ)=1(1+{log[1R(x)]}θ)ρ1(1η)(1+{log[1R(x)]}θ)ρ (1.1)

    and

    g(x;η,θ,ρ,ξ)=θρηr(x){log[1R(x)]}θ1(1+{log[1R(x)]}θ)ρ1{1R(x)}[1(1η)(1+{log[1R(x)]}θ)ρ]2. (1.2)

    Its hazard rate function (hrf) has the form

    h(x;η,θ,ρ,ξ)=θρr(x)1R(x){log[1R(x)]}θ1(1+{log[1R(x)]}θ)[1(1η)(1+{log[1R(x)]}θ)ρ]. (1.3)

    The random variable (rv) X having the density (1.2)is denoted by XMOB-R(η,θ,ρ). The quantile function (qf), Qx(u), of the MOB-R family reduces to

    QX(u)=R1(1exp [{(u11ˉηu)1ρ1}1θ]), (1.4)

    where x=Q(u) follows the MOBG family with Uuniform(0,1).

    Motivation

    The MOB-R family can be justified physically as follows. Cosider N independent components which are related by a series system and suppose that N is a rv with a probability mass function P(N=n)=δ(1δ)n, 0<θ<1 and n=0,1,.... Let X1,X2,...,Xn represents the lifetimes of the components, which are assumed to be independently and identically distributed rv's with cdf F(x;θ,ρ,ξ). The rv Y=min(X1,...,XN) refers to time of the first failure and its cdf takes the form

    G(y)=1n=0P[min(X1,...,Xn)>Y]δ(1δ)n=1δˉF(x)n=0{(1δ)ˉF(x)}n=1(1+{log[1R(x)]}θ)ρ1(1η)(1+{log[1R(x)]}θ)ρ,

    where R(x) is any baseline cdf, i.e 0<X< such as Lomax, Frechet, log-logistic and Weibull distributions.

    The rest of the article is followed as. In Section 2, a useful linear mixture representation. The general properties of the MOB-R class are provided in Section 3. In Section 4, estimation of the MOB-R parameters is carried out by maximum likelihood. In Section 5, we study the MOB-Lomax model as a special model of the proposed family. In Section 6, a modified chi-square test statistic for right censored data is applied to validate the MOB-L model. In Section 7, we provide simulation results to check the performance of the maximum likelihood and to validate the test statistic. Real data applications are analyzed to show the flexibility of the MOB-Lomax model in Section 8. Further, a real censored data is analyzed to validate the considered test statistic. Some concluding remarks are reported in Section 9.

    This section provides an infinite linear mixture for the cdf and pdf of the MOB-R class in (1.1) and (1.2). By using following two series expansions

    (1z)ρ=i=0(ρ+i1i)zi.

    The log-power expansion has the form

    [log (1+z)]β=βρ=0(ρβρ)ρj=0(1)jβj(ρj)ϱj,ρzρ,

    where ϱj,ρ=1ρρm=1(jmρ+m)θmϱj,ρm, pj,0=1 and θρ=(1)ρρ+1.

    The equation (1.1) reduce to

    G(x;η,θ,ρ,ξ)=1m=0amG(x)m, (2.1)

    where am=ηi=0ˉηij=0(ρ(i+1)+j1j),θj(mθjm)ml=0(1)l+cj+m+jcjl(ml)ϱl,m

    and am are weights and am=1. Equation (2.1) can be written as

    G(x;η,θ,ρ,ξ)=m=0bmHm(x), (2.2)

    where b0=1a0, bm=am and Hm(x)=R(x)m.

    Similarly

    g(x;η,θ,ρ,ξ)=m=0bm+1hm+1(x), (2.3)

    where hm+1(x)=(m+1)r(x)R(x)m. Equations (2.3) and (2.2) refer to the infinite linear representation for the cdf and pdf of the MOB-R class in terms of their baseline distributions, and they are helpful to derive the MOB-R properties.

    In this section, we derive some properties of MOB-R family, such as moments, generating function (mgf), stochastic ordering, reliability parameter, and order statistics.

    The rth moments of MOB-R family is derived using the following expression

    μr=E(xr)=xrg(x;η,θ,ρ,ξ)dx.

    Using mixture representation given in (2.3), we have

    μr=bm+1xrhm+1(x)dx.
    μr=bm+1Δr, (3.1)

    where Δr=xrhm+1(x)dx. Similarly, incomplete moments of MOB-R family can be calculated using the following formula

    μmr=bm+1m0xrhm+1(x)dx. (3.2)

    Applications of Equation (3.2) are related to mean deviation, Zenga index, income quantiles such as Lorenz and Bonferroni curves, mean waiting time, and mean residual life.

    The mgf of the MOB-G family can be obtained by using the following expression

    M0(t)=E(etx)=etxg(x;η,θ,ρ,ξ)dx. (3.3)

    Using mixture representation given in (2.3), we have

    M0(t)=bm+1etxhm+1(x)dx,

    Note that these integrals only depends only for the arbitrary base line distribution.

