One of the key research areas in the concept of distribution theory is the evolution of suggesting new statistical distributions. Such generalized distributions allow modelling for a range of disciplines, including reliability, engineering and medicine with even greater flexibility. The two-parameter Marshall-Olkin-Lindley (MOL) distribution suggested by Ghitany et al. [1] is one of the novel versions of Marshall-Olkin models that take the conventional Lindley distribution as a baseline distribution. Assume that X is a lifetime random variable of an experimental item that follows the MOL distribution, denoted by MOL(θ,σ), with shape parameter θ and scale parameter σ.

Hence, its probability density function (PDF), g(⋅), reliability function (RF), R(⋅), and hazard rate function (HRF), h(⋅), of x>0, are given, respectively, by:

(1)g(x;θ,σ)=θσ2e−σx(x+1)(σ+1)[1−θ¯e−σx(1+σxσ+1)]2, θ,σ>0, (2)R(x;θ,σ)=θe−σx(1+σxσ+1)1−θ¯e−σx(1+σxσ+1)

and

(3)h(x;θ,σ)=σ2(x+1)[σ(x+1)+1][1−θ¯e−σx(1+σxσ+1)],

where θ¯=1−θ. Obviously, the Lindley distribution can be obtained from (1) as a special case by setting θ=1. Using some specified values on the MOL’s parameters of θ and σ, via R 4.1.2 software, we plotted various shapes of the PDF and HRF of the MOL distribution, see Fig. 1. It shows that the density shapes are unimodal or decreasing while the HRF shapes are increasing, bathtub and unimodal then bathtub. Since these hazard rate shapes are quite beneficial in lifetime data modelling, hence the MOL distribution is justly flexible and can be considered to provide a good description of different plans of censored data, for details see Ghitany et al. [1]. A look at the literature reveals that just one study by do Espirito Santo et al. [2] used a complete sample to explore the estimations of the MOL distribution using six classical estimation approaches. On the other hand, no study considered the MOL distribution in the censoring case.

Frequently, life testing studies are stopped before all of the components fail. Due to financial or time restrictions, it occurs. The observations that emerge from this type of scenario are known as the censored sample. The literature has developed a number of filtering techniques for the evaluation of various life-testing strategies. The two most popular censorship techniques among the various techniques are Types I and II. The experimental units cannot be removed during a life-testing experiment, however, under any of these censorship techniques. This adaptability is featured in a life-testing experiment with progressive censoring. Since the publication of the book by Balakrishnan et al. [3], extensive research has been conducted on the various facets of progressive censoring. A recent book by Balakrishnan et al. [4] has an extensive compilation of different studies connected to the progressive censorship strategy.

In order to estimate the parameters of the MOL distribution, we work with the progressive Type-II censored (PT-IIC) sample in this study. The PT-IIC sample can be explained as follows: assume that a life testing experiment involving n units with a predetermined number progressive censoring scheme R=(R1,R2,…,Rm), where m<n is the desired number of observed failures. At the time of the first failure X1:m:n, the experiment is stopped and R1 working units are removed. The experiment then resumes using the remaining n−1−R1 units, and when it reaches the second failure, X2:m:n, it is stopped and R2 operating units are removed at random from the remaining n−2−R1 units, and so on. The experiment ends and all of the remaining units n−m−∑i=1m−1Ri are eliminated when it reaches the mth failure time Xm:m:n. The ordered observed failure times in this case are given by X1:m:n<X2:m:n<⋯<Xm:m:n, and the likelihood function for a PT-IIC sample can be expressed as: (4)L=A∏i=1mg(xi:m:n)[1−G(xi:m:n)]Ri,

where A is a constant that is independent of the parameters and G(x)=1−R(x). For several practical lifetime models, a number of inferential techniques based on the PT-IIC scheme have been introduced. For instance, see Sultan et al. [5], Guo et al. [6], Joukar et al. [7], Elshahhat et al. [8], Alotaibi et al. [9], Okasha et al. [10] and Maiti et al. [11], as well as the references therein.