    In the field of reliability, stress-strength model has an important role which defines the life time of a component which has a random strength, say X1, which is subject to an accidental stress, say X2. The component will still work when X1>X2. It has many application in engineering. Let X1 and X2 be two rv's follow the MOB-R family i.e. X1MOB-R(θ1,ρ1,η1) and X2MOB-R(θ2,ρ2,η2) with a common shape and scale parameters.

    R=P(X1<X2)=0f1(x)F2(x)dx. (3.4)

    Using Equations (2.2) and (2.3), we can write

    R=P(X1<X2)=m=0p=0bp+1bm0hp+1(x)Hm(x)dx, (3.5)

    where hp+1(x) and Hm(x) are already defined in the previous section.

    The stochastic ordering is commonly used in showing the ordering mechanism in life time distribution. A rv X is stochastically greater than the rv Y if FX(x)FY(x) for all x's. Further, there are some important stochastic orderings namely, stochastic order, hazard rate order, mean residual order, likelihood ratio order, and reversed hazard rate order which are related to each other according to the following chain of stochastic orders

    XrhrYXlrYXhrYXstYXmrlY

    Further details about different stochastic orderings and their definitions can be explored in Shaked and Shanthikumar [23]. Let X1MOBG(θ,ρ,η1) and X2MOBG(θ,ρ,η2). Then, using likelihood ratio ordering defined by [f(x)g(x)], we can write

    f(x)g(x)=[1ˉη1ξ1ˉη2ξ]2,

    where ξ=(1+(HR(x))θ)ρ and ξ=θρhR(x)HR(x)θ1(1+(HR(x))θ)ρ1. Therefore,

    ddxf(x)g(x)=2[1ˉη1ξ1ˉη2ξ]ξη1η2(1ˉη2ξ)2.

    From the above expression we see that ddxf(x)g(x)<0, if η1<η2, so the likelihood ratio ordering exists among the variables i.e. XlrY and the remaining stochastic orderings follow simply from the above chain.

    Let X1,...,Xn be a random sample for MOB-R family. The ith order statistic, Xi:n, has the following pdf

    fi:n(x)=n!(i1)!(ni)!nij=0(nij)(1)if(x)[F(x)]i+j1 (3.6)

    Using Equations (2.2), (2.3) and the power series expansion in [13] (Pages 17–18), we have

    =nij=0r,m=0vj,r,mhr+m(x), (3.7)

    where vj,r,m=ρ(m+1)n!(1)jbr+1ej+i1:r(i1)!(nij)!j!(r+m+1), hr+m(x)=(r+m+1)g(x)Gr+m(x) is already defined in previous section.

    In this section, the estimation of the MOB-R parameters are obtained using the maximum likelihood (ML) method. Let X1,X2,...,Xn be a random sample from the MOB-R class. The log-likelihood function for Θ=(η,θ,ρ,ξ)T takes the form

    (Θ)=nlog (θρη)+ni=1log [hR(xi)]+(θ1)ni=1log [HR(xi)]   (ρ+1)ni=1log (1+Si)2ni=1log [1ˉη(1+Si)ρ], (4.1)

    where Si={log[1R(xi)]}θ. The score vector elements take the forms

    Uη=nη+2ni=1[(1+Si)ρ1ˉη(1+Si)ρ],Uθ=nθ+(ρ+1)ni=1(´Si:θ1+Si)+ni=1log [HR(xi)]+2ni=1[ˉηρ(1+Si)ρ11ˉη(1+Si)ρ´Si:θ],Uρ=nρni=1log (1+Si)+2ni=1[ˉη(1+Si)ρlog (1+Si)1ˉη(1+Si)ρ],Uξ=ni=1[hξR(xi)hR(xi)]ni=1(´Si:ξ1+Si)+(θ1)ni=1[HξR(xi)HR(xi)]2ni=1[ˉηρ(1+Si)ρ11ˉη(1+Si)ρ´Si:ξ].

    Equating the above equations by zero and solving them simultaneously yields the ML estimates.

    If XLomax(a,b), then its cdf takes the form R(x)=1(1+xa)b. Using Equations (1.2) and (1.1), we obtain the cdf and pdf of the MOB-Lomax (MOB-L) distribution as follows

    G(x)=1{1+[blog (1+xa)]θ}ρ1ˉη{1+[blog (1+xa)]θ}ρ

    and

    g(x)=bθρη[blog (1+xa)]θ1{1+[blog (1+xa)]θ}ρ1a(1+xa)(1ˉη{1+[blog (1+xa)]θ}ρ)2.

    The hrf of the MOB-L distribution reduces to

    h(x)=bθρη[blog (1+xa)]θ1a(1+xa){1+[blog (1+xa)]θ}(1ˉη{1+[blog (1+xa)]θ}ρ)2.

    In Figure 1, the plots for pdf and hrf are presented for the MOB-L distribution for several values of parameters. As seen in Figure 1, MOB-L distribution is very flexible with skewed shapes.

    Figure 1.  Plots of the pdf and hrf of the MOB-L distribution.

    The mixture representations for the cdf and pdf of the MOB-L distribution follow from Equations (2.2) and (2.3) as

    G(x)=m=0bm{1(1+xa)b}m

    and

    g(x)=m=0bm+1(m+1)ba(1+xa)b1{1(1+xa)b}m+1.

    The qf of the MOB-L distribution follows from Equation (1.4) as

    Qx(u)=a[(1A)1b1],

    where A=1exp[{(u11ˉη)1ρ1}1θ].

    The rth moment of the MOB-L model can be calculated using (3.1) as

    μr=m=0bm+1(m+1)arbj=0(m+1j)(1)jB(r+1,β(j+1)r)

    Its rth incomplete moment takes the form

    mr=j=0vj,marbBxa(r+1,β(j+1)r), (5.1)

    where vj,m=m=0bm+1(m+1)(m+1j)(1)j.

    Setting r=1 in (5.1), the first incomplete moment reduces to

    m1=j=0vj,mabBxa(2,β(j+1)1).

    The mgf of MOB-L distribution follows from Equation (3.3) as

    M0(t)=i=0vi,j,mΓ(i+1)(1t)i+1,

    where vi,j,m=m,j=0bm+1(m+1)(m+1j)(β(j+1)+ii)(1)i+j.

    The pdf of ith order statistic for the MOB-L model is

    fi:n(x)=j=0r,m=0vj,r,m(1+xa)b1{1(1+xa)b}m+r+1,

    where

    vj,r,m=ρ(m+1)n!(1)jbm+1ej+i1:r(i1)!(nij)!j!.

    If we have two MOB-L distributions, such as MOB-L(η1,θ1,ρ1,a,b1) and MOB-L(η2,θ2,ρ2,a,b2), with a common parameter a then from Equation (3.5), we obtain the reliability function as

    R=m=0p=0bp+1bm(p+1i)(m+1j)(1)i+j(p+1)b2{b2(i+1)+b1j}.

    The log-likelihood function for the MOB-L model takes the form

    l(Θ)=log{θρbηa}ni=1log(di)+(θ1)ni=1log{blog(di)}
    (ρ+1)ni=1log(1+Bi)2ni=1log[1ˉη(1+Bi)ρ],

    where Bi=[blog(di)]θ and di=(1+xia).

    The components of score vector are

    Uη=nηni=1(1+Bi)ρ1ˉη(1+Bi)ρ,Uρ=nρni=1log(1+Bi)2ni=1ˉη(1+Bi)ρlog(1+Bi)1ˉη(1+Bi)ρ,Uθ=nθ+ni=1log{blog(di)}(ρ+1)ni=1Bi:θ1+Bi2ni=1ˉηρ(1+Bi)ρ1Bi:θ1ˉη(1+Bi)ρ,Ub=nb+n(θ1)b(ρ+1)ni=1Bi:b1+Bi2ni=1ˉηρ(1+Bi)ρ1Bi:b1ˉη(1+Bi)ρ,Ua=na+ni=1xia2(1+xia)(θ1)ni=1xia2(di){log(di)}(ρ+1)ni=1Bi:a1+Bi   2ni=1ˉηρ(1+Bi)ρ1Bi:a1ˉη(1+Bi)ρ.

    The log-likelihood function can be maximized directly using the R-package (AdequecyModel). In AdequecyModel package, there are some maximization algorithms such as NR (Newton-Raphson), BFGS (Broyden-Fletcher-Goldfarb-Shanno), BHHH (BerndtHall-Hall-Hausman), NM (Nelder-Mead), SANN (Simulated-Annealing) and limited memory quasi-Newton code for Bound-constrained optimization (L-BFGS-B). However, the MLEs here are computed using the BFGS algorithm.

    In this section, we provide a modified chi-square test statistic for right censored data from the MOB-L distribution based on the modified chi-square type test which is proposed by Bagdonavicius et al. [8] and Bagdonavicius and Nikulin [9], for parametric models with right censored data. Using the maximum likelihood estimators (MLEs) for non-grouped data, this test statistic is also based on the differences between the numbers of observed failures and the numbers of expected failures in the chosen grouped intervals. Here, random grouping intervals are considered as data functions. Voinov et al. [24] developed the description of construction of this chi-square type test. The test statistic can be defined as follows.

    Suppose that X1,X2,.....,Xn is a random sample with right censoring from a parametric model, and a finite time τ.