Due to the MOL distribution’s flexibility and the PT-IIC scheme’s effectiveness in gathering sample data, no study investigated the estimation problems of the MOL distribution in the case of the PT-IIC sample. Also, in the original work of Ghitany et al. [1], they just used the maximum likelihood approach to estimate the parameters of the MOL distribution without saying anything about the Bayesian estimation method. In addition, they estimated only the unknown parameters, while it is of interest to reliability engineers and other practitioners to see the performance of the reliability measures of the used distribution. Therefore, this paper’s main goal is to examine frequentist and Bayesian inferences of the MOL distribution’s unknown parameters under the PT-IIC, along with the related reliability indices, such as the RF and HRF. As expected, it is found that the maximum likelihood estimators (MLEs) of θ and σ cannot be derived in closed form; instead, they must be obtained by simultaneously solving two non-linear equations. The MLEs of the RF and HRF are obtained using the invariance property. We suggest constructing the approximative confidence intervals (ACIs) for the various parameters, including RF and HRF, using the asymptotic distribution of the MLEs. We also take into account the Bayesian inference based on independent gamma priors and use two loss functions: squared error (SE) and general entropy (GEnt), which serve as symmetric and asymmetric loss functions, respectively. Due to the fact that the Bayesian estimators cannot be derived in closed form, we suggest using the Markov Chain Monte Carlo approach to get point estimates and the highest posterior density (HPD) credible intervals. The overall performance of the various techniques is compared using Monte Carlo simulations, and two data sets with various progressive censoring plans are examined for illustration.

The remainder of the article is structured as follows. We present the MLEs and ACIs of the unknown parameters, RF and HRF, in Section 2. We acquire the Bayesian inference in Section 3. Sections 4 and 5 separately describe the findings of the Monte Carlo simulation and the analysis of two data sets, respectively. At last, we sum up the paper in Section 6.

In this part, we estimate the unknown parameters, RF and HRF of the MOL distribution using the method of maximum likelihood based on the PT-IIC sample. The ACIs of the different parameters are explained in addition to the point estimators. Assume that x1:m:n,x2:m:n,...,xm:m:n is a PT-IIC sample of size m with progressive pattern R1,…,Rm taken from the MOL population with PDF and RF as displayed in (1) and (2), respectively. Then the likelihood function, without the constant term, takes the following form based on (1) and (2), and (4): (5)L(θ,σ)=θnσ2m(σ+1)me−σ∑i=1m(1+Ri)xi∏i=1m(1+σyi)Ri[1+σ−θ¯e−σxi(1+σyi)]2+Ri,

where xi=xi:m:n for simplicity and yi=1+xi. The log-likelihood function is expressed as follows, using Eq. (5): (6)ℓ(θ,σ)=nlog⁡(θ)+2mlog⁡(σ)+mlog⁡(σ¯)−σ∑i=1m(1+Ri)xi+∑i=1mRilog⁡(1+σyi)−∑i=1m(2+Ri)log⁡[σ¯−θ¯e−σxi(1+σyi)],

where σ¯=σ+1. The likelihood equations are derived by calculating the first partial derivatives of (6) with regard to θ and σ and equating each one to zero as shown below: (7)∂ℓ(θ,σ)∂θ=nθ−∑i=1m(2+Ri)e−σxi(1+σyi)σ¯−θ¯e−σxi(1+σyi)=0

and

(8)∂ℓ(θ,σ)∂σ=2mσ+mσ¯−∑i=1m(1+Ri)xi+∑i=1mRiyi1+σyi−∑i=1m(2+Ri)ψiσ¯−θ¯e−σxi(1+σyi)=0,

where ψi=1−θ¯e−σxi[yi−xi(1+σyi)]. It is obvious that analytical solutions to the likelihood equations in (7) and (8) in order to obtain the MLEs of θ and σ, denoted by θ^ and σ^, are not possible. Therefore, to obtain the needed MLEs, any iteration process may be used, including the Newton-Raphson procedure. The MLEs of RF and HRF at a given time t can then be calculated using the invariance property of MLEs once the MLEs θ^ and σ^ have been obtained. The MLEs of G(t) and h(t) in this instance are derived from (2) and (3) as follows: R^(t)=θ^e−σ^t(1+σ^tσ^+1)1−θ¯^e−σ^t(1+σ^tσ^+1)

and

h^(t)=σ^2(t+1)[σ^(t+1)+1][1−θ¯^e−σ^t(1+σ^tσ^+1)].