    The test statistic takes the form

    Y2n=nj=1(Ujej)2Uj+Q,

    where Uj and ej are the observed and expected numbers of failure in grouping intervals, and Q has the form

    Q=WTˆGW                       ˆAj=Uj/n,          Uj=i:XiIjδi,W=(W1,....,Ws)T,     ˆG=[ˆgll]sxs,       ˆgll=ˆillrj=1ˆCljˆCljˆA1j,ˆClj=1ni:XiIjδiξlnh(xi,ˆξ),         ˆill=1nni=1δilnh(xi,ˆξ)ξllnh(xi,ˆξ)ξl,ˆWl=rj=1ˆCljˆA1jZj,    l,l=1,....,s,

    where h(xi,ˆξ) is the hrf ξ=(b,θ,ρ,η,ˉη) and ˆξ is the MLE of ξ on initial non-grouped data.

    The limits aj of r random grouping intervals Ij=[aj1,aj[ are chosen such as the expected failure times to fall into these intervals which are the same for each j=1,..,r1, ˆar=max(X(l),τ). The estimated ˆaj is

    ˆaj=H1(Eji1l=1H(xl,ξ)ni+1,ˆξ),        ˆar=max(X(n),τ),

    where H(x) is the cdf of the considered distribution. This test statistic Y2n follows a chi-square distribution.

    The expected failure times ej to fall into these intervals are ej=Err for any j, with Er=ni=1H(xi,ˆγ).

    The limit intervals aj are considered as random variables such that the expected numbers of failures in each interval Ij are the same, so the expected numbers of failures ej can be calculated by the following formula

    Ej=jk1ni=1ln(11(1+ϖθi)ρ1ˉη(1+ϖθi)ρ),   j=1,..r1.

    To calculate the quadratic form Q of the statistic Y2n, and as its distribution does not depend on the parameters, so we can use the estimated matrices ˆW ˆC and the estimated information matrix ˆI. The elements of ˆC are defined by

    ˆClj=1nni:xiIjδiˆξllnh(xi;ˆξ),

    where ϖi=blog(1+xa) and

    lnh(xi)=ln(bθρη)+(θ1)lnϖilnaln(1+xa)ln(1+ϖθi)2ln(1ˉη(1+ϖθi)ρ)

    The elements of ˆC take the forms

    ˆC1j=1nni:xiIjδi[1b+θ1bθϖθib(1+ϖθi)2ˉηθρϖθi(1+ϖθi)ρ1b(1ˉη(1+ϖθi)ρ)],
    ˆC2j=1nni:xiIjδi[1θ+lnϖi1+zbi2ˉηρϖθilnϖi(1+ϖθi)ρ11ˉη(1+ϖθi)ρ],           
    ˆC3j=1nni:xiIjδi[1ρ2ˉη(1+ϖθi)ρln(1+ϖθi)1ˉη(1+ϖθi)ρ],                         
    ˆC4j=1nni:xiIjδiη,
    ˆC5j=1nni:xiIjδi[2(1+ϖθi)ρ1ˉη(1+ϖθi)ρ]

    and

    ˆWl=rj=1ˆCljA1jZj,   l=1,..,m      j=1,..,r.

    As the above components of the statistic have explicit forms, then we can obtain the test statistic for the MOB-L(ˆξ) distribution with unknown parameters and right censored data. This statistic, follows a chi-square distribution with r degrees of freedom, takes the form

    Y2n(ˆξ) =rj=1(Ujej)2Uj+ˆWT[ˆıllrj=1ˆCljˆCljˆA1j]1ˆW.

    In this section, we provide two simulation studies to assess the performance of the MLEs and to validate the test statistic Y2n(ξ).

    Now, we will study the performance of the maximum likelihood in estimating the MOB-L parameters using simulations which are conducted for sample sizes n=50,150,300, and for different parameter combinations (I: a=3,b=4.5,θ=0.5,ρ=4,η=2), (II: a=0.2,b=0.8,θ=1.5,ρ=8,η=7) and (III: a=0.2,b=0.8,θ=0.5,ρ=5,η=5). To obtain the average values of estimates (AEs), mean square errors (MSEs) and absolute biases (ABs) of the parameters, we generated N=3000 samples from the MOB-L model using the R program.

    The MSEs and ABs were determined by the following equations:

    MSEs(^ϑϑ)=1NNi=1(^ϑϑϑϑ)2,ABs(^ϑϑ)=1NNi=1|^ϑϑϑϑ|,

    where ϑϑ=(a,b,θ,ρ,η).

    The simulation results are shown in Table 1. The small values of ABs and MSEs prove that the maximum likelihood performs very well in estimating the MOB-L parameters.