Utilizing the asymptotic normality of the MLEs is the most common approach for establishing confidence bounds for the parameters. The MLEs’ asymptotic distribution can be expressed as (θ^,σ^)∼N[(θ,σ),J−1(θ,σ)], where J−1(θ,σ) stands for the variance-covariance matrix obtained based on the Fisher information matrix denoted by J. In practice, we use J−1(θ^,σ^) to estimate J−1(θ,σ) due to the challenging second derivative expressions. In this case, we can write J−1(θ^,σ^) as follows: (9)J−1(θ^,σ^)=(−∂2ℓ(θ,σ)∂θ2−∂2ℓ(θ,σ)∂θ∂σ−∂2ℓ(θ,σ)∂σ∂θ−∂2ℓ(θ,σ)∂σ2)(θ,σ)=(θ^,σ^)−1=(var^(θ^)cov^(θ^,σ^)cov^(σ^,θ^)var^(σ^)),

with

∂2ℓ(θ,σ)∂θ2=−nθ2−∑i=1m(2+Ri)e−2σxi(1+σyi)2ϕi2, ∂2ℓ(θ,σ)∂σ2=−2mσ2−mσ¯2^+∑i=1mRiyi2(1+σyi)2−∑i=1m(2+Ri)ϖiϕi−∑i=1m(2+Ri)ψi2ϕi2

and

∂2ℓ(θ,σ)∂θ∂σ=∑i=1m(2+Ri)ψ´iϕi−∑i=1m(2+Ri)e−σxi(1+σyi)ψiϕi2,

where ϕi=σ¯−θ¯e−σxi(1+σyi) and ϖi=xiθ¯e−σxi[xi(1+σyi)−2yi] and ψ´i=e−σxi[yi−xi(1+σyi)]. Then, the 100(1−α)% ACIs of θ and σ can be computed, respectively, as: θ^±zα/2var^(θ^),andσ^±zα/2var^(σ^),

where var^(θ^) and var^(σ^) are the main diagonal elements of (9), respectively, and zα/2 is the upper (α/2)th percentile point of the standard normal distribution. On the other hand, we must first determine the variances of respective estimators for the RF and HRF in order to construct such intervals. Here, we approximate these variances using the delta approach. The following derivatives obtained from (2) and (3), respectively, are required in order to use the delta method: Rθ=∂R(t;θ,σ)∂θ=G(t;θ,σ)θ[1−R(t;θ,σ)], Rσ=∂R(t;θ,σ)∂σ=θte−σtσ¯ϕt−θ¯tσR(t;θ,σ)e−σt(1+σ¯yt)σ¯2ϕt−tR(t;θ,σ), hθ=∂h(t;θ,σ)∂θ=−h(t;θ,σ)e−σt(1+σyt)ϕtσ¯

and

hσ=∂h(t;θ,σ)∂σ=σytϕt(1+σyt)[2−σyt(1+σyt)]−θ¯tσh(t;θ,σ)e−σt(1+σ¯yt)σ¯2ϕt,

where ϕt=σ¯−θ¯e−σt(1+σ(1+t)) and yt=1+t.

Suppose that ΔR=(Rθ,Rσ)|(θ,σ)=(θ^,σ^) and Δh=(hθ,hσ)|(θ,σ)=(θ^,σ^), then we can obtain the approximated estimated variances of R^(t) and h^(t), respectively, as follows: var^(R^)≈[ΔRJ−1(θ^,σ^)ΔR⊤]andvar^(h^)≈[ΔhJ−1(θ^,σ^)Δh⊤],

As a result, with 100(1−α)% confidence level, the ACIs that align to G(t) and h(t) can be acquired, respectively, as follows: R^(t)±zα/2var^(R^),andh^(t)±zα/2var^(h^).

For analyzing failure time data, the Bayesian estimation approach has attracted a lot of attention. It uses one’s past knowledge of the parameters and also takes into account the information that is readily available. In this section, the Bayesian estimators of θ,σ,R(t) and h(t) are considered under the assumption that the two unknown parameters are independent and have gamma prior distributions. We consider the use of independent gamma priors due to the flexibility of gamma distribution and to avoid adding more complexity to the posterior distribution. Moreover, independent priors are considered because they are rather straightforward and concise, which may not produce many challenging computational and inferential problems. Despite dependent priors appearing more appealing in some practical contexts, the dependent property between parameters cannot be justified subjectively based on historical data and expert knowledge where such prior information may be extremely rare. Hence, for the sake of simplicity, independent priors are more widely used in statistics under the Bayesian method. In order to get the Bayesian estimators, two loss functions are offered to get the point estimators, namely SE and GEnt loss functions. Besides acquiring the point estimators, the HPD credible intervals are also obtained. Suppose that θ∼G(a1,b1) and σ∼G(a2,b2), where aj,bj,j=1,2 are the hyper-parameters. Therefore, the joint prior of θ and σ can be expressed as shown below: (10)q(θ,σ)∝θa1−1σa2−1e−(b1θ+b2σ),θ,σ>0.

The posterior distribution of the unknown parameters θ and σ can be obtained by combining the likelihood function in (5) with the joint prior distribution provided (10) and by applying the Bayes theorem as follows: (11)g(θ,σ|x_)=θn+a1−1σ2m+a2−1σ¯me−σ[∑i=1m(1+Ri)xi+b2]−b1θC∏i=1m(1+σyi)Ri[σ¯−θ¯e−σxi(1+σyi)]2+Ri,

where x_=(x1,…,xm) and C is the normalized constant.

If one setting aj=bj=0 for j=1,2 in (10), the joint posterior density (11) will then be in proportion to the likelihood function (5), i.e., g(θ,σ|x_)∝(θσ)−1L(θ,σ), which is the non-informative case. From a Bayesian viewpoint, there is clearly no way in which one can say that one prior is better than any other. Generally, if the proper prior information is available, it is better to use the informative prior(s) than the non-informative prior(s). Otherwise, if one does not have sufficient prior information, it is better to use a non-informative prior distribution. Since the Bayesian estimates using SE loss function with non-informative priors behave like the maximum likelihood estimates whereas those with informative priors behave much better than others, it is always better to use the frequentist estimates rather than the Bayesian estimates because the latter are computationally more expensive when the MCMC procedure is used. In Section 4, to evaluate the sensitivity of the priors, some discussions about various sets of prior distributions are reported.

Now, in order to derive the Bayesian estimators, we take into account the SE and GEnt loss functions. The Bayesian estimator for the SE loss function is the posterior mean, which considers overestimation and underestimation equally. In contrast hand, the GEnt loss function offers different influences for overestimation and underestimation. Calabria et al. [12] introduced the GEnt loss function, which is defined as: GEnt(λ~,λ)∝(λ~λ)μ−μlog⁡(λ~λ)−1,

where μ is a parameter that controls the level of asymmetry and λ~ is the Bayesian estimator of λ. Below is the Bayesian estimator of λ using the GEnt loss function: (12)λ~GEnt=[Eλ(λ−μ)]−1μ,

given that Eδ(δ−κ) exists and is finite. Assume that ϑ(θ,σ) is a function of the unknown parameters, we may easily derive its Bayesian estimator using SE and GEnt loss functions, respectively, as follows: (13)ϑ~SE(θ,σ)=∫0∞∫0∞ϑ(θ,σ)g(θ,σ|x_)dθdσ

and

(14)ϑ~GEnt(θ,σ)=[∫0∞∫0∞[ϑ(θ,σ)]−μg(θ,σ|x_)dθdσ]−1μ

It is obvious that it is difficult to determine the Bayesian estimators using (13) and (14) analytically. In order to acquire the Bayesian estimates of θ and σ and the related HPD credible intervals, we suggest using the MCMC procedure. We must first determine the full conditional distributions of the unknown parameters from (11) as follows: (15)g(θ|σ,x_)∝θn+a1−1exp⁡{−∑i=1m(2+Ri)log⁡[σ¯−θ¯e−σxi(1+σyi)]−b1θ}

and

(16)g(σ|θ,x_)∝σ2m+a2−1σ¯me−σ[∑i=1m(1+Ri)xi+b2]∏i=1m(1+σyi)Ri[σ¯−θ¯e−σxi(1+σyi)]2+Ri.