    Table 1.  Estimated AEs, ABs, and MSEs of the MLEs of the MOB-L parameters.
    I II III
    n Parameters AEs ABs MSEs AEs ABs MSEs AEs ABs MSEs
    50 a 2.251 0.749 1.763 0.224 0.024 0.020 0.251 0.224 0.089
    b 5.010 0.510 0.351 0.885 0.085 0.076 0.921 0.194 0.100
    θ 0.735 0.235 0.272 2.222 0.722 0.541 0.823 0.624 0.320
    ρ 4.847 0.847 1.934 8.016 0.016 0.006 5.614 0.094 0.040
    η 2.593 2.407 1.240 6.989 0.015 1.038 5.824 0.324 0.192
    150 a 2.549 0.451 0.651 0.157 0.015 0.018 0.230 0.222 0.077
    b 4.953 0.453 0.244 0.708 0.072 0.035 0.920 0.171 0.091
    θ 0.776 0.206 0.102 2.218 0.718 0.531 0.853 0.522 0.211
    ρ 4.631 0.631 1.113 7.980 0.08 0.002 5.714 0.082 0.034
    η 1.918 1.082 0.851 6.985 0.011 0.961 5.124 0.291 0.131
    300 a 2.568 0.232 0.539 0.184 0.012 0.013 0.241 0.181 0.031
    b 4.996 0.406 0.210 0.767 0.033 0.017 0.811 0.123 0.052
    θ 0.754 0.154 0.082 2.177 0.677 0.490 0.639 0.031 0.021
    ρ 4.689 0.589 1.106 7.784 0.002 0.001 5.219 0.011 0.009
    η 2.062 1.009 0.818 7.023 0.003 0.910 5.277 0.091 0.035
    500 a 2.988 0.202 0.511 0.204 0.008 0.002 0.204 0.125 0.010
    b 4.496 0.316 0.111 0.807 0.031 0.009 0.801 0.029 0.002
    θ 0.554 0.124 0.051 1.537 0.677 0.240 0.530 0.025 0.019
    ρ 4.029 0.511 0.096 7.989 0.002 0.001 5.111 0.011 0.008
    η 2.010 0.077 0.515 7.001 0.003 0.410 5.277 0.058 0.022

     | Show Table
    DownLoad: CSV
    Table 2.  Simulated significance levels for Y2n(ξ) statistic of the MOB-L model against their theoretical values.
    N=10,000 n=15 n=25 n=50 n=130 n=350 n=500 n=1000
    ε=1% 0.0059 0.0062 0.0067 0.0073 0.0086 0.0094 0.0109
    ε=5% 0.0432 0.0455 0.0469 0.0478 0.0488 0.0492 0.0508
    ε=10% 0.0922 0.0934 0.0954 0.0962 0.0979 0.0995 0.1001

     | Show Table
    DownLoad: CSV

    For testing the null hypothesis H0 that right censored data are from MOB-L model, we calculate the test statistic Y2n(ξ), defined above, using 10,000 simulated samples from the hypothesized distribution with different sizes n=15,25,50,130,350,500,1000 using the package "bb solve algorithm" in the R software. Then, the empirical levels of significance are calculated, for Y2>χ2ε(r), with theoretical levels of significance ε=0.10,0.05,0.01, and r=5. The simulated levels of significance for Y2n(ξ) of the MOB-L model are reported in Table 1.

    The null hypothesis H0 for which simulated samples are fitted by MOB-L distribution, is widely validated for different significance levels. Therefore, the proposed test statistic can be used to fit the data from the MOB-L distribution.

    In this section, we provide two real data analysis to prove the importance and flexibility of the MOB-L distribution and another censored data to validate the MOB-L model using the modified test statistic.

    In this section, we illustrate the performance and flexibility of the MOB-L distribution, as a sub-model of the MOB-R class, using two real-life data sets. The first data on 63 of strengths of 1.5 cm glass fibres which are used by [6] and [19]. The data are: 0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.77, 1.84, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76, 1.48, 1.5, 1.55, 1.61, 1.62, 1.81, 2, 1.82, 2.01, 0.77, 1.61, 0.74, 1.04, 1.62, 1.66, 1.7, 1.64, 1.68, 1.73, 1.11, 1.28, 1.42, 1.5, 1.54, 1.3, 1.48, 1.51, 1.55, 1.61, 1.6, 0.84, 1.24, 1.63, 1.67, 1.7, 1.78, 1.89. The second data on 128 bladder cancer patients on their remission times (in months) which are reported in [15] and are analyzed by [4] and [5]. The data are: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 6.97, 9.02, 13.29, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 0.40, 2.26, 13.80, 25.74, 0.50, 2.46, 3.57, 5.06, 7.09, 9.22, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 3.64, 5.09, 7.26, 9.47, 14.24, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 10.75, 16.62, 43.01, 1.19, 2.75, 1.05, 2.69, 4.23, 5.41, 7.62, 4.26, 5.41, 7.63, 4.33, 5.49, 7.66, 11.25, 17.14, 17.12, 46.12, 1.26, 2.83, 79.05, 11.64, 17.36, 1.40, 3.02, 1.35, 2.87, 5.62, 7.87, 4.34, 5.71, 7.93, 11.79, 8.26, 11.98, 19.13, 1.76, 18.10, 1.46, 4.40, 5.85, 3.25, 4.50, 6.25, 6.54, 8.53, 12.03, 20.28, 2.02, 8.37, 12.02, 2.02, 3.31, 4.51, 3.36, 6.76, 2.07, 3.36, 12.07, 21.73, 6.93, 8.65, 12.63, 22.69.