It is evident that the conditional distributions of θ and σ as provided in (15) and (16) cannot be represented in standard forms, but their graphs are equivalent to the normal distribution. As a result, we employ the Metropolis-Hastings (M-H) algorithm with normal proposal distribution with asymptotic variances to produce random samples from these distributions. The steps that follow now demonstrate how to get the required samples.

Step 1. Set k=1.

Step 2. Begin with the initial guesses (θ(0),σ(0))=(θ^,σ^).

Step 3. From (15), generate θ(k) using normal proposal distribution, i.e., N(θ(0),var^(θ(0))), by using the M-H steps, where var^(θ(0))≅var^(θ^) is given by (9).

Step 4. Use (16) to get σ(k) using the M-H steps with normal proposal distribution, i.e., N(σ(0),var^(σ(0))), where var^(σ(0))≅var^(σ^) is given by (9).

Step 5. Based on the generated θ(k) and σ(k), obtain R(k)(t)=θ(k)e−σ(k)t(1+σ(k)tσ(k)+1)1−θ¯(k)e−σ(k)t(1+σ(k)tσ(k)+1)

and

h(k)(t)=σ2(k)(t+1)[σ(k)(t+1)+1][1−θ¯(k)e−σ(k)t(1+σ(k)tσ(k)+1)],

Step 6. Put k=k+1.

Step 7. Repeat steps 3–6, M times to compute [β(1),…,β(M)],

where β=θ,σ,R(t) or h(t).

In this study, the first B generated samples are discarded in order to ensure convergence and remove the appeal of initial guesses. In this situation, we possess β(k),k=B+1,…,M. The Bayesian estimate of β based on the SE and GEnt loss functions can be calculated using large M, respectively, as: β~SE=1M−B∑k=B+1Mβ(k)andβ~GEnt={1M−B∑k=B+1M[β(k)]−μ}−1μ.

To construct the HPD credible intervals of β, order β(k),k=B+1,…,M. Therefore, the 100(1−α)% HPD credible interval of β will be [β(k∗),β(k∗+(1−α)(M−B))], where k∗=B+1,B+2,…,M is determined such that:

β(k∗+[(1−α)(M−B)])−β(k∗)=min1⩽l⩽α(M−B)⁡[β(l+[(1−α)(M−B)])−β(k))],

where [ν] stands for the biggest integer that is less than or equal to ν.

To evaluate the behavior of the proposed estimators of θ, σ, R(t) and h(t) developed in the proceeding sections, Monte Carlo simulation experiments are conducted. For this target, we simulate 1,000 PT-IIC samples from MOL(0.8,0.4) based on various choices of n, m and progressive pattern R. Taking t=0.2, the actual values of R(t) and h(t) are 0.970 and 0.104, respectively. Using n(=50,90), the proposed numerical experiments are performed by taking m as a failure percentage (FP) of each n as mn(=40, 80)%. Moreover, for each set of (n,m), different progressive patterns R=(R1,R2,…,Rm) are considered as:

Scheme-1: R=(n−m,0∗(m−1)),Scheme-2: R=(0∗(m2−1),n−m,0∗(m2)),Scheme-3: R=(0∗(m−1),n−m),

where R=(1,0,0,0,1) is denoted by R=(1,0∗3,1) for short notation.

Once the 1,000 PT-IIC samples collected, the maximum likelihood and 95% ACI estimates of θ, σ, R(t) and h(t) are calculated utilizing via ’maxLik’ package (by Henningsen et al. [13]) in R 4.1.2 software. To develop the Bayesian MCMC inferences of the same unknown MOL parameters, by according to the mean and variance of the gamma density, two informative sets of the hyperparameters ai and bi for i=1,2 of θ and σ are used namely: prior-1: (a1,a2,b1,b2)=(4,2,5,5) and prior-2: (a1,a2,b1,b2)=(8,4,10,10). Following Kundu [14] and Dey et al. [15], the given hyperparameter values of ai,bi, i=1,2 of the unknown MOL parameters are chosen in such a way that the prior mean becomes the expected value of the corresponding parameter. Using the M-H algorithm sampler, from SE and GEnt (for μ(=−2,+2)) loss functions, the Bayesian point estimates of θ, σ, R(t) and h(t) are calculated based on 12,000 MCMC samples after ignoring the first 2,000 variates as burn-in. Further, from 10,000 MCMC samples, the 95% HPD credible intervals of θ, σ, R(t) or h(t) are calculated also. All Bayesian evaluations of the unknown parameters θ and σ or the reliability time parameters R(t) and h(t) are performed via ’coda’ package (by Plummer et al. [16]) in R 4.1.2 software. All necessary computational algorithms were performed on a laptop with Core(TM) i5-2410M processor and 4.00 GB of RAM. It is better to be noted, regarding the benefits of both ‘maxLik’ and ‘coda’ programming packages, that the CPU time required per iteration is not expensive.