    The MOB-L model is compared with some existing models namely, Kumaraswamy-Lomax (Kw-L) [17], generalized-exponentiated exponential-Weibull (GE-EW), beta-Lomax (B-L) [17], generalized exponentiated-exponential (GEE), exponentiated-Weibull (EW) [21] and Lomax (L) distributions. The fitted competing models are assessed using the Anderson-Darling (AnD) and Cramer von Mises (CvM) measures.

    The MLEs and the discrimination measures for all competing models are listed in Tables 3 and 4 for both data sets, respectively. It is clear from Tables 3 and 4, that the MOB-L model provides better fit for both data sets as compared with other competing models.

    Table 3.  Parameter estimates, AnD and CvM statistics for carbon fibres data.
    Distribution θ ρ η a b AnD CvM
    MOB-L 1.92000 33.30000 20.99000 18.83000 2.15000 0.26360 0.04240
    (1.2500) (1.0980) (1.7870) (0.7526) (0.0946)
    GE-EW 0.15704 0.03692 3.22861 1.77021 - 0.37840 0.05954
    (0.3778) (0.0389) (0.6367) (1.3850)
    Kw-L 103.18000 8.72000 - 3.90000 345.35000 0.58070 0.105900
    (31.2200) (26.5700) - (0.6030) (72.1100)
    B-L 181.89000 7.02000 - 7.57000 68.44000 1.33900 0.24740
    (38.4600) (40.6400) - (1.3000) (38.3300)
    L 109.20000 39.67000 - - - 1.36400 0.25160
    (19.5500) (12.8070)
    GEE 0.26555 10.03650 7.23658 - - 1.43415 0.26682
    (0.2162) (2.5950) (7.0528)
    EW 3.73666 0.01709 0.01402 - - 0.40365 0.06479
    (0.4457) (0.0213) (0.0084)

     | Show Table
    DownLoad: CSV
    Table 4.  Parameter estimates, AnD and CvM statistics for remission times data.
    Distribution θ ρ η a b AnD CvM
    MOB-L 1.64953 0.08757 1.15492 32.19600 21.31120 0.09018 0.01391
    (0.0144) (0.1735) (0.8478) (5.6221) (1.8282)
    GE-EW 1×1010 1.30988 0.52009 3.74791 - 0.29907 0.04526
    (0.0983) (1.9112) (0.3223) (3.3941)
    Kw-L 13.19000 0.53900 - 1.51800 8.28900 0.17240 0.02580
    (17.6800) (2.7120) - (0.2667) (47.4700)
    B-L 20.63000 0.08670 - 1.58500 54.60000 0.19230 0.02860
    (14.18000) (0.31350) - (0.28360) (19.9300)
    L 121.04100 13.94000 - - - 0.48730 0.08060
    (42.76000) (15.39000)
    GEE 0.12117 1.21795 1.00156 - - 0.71819 0.12840
    (0.1068) (0.1877) (0.8659)
    EW 1.04783 1.00500×107 0.09389 - - 0.96345 0.15430
    (0.31424) (0.3013) (0.1179)

     | Show Table
    DownLoad: CSV

    The hrf plots of the MOB-L model for the data sets are depicted in Figure 2. The TTT plots for the two data sets are shown in Figure 3. The TTT plot for glass fibres data is concave that refers to increasing failure rate, whereas the TTT plot for remission times data is concave then convex which refers to a unimodal hazard rate. As shown, Figures 2 and 3 are consistent, where the hrf of the MOB-L model is increasing for glass fibers data and unimodal for remission times data, hence we conclude that the MOB-L is a suitable distribution for fitting the two data sets. Further, the estimated pdf, cdf, sf and pp plots of the MOB-L model are displayed in Figures 4 and 5, for the two data sets.

    Figure 2.  The hrf plots of the MOB-L model (left) for glass fibres data and (right) for remission times data.
    Figure 3.  TTT plots (left) for glass fibres data and (right) for remission times data.
    Figure 4.  Estimated pdf, cdf, sf and pp of the MOB-L model for glass fibres data.
    Figure 5.  Estimated pdf, cdf, sf and pp of the MOB-L model for remission times data.

    For now, we analyze the lymphoma data which represent times from diagnosis to death (in months) for 31 individuals with advanced non Hodgkin's lymphoma clinical symptoms, using the MOB-L model. This data have been analyzed by Matthews et al. [18] and Gijbels and Gurler [12]. Among these 31 observations 11 of the times are censored, because the patients were alive at the last time of follow-up. The data are: 2.5, 4.1, 4.6, 6.4, 6.7, 7.4, 7.6, 7.7, 7.8, 8.8, 13.3, 13.4, 18.3, 19.7, 21.9, 24.7, 27.5, 29.7, 30.1*, 32.9, 33.5, 35.4*, 37.7*, 40.9*, 42.6*, 45.4*, 48.5*, 48.9*, 60.4*, 64.4*, 66.4*. The * denotes a censored observation.

    The test statistic provided in Section 7 is used to verify if these data can be modeled by the MOB-L distribution. To this end, we first calculate the MLEs of the MOB-L parameters

    ˆξ=(b,θ,ρ,η,ˉη)T=(2.673,1.983,5.124,4.286,3.463)T.

    Data are grouped into r=5 intervals Ij. The numerical result are listed in Table 5.

    Table 5.  The values of ˆaj,ej,Uj,ˆC1j,ˆC2j,ˆC3j,ˆC4j,ˆC5j.
    ˆaj 7.1 8.3 16.4 32.5 66.4
    UJ 5 4 3 7 12
    ˆC1j 0.021 0.074 0.526 0.432 0.236
    ˆC2j 0.564 0.013 0.008 0.718 0.936
    ˆC3j 0.536 0.895 0.921 0.748 1.103
    ˆC4j 1.166 0.933 0.699 1.399 0.466
    ˆC5j 0.852 0.763 0.236 0.974 0.125
    ej 2.1935 2.1935 2.1935 2.1935 2.1935

     | Show Table
    DownLoad: CSV

    Then, we obtain the value of the test statistic Y2n as

    Y2n=X2+Q=4.623+2.635=7.258

    For significance level ε=0.05, the critical value χ25=11.0705 is superior than the value of Y2n=7.258, so we can conclude that the proposed MOB-L model fit these data very well.

    In this paper, we present a new family called, Marshall-Olkin Burr-R family. Some general properties of this family are studied. The estimation of its parameters is carried out by the maximum likelihood approach. One special sub-model namely, Marshall-Olkin Burr-Lomax (MOB-L) distribution is discussed in detail. Two applications are used to check the performance of the MOB-L model. A modified chi-square test statistic for censored data is used to verify the validity of the MOB-Lomax distribution. This test statistic shows that the MOB-Lomax model can be used as a good candidate for analyzing real censored data. The work in the present paper can be extended in some ways. For example, bivariate Marshall-Olkin Burr family can be studied. Further, the parameters of the MOB-L distribution can be estimated using classical and Bayesian estimation methods and compare between them to determine the best estimation method.

    This project is supported by Researchers Supporting Project number (RSP-2020/156) King Saud University, Riyadh, Saudi Arabia. The first author, therefore, gratefully acknowledges the KSU for technical and financial support. The authors would like to thank the Editor and reviewer for their constructive comments that improved the final version of the paper.

    There is no conflict of interest declared by the authors.