To monitor whether the simulated Markovian sample is sufficiently close to the target posterior, beside the trace and autocorrelation plots, we purpose to consider the Brooks-Gelman-Rubin (BGR) diagnostic statistic, which evaluates the convergence by analyzing the difference between the variance-within chains and the variance-between chains for each model parameter, for details see [17]. To establish this purpose, by running two chains using n[FP%] = 50[40%], Scheme-1, and prior-1 (as an example), we plotted the suggested convergence diagnostics in Figs. 2 and 3 via R 4.1.2 software. It is clear, from Fig. 2, that the MCMC iterations are converged well. On the other hand, Fig. 3 shows that the proposed BGR diagnoses close to one after simulating the first 2,000 iterations, which means that the combustion sample has an adequate size to ignore the influence of the initial guesses, and thus the simulated chains converged well.

The average estimates (Av.Es) from classical (or Bayesian) approach of θ, σ, R(t) and h(t) (say ω) are given by:

ωˇd¯=11,000∑i=11,000ωˇd(i), d=1,2,3,4,

where ωˇ(i) is the calculated estimate of ω at the ith simulated sample, ω1=θ, ω2=σ, ω3=R(t) and ω4=h(t).

Further, the comparison between point estimates of ω is made based on their root mean squared-errors (RMSEs) and mean relative absolute biases (MRABs) as:

RMSE(ωˇd)=11000∑i=11000(ωˇd(i)−ωd)2, d=1,2,3,4,

and

MRAB(ωˇd)=11000∑i=110001ωd|ωˇd(i)−ωd|, d=1,2,3,4,

respectively.

Furthermore, the comparison between interval estimates of the same unknown parameters is made using their average confidence lengths (ACLs) and coverage percentages (CPs) which can be computed as

ACL(1−α)%(ωd)=11000∑i=11000(𝒰ωˇd(i)−ℒωˇd(i)), d=1,2,3,4,

and

CP(1−α)%(ωd)=11000∑i=110001(ℒωˇd(i);𝒰ωˇd(i))(ωd), d=1,2,3,4,

respectively, where 1(⋅) is the indicator function and ℒ(⋅) and 𝒰(⋅) denote the lower and upper bounds, respectively, of (1−α)% asymptotic (or HPD credible) interval of ωd.

Heatmap is a method of representing data graphically where values are depicted by color, making it easy to visualize complex data and understand it at a glance. So, via R data visualization, all numerical results of θ, σ, R(t) and h(t) are displayed with heatmap plots in Figs. 4–7, respectively. Here, following the graphical tools reported in Elshahhat [18], the suggested heatmaps are plotted via R 4.1.2 software. Each heatmap range is classified from lowest to highest values with the colors cyan, red, and yellow, respectively. Each heatmap also displays the proposed estimation methods and the specified test settings on the “x-axis” and “y-axis” lines, respectively. In the supplementary file, all simulation tables of θ, σ, R(t) and h(t) are listed. Furthermore, for specification, several notations of the estimation methods have been used in Figs. 4–7 such as (based on Prior 1 (say P1) as an example) the Bayesian estimates based on SE loss mentioned as “SE-P1” as well as the Bayesian estimates based on GEnt loss for μ=−2 and +2 mentioned as “GE1-P1” and “GE2-P1”, respectively.

From Figs. 4–7, in terms of the lowest RMSE, MRAB and ACL values as well as the highest CP values, the following comments can be drawn:

Generally, the proposed point and interval estimates of θ, σ, R(t) and h(t) of the MOL model in presence of PT-IIC data behave satisfactorily.

In order to demonstrate the significance of the suggested inferential methodologies and the applicability of study objectives to actual phenomena, this part presents two practical applications from the domains of engineering and chemistry.