    [1] A. Z. Afify, G. M. Cordeiro, N. A. Ibrahim, F. Jamal, M. Elgarhy, M. A. Nasir, The Marshall–Olkin odd Burr III-G family: theory, estimation, and engineering applications, IEEE Access, (2020).
    [2] A. Z. Afify, G. M. Cordeiro, H. M. Yousof, A. Saboor, E. M. M. Ortega, The Marshall–Olkin additive Weibull distribution with variable shapes for the hazard rate, Hacettepe J. Math. Stat., 47 (2018), 365–381.
    [3] A. Z. Afify, D. Kumar, I. Elbatal, Marshall–Olkin power generalized Weibull distribution with applications in engineering and medicine, J. Stat. Theory Appl., 19 (2020), 223–237. doi: 10.2991/jsta.d.200507.004
    [4] A. Z. Afify, O. A. Mohamed, A new three–parameter exponential distribution with variable shapes for the hazard rate: estimation and applications, Mathematics, 8 (2020), 1–17. doi: 10.3390/math8101793
    [5] A. Z. Afify, M. Nassar, G. M. Cordeiro, D. Kumar, The Weibull Marshall–Olkin Lindley distribution: properties and estimation, J. Taibah Univ. Sci., 14 (2020), 192–204. doi: 10.1080/16583655.2020.1715017
    [6] A. Z. Afify, M. Zayed, M. Ahsanullah, The extended exponential distribution and its applications, J. Stat. Theory Appl., 17 (2018), 213–229. doi: 10.2991/jsta.2018.17.2.3
    [7] T. Alice, K. K. Jose, Marshall–Olkin Pareto distributions and its reliability applications, IAPQR Trans., 29 (2004), 1–9.
    [8] V. B. Bagdonavicius, R. J. Levuliene, M. S. Nikulin, Chi–square goodness-of-fit tests for parametric accelerated failure time models, Commun. Stat. Theory Methods, 42 (2013), 2768–2785. doi: 10.1080/03610926.2011.617483
    [9] V. Bagdonavicius, M. Nikulin, Chi–square goodness-of-fit test for right censored data, Int. J. Appl. Math. Stat., 24 (2011), 30–50.
    [10] G. M. Cordeiro, A. J. Lemonte, On the Marshall–Olkin extended weibull distribution, Stat. pap., 54 (2013), 333–353. doi: 10.1007/s00362-012-0431-8
    [11] G. Cordeiro, M. Mead, A. Z. Afify, A. Suzuki, A. Abd El-Gaied, An extended Burr XII distribution: properties, inference and applications, Pak. J. Stat. Oper. Res., 13 (2017), 809–828.
    [12] I. Gijbels, U. Gurler, Estimation of a change-point in a hazard function based on censored data, Lifetime Data Anal., 9 (2003), 395–411. doi: 10.1023/B:LIDA.0000012424.71723.9d
    [13] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals: Series, and Products, sixth ed., Academic Press, San Diego, 2000.
    [14] K. K. Jose, A. Joseph, M. M. Ristić, A Marshall-Olkin beta distribution and its applications, J. Prob. Stat. Sci., 7 (2009), 173–186.
    [15] E. T. Lee, J. W. Wang, Statistical Methods for Survival Data Analysis, 3rd ed., Wiley, New York, 2003.
    [16] A. W. Marshall, I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, 84 (1997), 641–652. doi: 10.1093/biomet/84.3.641
    [17] A. J. Lemonte, G. M. Cordeiro, An extended Lomax distribution, Stat. J. Theor. Appl. Stat., 47 (2013), 800–816.
    [18] D. E. Matthews, V. T. Farewell, R. Pyke, Asymptotic score-statistic processes and tests for constant hazard against a change-point alternative, Ann. Statist., 13 (1985), 583–591.
    [19] M. E. Mead, G. M. Cordeiro, A. Z. Afify, H. Al Mofleh, The alpha power transformation family: properties and applications, Pak. J. Stat. Oper. Res., 15 (2019), 525–545.
    [20] M. A. Nasir, M. H. Tahir, F. Jamal, G. Ozel, A new generalized Burr family of distributions for the lifetime data, J. Stat. Appl. Prob., 6 (2017), 1–17. doi: 10.18576/jsap/060101
    [21] G. S. Mudholkar, D. K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Trans. Reliab., 42 (1993), 299–302. doi: 10.1109/24.229504
    [22] M. Nassar, D. Kumar, S. Dey, G. M. Cordeiro, A. Z. Afify, The Marshall–Olkin alpha power family of distributions with applications, J. Comput. Appl. Math., 351 (2019), 41–53. doi: 10.1016/j.cam.2018.10.052
    [23] M. Shaked, J. G. Shanthikumar, Stochastic Orders, Springer: New York, NY, USA, 2007.
    [24] V. Voinov, M. Nikulin, N. Balakrishnan, Chi-square goodness of fit tests with applications, Academic Press, Elsevier, 2013.
  • This article has been cited by:

    1. Tariq Iqbal, Nada M. Alfaer, Muhammad H. Tahir, Hassan M. Aljohani, Farrukh Jamal, Ahmed Z. Afify, Properties and estimation approaches of the odd JCA family with applications, 2023, 35, 1532-0626, 10.1002/cpe.7417
    2. Muhammad Ahsan ul Haq, Ahmed Z. Afify, Hazem Al- Mofleh, Rana Muhammad Usman, Mohammed Alqawba, Abdullah M. Sarg, The Extended Marshall-Olkin Burr III Distribution: Properties and Applications, 2021, 2220-5810, 1, 10.18187/pjsor.v17i1.3649
    3. Isidro Jesús González-Hernández, Rafael Granillo-Macías, Carlos Rondero-Guerrero, Isaías Simón-Marmolejo, Marshall-Olkin distributions: a bibliometric study, 2021, 126, 0138-9130, 9005, 10.1007/s11192-021-04156-x
    4. Ahmed Z. Afify, Hassan M. Aljohani, Abdulaziz S. Alghamdi, Ahmed M. Gemeay, Abdullah M. Sarg, Barbara Martinucci, A New Two-Parameter Burr-Hatke Distribution: Properties and Bayesian and Non-Bayesian Inference with Applications, 2021, 2021, 2314-4785, 1, 10.1155/2021/1061083
    5. Abdelaziz Alsubie, Mostafa Abdelhamid, Abdul Hadi N. Ahmed, Mohammed Alqawba, Ahmed Z. Afify, Inference on Generalized Inverse-Pareto Distribution under Complete and Censored Samples, 2021, 29, 1079-8587, 213, 10.32604/iasc.2021.018111
    6. Tsvetelin Zaevski, Nikolay Kyurkchiev, On some composite Kies families: distributional properties and saturation in Hausdorff sense, 2023, 2351-6046, 1, 10.15559/23-VMSTA227
    7. Getachew Tekle, Rasool Roozegar, Zubair Ahmad, Alessandro De Gregorio, A New Type 1 Alpha Power Family of Distributions and Modeling Data with Correlation, Overdispersion, and Zero-Inflation in the Health Data Sets, 2023, 2023, 1687-9538, 1, 10.1155/2023/6611108
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1750) PDF downloads(35) Cited by(7)

Figures and Tables

Figures(5)  /  Tables(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog