Statistical Inference for Kumaraswamy Distribution under Generalized Progressive Hybrid Censoring Scheme with Application

1  Introduction

Kumaraswamy [1] created a novel two-parameter distribution with hydrological concerns in mind because the conventional probability distributions functions such as beta, normal, log-normal, Student-t, and other empirical distributions do not suit hydrological data well. For further details on how the Kumaraswamy distribution fits well with several natural occurrences, including engineering modeling, daily rainfall, water flows, and other pertinent fields, see [2–4], and [5]. This is particularly true for results that have upper and lower boundaries, such as economic statistics, test scores, human height, and air temperatures. Although the Kumaraswamy distribution has a closed form cumulative distribution function (CDF) in the closed interval [0, 1], it is comparable to the beta distribution. The probability density function (PDF) of the Kumaraswamy distribution, CDF, reliability function (RF), and hazard rate function (HRF) can be written in exponential form, respectively, as follows:

f(z;λ,δ)=λδzδ−1(1−zδ)λ−1=λδzδ−11−zδexp⁡[λln⁡(1−zδ)],0 ≤ z ≤ 1,(1)

F(z;λ,δ)=1−(1−zδ)λ=1−exp⁡[λln⁡(1−zδ)],0 ≤ z ≤ 1,(2)

R(t)=(1−tδ)λ=exp⁡[λln⁡(1−tδ)],t > 0(3)

H(t)=λδtδ−11−tδ,t > 0,(4)

where λ and δ are non-negative, shape and scale parameters, respectively. In recent years, estimation for the Kumaraswamy distribution has increasingly come to the attention of academics, see for more information [6–10]. Fig. 1a–d shows the PDF, CDF, RF and HRF for the Kumaraswamy distribution, respectively, with different values for shape and scale parameters.

Figure 1: The PDF, CDF, RF and HRF for the Kumaraswamy distribution with different values of shape and scale parameters

Due to time, cost, and resource limitations, quality assessments that use lifetime data may encounter issues in practice. To address these problems, censoring schemes (CSs) have been employed. The progressive censoring schemes (PCSs) technique has become the norm due to its versatility and the range of data it provides. PCS states that in an experiment on n experimental units, the R1 surviving items are randomly selected from (n−1) surviving items at the moment of the first failure in the test. Also, at the moment of the second failure in the test, R2 surviving items are randomly selected from (n−R1−2) surviving items. The procedure keeps going until the mth failure time; at the mth failure, every last surviving Rm item is arbitrarily eliminated from the experiment. The m ordered observed failure times will be designated by (Z1:m:nR1,Z2:m:nR2,…,Zm:m:nRm) and referred to as progressively censored samples of size m from a sample of size n with PCS (R1,R2,…,Rm),m≤n. The use of PCS is often motivated by practical considerations, such as cost constraints or limited resources, where it may not be feasible or efficient to monitor the subjects continuously or until the failure occurs. Data analysis using PCS poses challenges as the censoring mechanism relies on observed failure times, which may be long, and the time intervals between successive observations may vary. This requires the development of specialized statistical methods for estimation and inference. A generalized progressive hybrid censored scheme (GPHCS) was recently presented by Cho et al. [11]. He may handle the complete test duration inside the allocated time after determining the proper total test time T. Fig. 2 shows the three different cases of GPHCS.

Figure 2: The Layout representation of generalized progressive hybrid censoring scheme

In the recent years, GPHCS has gained a lot of attention in reliability and survival analysis. Based on GPHCS from various distributions, Bayesian estimation has been studied by Nagy et al. in [12,13], and Nagy et al. in [14].

Bayesian estimation involves updating prior beliefs with observed data to obtain a posterior distribution. In expected Bayesian (E-Bayesian), we impose certain restrictions on our hyperparameters; it provides more stable and reliable estimates, especially in situations with uncertainty about the appropriate prior specification. The hierarchical Bayesian (H-Bayesian) method is more robust because it entails a two-stage process for constructing the prior distribution. However, a drawback of this approach lies in the intricate integrals within the estimator expression. These integrals often necessitate resolution through numerous numerical approximation techniques, making calculations tedious and time-consuming.

In the recent past, E- and H-Bayesian estimations have been considered for the parameters and the reliability characteristics under different censored data. Han [15] discussed the E-Bayesian and H-Bayesian estimations of the parameter derived from the Pareto distribution under different loss functions. Okasha et al. [16] discussed the E-Bayesian estimation for geometric distribution. Yousefzadeh [17] considered the E- and H-Bayesian estimates for Pascal distribution. Rabie et al. [18] discussed the Bayesian and E-Bayesian estimation methods of the parameter and the reliability function of the Burr-X distribution based on a generalized Type-I hybrid censoring scheme. Yaghoobzadeh [19] conducted research on E-Bayesian and H-Bayesian estimation of a scalar parameter in the Gompertz distribution under type-II censoring schemes based on fuzzy data. Their study focused on developing estimation techniques that account for uncertainty and imprecision in the data. Nassar et al. [20] conducted research on E-Bayesian estimation and associated properties of the simple step-stress model for the exponential distribution based on type-II censoring. Their study focused on developing estimation methods that account for the censoring scheme employed and investigating the properties of the estimated parameters. Nagy et al. [21] provided an E-Bayesian estimation for an exponential model based on simple step stress with Type I hybrid censored data. They developed estimation methods to obtain reliable parameter estimates considering the specific censoring scheme employed. Balakrishnan et al. [22] discussed the best linear unbiased estimation (BLUE) and maximum likelihood estimation (MLE) techniques for exponential distributions under general progressive type-II censored samples. They provided statistical methodologies for parameter estimation under this type of censoring scenario. They developed methodologies to analyze this specific censoring scheme and its impact on the estimation of the distribution’s parameters. Mohie El-Din et al. [23] proposed an E-Bayesian estimation approach for the parameters and HRF of the Gompertz distribution using type-II PCS. Their research aimed to provide reliable estimation techniques for this specific censoring scheme and demonstrated the application of these methods in practical scenarios.

Recent research has demonstrated that E- and H-Bayesian estimates are more effective than both classical and Bayesian approaches. To the best of our knowledge, no article can be found in the literature having both E- and H-Bayesian estimation for the Kumaraswamy distribution under GPHCS. Motivated by the effectiveness of E- and H-Bayesian estimates, the usefulness of the GPHCS, and the importance of the Kumaraswamy distribution in reliability applications. Our main objective in this paper is to estimate the shape parameter and the HRF of the Kumaraswamy distribution based on GPHCS using E-Bayesian and H-Bayesian approaches, which we believe would be of profound interest to applied statisticians and quality control engineers.

The structure of the article is as follows: Section 1 provides an introduction, outlining the research problem and objectives. Section 2 focuses on the MLE and Bayesian estimation techniques for the shape parameter and HRF, considering different loss functions. Section 3 presents the derivation of E-Bayesian estimators for the shape parameter under different loss functions. In Section 4, H-Bayesian estimators are obtained, again considering different loss functions. To assess the performance of the estimators, a simulation study is conducted in Section 5, where different loss functions are considered and the estimators are compared. Section 6 presents the analysis of a real-life data to demonstrate the practical application of the proposed estimators. In Section 7, the article concludes by summarizing the key findings and implications of the study. Finally, the properties of the E- and H-Bayesian estimators are discussed in the Appendix A of this paper.

2  Maximum Likelihood and Bayesian Estimation

In this section, we estimate the shape parameter λ and HRF H(t) of the Kumaraswamy distribution using the MLE method. Let Z_=(Z1:n∗:nR1,Z2:n∗:nR2,…,Zn∗:n∗:nRn∗)=(Z1,Z2,…,Zn∗) where Zj:n∗:nRj=Zj, for simplicity of notation, be the ordered observed failure times based on GPHCS follow from absolutely continuous PDF F(.) and RF R(.), the joint PDF is given by

fZ_(z_)=[∏i=1n∗∑j=im(Rj∗+1)]∏i=1n∗f(zi:n∗:n)[R(zi:n∗:n)]Ri∗[R(T)]Rτ∗,(5)

0<z1<z2<…<zn∗<∞.

where Rj∗ is the jth value of the vector R∗.

[z_,R∗]={[(z1,R1),…,(zs,Rs),(zs+1,0),…,(zk−1,0),(zk,Rk∗=n−k−∑j=1sRj)],Case-I,[(z1,R1),…,(zs,Rs)],Case-II,[(z1,R1),…,(zm,Rm)],Case-III,(6)

and Rτ∗ is the number of units eliminated at time T, as determined by

Rτ∗={0,Case-I,n−s−∑j=1sRj,Case-II,0,Case-III, and n∗={kCase-I,sCase-II,mCase-III.(7)

Let (Z1,Z2,…,Zn∗) be the observed ordered data under the GPHCS follow from the Kumaraswamy distribution with PDF and RF as in Eqs. (1) and (3), the likelihood function of λ,δ can be derived by Eq. (5), as

L(λ,δ;z_)=C(λδ)n∗∏j=1n∗zjδ−11−zjδexp⁡[−λD(δ,z_)],(8)

where C=∏j=1n∗∑j=jm(Rj∗+1) and D (δ,z_) =−[∑j=1n∗(Rj∗+1)ln⁡(1−zjδ)+Rτ∗ln⁡(1−Tδ)]. Taking log of Eq. (8), we get

log⁡L(λ|z_)∝n∗log⁡(λδ)+(δ−1)∑j=1n∗log⁡zj−∑j=1n∗log⁡(1−zjδ)−λD(δ,z_).

We differentiate the above equation with respect to λ and equating to zero, as

∂ln⁡L(λ,δ|z_)∂λ=n∗λ−D(δ,z_)=0,

λ^ML=n∗D(δ, z_).(9)

The ML estimate of H(t) for given δ is given as

H^ML(t)=λ^MLδtδ−11−tδ.(10)

2.1 Bayesian Estimation

In calculating statistical inference and estimation of scale parameters using the Bayesian method, this is done through integrations of non-closed formulas. Then, calculating the inference and estimation of these parameters using the E- and H-Bayesian approaches is a more complex integral formula. Therefore, methods are used to approximate these quantities, which often lead to large errors. Therefore, in this manuscript, we calculate the estimate of the shape parameter for the Kumaraswamy distribution because the Bayesian estimate of this parameter is a closed form. Therefore, calculating the estimates using the E- and H-Bayesian approaches is more accurate and efficient. Here, we have developed Bayesian estimators for the shape parameter λ and the HRF of the Kumaraswamy distribution using various loss functions. These loss functions include the squared error loss function (SELF), entropy loss function (ELF), weighted balance loss function (WBLF), and minimum expected loss function (MELF).

We are using gamma distribution as the prior distribution of λ as it is a conjugate prior. The gamma prior distribution with shape and scale parameters a and b, respectively, has the following PDF:

P(λ|a,b)=baλa−1Γ(a)e−bλ;λ,a,b > 0.(11)

Here the hyper-parameters a and b have been chosen based on a formula, which can be found in Dutta et al. [24]. Using the likelihood function and the prior in Eq. (11), the posterior distribution for the parameter λ becomes

P∗(λ|z_)= L(λ|z_)P(λ|a,b)∫0∞L(λ|z_)P(λ|a,b)dλ= [g1(δ,z_)]n∗+aλ(n∗+a)−1exp⁡[−λg1(δ,z_)]Γ(n∗+a),(12)

where g1(δ,z_) =b+D(δ,z_) and we will write it as g1 for simplicity of notation.

2.2 Bayesian Estimate with SELF

The Bayes estimator of λ under the squared error loss function (SELF) λ^BS, is given by

λ^BS=E[λ|z_],(13)

provided if E[λ|z_] exists and finite. Then, by using Eqs. (12) and (13) with the sample  z_=(z1,⋯,zn∗), the Bayes estimator of λ under SELF is given by

λ^BS=E[λ|z_]=∫0∞λP∗(λ|z_)dλ=n∗+ag1.(14)

The Bayes estimate of H(t) for given δ is given as

H^BS(t)=λ^BSδtδ−11−tδ.(15)

2.3 Bayesian Estimate with ELF

Dey et al. [25] discussed the ELF and the Bayes estimator of λ under ELF λ^BE, is given by

λ^BE=E[λ−1|z_]−1,(16)

provided if E[λ−1|z_]−1 exists and finite. Then, by using Eqs. (12) and (16) with the sample z_=(z1,⋯,zn∗), the Bayes estimator of λ under ELF is given by

λ^BE=E[λ−1|z_]−1=∫0∞P∗(λ|z_)dλλ=n∗+a−1g1.(17)

The Bayes estimate of H(t) for given δ under ELF is given as

H^BE(t)=λ^BEδtδ−11−tδ.(18)

2.4 Bayesian Estimate with WBLF

The WBLF can be expressed as (see Nasir et al. [26]) where the Bayes estimator of λ under WBLF λ^BW, is given by

λ^BW=E[λ2|z_]E[λ|z_],(19)

provided if E[λ2|z_] and E[λ|z_] exist and finite. Then, by using Eqs. (12) and (19) with the sample z_=(z1,⋯,zn∗), we have

E[λ2|z_]=∫0∞λ2P∗(λ|z_)dλ=(n∗+a)(n∗+a+1)[g1]2andE[λ|z_]=n∗+ag1.

Hence, the Bayes estimator of λ under WBLF is given by

λ^BW=(n∗+a+1)g1.(20)

The Bayes estimate of H(t) for given δ under WBLF is given as

H^BW(t)=λ^BWδtδ−11−tδ.(21)

2.5 Bayesian Estimate with MELF

Tummala et al. [27] defined the MELF where the Bayes estimator of λ under WBLF λ^BM, is given by

λ^BM=E[λ−1|z_]E[λ−2|z_],(22)

provided that E[λ−1|z_] and E[λ−2|z_] exist and finite. Then, by using Eqs. (12) and (22) with the sample z_=(z1,⋯,zn∗), we have

E[λ−2|z_]=∫0∞1λ2P∗(λ|z_)dλ=[g1]n∗+aΓ(n∗+a)Γ(n∗+a−2)[g1]n∗+a−2,andE[λ−1|z_]=g1n∗+a−1.

Hence, by using MELF, the Bayes estimate is obtained as

λ^BM=(n∗+a−2)g1.(23)

The Bayes estimate of H(t) for given δ under MELF is given as

H^BM(t)=λ^BWδtδ−11−tδ.(24)

3  E-Bayesian Estimation

According to Han [28], it is recommended to select the prior parameters a and b in such a way that the prior distribution in Eq. (11) is a decreasing function of λ.

ddλP(λ|a,b)=(ab)Γ(a)λ(b−2)e−λa[(b−1)−aλ].

Specifically, it is suggested that for 0 < a < 1 and b > 0, the prior distribution becomes a decreasing function of λ. The E-Bayesian estimate of λ is given by

λ^EB=∫01∫0kλ^BP(λ,a,b) dadb.

The E-Bayesian estimate of λ is then given by the integral of the Bayesian estimate of λ, denoted as λ^B, multiplied by the prior distribution, over suitable domains for the hyper parameters a and b. The domains for the first and second integrals correspond to the regions where the prior density function is a decreasing function of λ.

The specific distributions chosen for the hyper parameters a and b are as follows:

P1(a,b)=2(k−b)k2,0 < a < 1,0 < b < k,(25)

P2(a,b)=1k,0 < a < 1,0 < b < k,(26)

and

P3(a,b)=2bk2,0 < a < 1,0 < b < k.(27)

3.1 E-Bayesian Estimation under SELF

For a sample z_=(z1,⋯,zn∗) of the Kumaraswamy distribution, using the Bayesian estimator of λ under SELF in Eq. (14) with the priors given in Eqs. (25)–(27), the E-Bayesian estimators of λ under SELF are given by

λ^EBS1= 2n∗+1k2[(k+g2(δ,z_))log(k+g2(δ,z_)g2(δ,z_))−k],λ^EBS2= 2n∗+12klog⁡(k+g2(δ,z_)g2(δ,z_)),λ^EBS3= 2n∗+1k2[g2(δ,z_)log⁡(g2(δ,z_)k+g2(δ,z_))+k].(28)

For a time t with Eq. (15), using the priors given in Eqs. (25)–(27), the E-Bayesian estimators of H(t) under SELF are given by

H^EBS1= δtδ−11−tδ(2n∗+1)k2[(k+g2)log⁡(k+g2g2)−k],H^EBS2= δtδ−11−tδ(2n∗+1)2klog⁡(k+g2g2),H^EBS3= δtδ−11−tδ(2n∗+1)k2[g2log⁡(g2k+g2)+k],(29)

where g2(δ,z_) =D(δ,z_)=−[∑j=1n∗(Rj∗+1)ln⁡(1−zjδ)+Rτ∗ln⁡(1−Tδ)] and we will write it as g2 for simplicity of notation.

3.2 E-Bayesian Estimation under ELF

For a sample z_=(z1,⋯,zn∗) from the Kumaraswamy distribution, using the Bayesian estimator of λ under SELF in Eq. (17) with the priors given in Eqs. (25)–(27), the E-Bayesian estimators of λ under ELF are given by

λ^EBE1= 2n∗−1k2[(k+g2)log⁡(k+g2g2)−k],λ^EBE2= 2n∗−12klog⁡(k+g2g2),λ^EBE3= 2n∗−1k2[g2log⁡(g2k+g2)+k].(30)

For a time t with Eq. (18), using the priors given in Eqs. (25)–(27), the E-Bayesian estimators of H(t) under ELF are given by

H^EBE1= δtδ−11−tδ(2n∗−1)k2[(k+g2)log⁡(k+g2g2)−k],H^EBE2= δtδ−11−tδ(2n∗−1)2klog⁡(k+g2g2),H^EBE3= δtδ−11−tδ(2n∗−1)k2[g2log⁡(g2k+g2)+k].(31)

3.3 E-Bayesian Estimation under WBLF

For a sample z_=(z1,⋯,zn∗) from the Kumaraswamy distribution, using the Bayesian estimator of λ under WBLF in Eq. (20) with the priors given in Eqs. (25)–(27), the E-Bayesian estimators of λ under WBLF are given by

λ^EBW1= 2n∗+3k2[(k+g2)log⁡(k+g2g2)−k],λ^EBW2= 2n∗+32klog⁡(k+g2g2),λ^EBW3= 2n∗+3k2.[g2log⁡(g2k+g2)+k].(32)

For a time t with Eq. (21), using the priors given in Eqs. (25)–(27), the E-Bayesian estimators of H(t) under WBLF are given by

H^EBW1= δtδ−1(1−tδ)(2n∗+3)k2[(k+g2)log⁡(k+g2g2)−k],H^EBW2= δtδ−1(1−tδ)(2n∗+3)2klog⁡(k+g2g2),H^EBW3= δtδ−1(1−tδ)(2n∗+3)k2[g2log⁡(g2k+g2)+k].(33)

3.4 E-Bayesian Estimation under MELF

For a sample z_=(z1,⋯,zn∗) of the Kumaraswamy distribution, using the Bayesian estimator of λ under MELF in Eq. (23) with the priors given in Eqs. (25)–(27), the E-Bayesian estimators of λ under MELF are given by

λ^EBM1= 2n∗−3k2[(k+g2)log⁡(k+g2g2)−k],λ^EBM2= 2n∗−32klog⁡(k+g2g2),λ^EBM3= 2n∗−3k2[g2log⁡(g2k+g2)+k].(34)

For a time t with Eq. (24), using the priors given in Eqs. (25)–(27), the E-Bayesian estimators of H(t) under MELF are given by

H^EBM1= δtδ−11−tδ(2n∗−3)k2[(k+g2)log⁡(k+g2g2)−k],H^EBM2= δtδ−11−tδ(2n∗−3)2klog⁡(k+g2g2),H^EBM3= δtδ−11−tδ(2n∗−3)k2[g2log⁡(g2k+g2)+k].(35)

4  H-Bayesian Estimation

In this part, the H-Bayesian estimates of the shape parameter of the Kumaraswamy distribution are obtained using different loss functions, namely SELF, ELF, WBLF, and MELF. Following the methodology proposed by Lindley et al. [29], we introduce hyper parameters denoted as a and b in the prior distribution Prior(λ|a,b) as given in Eq. (11). The hyper prior distributions of a and b are defined in Eqs. (25)–(27). Then the corresponding hierarchical prior distributions of λ are derived based on these hyper priors.

P4(λ)=∫01∫0kP(λ|a,b)P1(a,b)dbda=∫01∫0kbaλa−1Γ(a)exp⁡(−bλ)2(k−b)k2 dbda,(36)

P5(λ)=∫01∫0kP(λ|a,b)P2(a,b)dbda=∫01∫0kbaλa−1Γ(a)exp⁡(−bλ)1k dbda,(37)

and

P6(λ)=∫01∫0kP(λ|a,b)P3(a,b)dbda=∫01∫0kbaλa−1Γ(a)exp⁡(−bλ)2bk2 dbda.(38)

Using Bayes theorem, likelihood function and Eqs. (36)–(38), the hierarchical posterior distributions of λ can be written as

P1(λ|z_)=L(λ|z_)P4(λ)∫0∞L(λ|z_)P4(λ) dλ=∫01∫0k(k−b)baΓ(a)λn∗+a−1exp⁡[−λg1] dbda∫01∫0k(k−b)baΓ(a)Γ(n∗+a)[g1]n∗+a dbda,

P2(λ|z_)=L(λ|z_)P5(λ)∫0∞L(λ|z_)P5(λ) dλ=∫01∫0kbaΓ(a)λn∗+a−1exp⁡[−λg1] dbda∫01∫0kbaΓ(a)Γ(n∗+a)[g1]n∗+a dbda,

and

P3(λ|z_)=L(λ|z_)P6(λ)∫0∞L(λ|z_)P6(λ) dλ=∫01∫0kba+1Γ(a)λn∗+a−1exp⁡[−λg1]dbda∫01∫0kba+1Γ(a)Γ(n∗+a)[g1]n∗+a dbda.

4.1 H-Bayesian Estimation Based on SELF

Using SELF and the H-posterior distributions which are defined respectively in Eqs. (13), (36)–(38), the H-Bayesian estimates λ^HBS1, λ^HBS2, λ^HBS3 of λ can be defined as

λ^HBSj=E[(λ|z_)];j=1,2,3.

Here,

λ^HBS1=E[(λ|z_)]=∫0∞λP1(λ|z_) dλ=∫01∫0k(k−b)baΓ(a)Γ(n∗+a+1)[g1]n∗+a+1 dbda∫01∫0k(k−b)baΓ(a)Γ(n∗+a)g1n∗+a dbda,λ^HBS2=E[(λ|z_)]= ∫0∞λP2(λ|z_) dλ=∫01∫0kbaΓ(a)Γ(n∗+a+1)[g1]n∗+a+1dbda∫01∫0kbaΓ(a)Γ(n∗+a)[g1]n∗+adbda,λ^HBS3=E[(λ|z_)]=∫0∞λP3(λ|z_) dλ=∫01∫0kba+1Γ(a)Γ(n∗+a+1)g1n∗+a+1 dbda∫01∫0kba+1Γ(a)Γ(n∗+a)g1n∗+a dbda.(39)

For a time t, the H-Bayesian estimators of H(t) under SELF are given by

H^HBS1= δtδ−1(1−tδ)∫01∫0k(k−b)baΓ(a)Γ(n∗+a+1)[g1]n∗+a+1 dbda∫01∫0k(k−b)baΓ(a)Γ(n∗+a)[g1]n∗+a dbda,H^HBS2= δtδ−1(1−tδ)∫01∫0kbaΓ(a)Γ(n∗+a+1)[g1]n∗+a+1 dbda∫01∫0kbaΓ(a)Γ(n∗+a)[g1]n∗+a dbda,H^HBS3= δtδ−1(1−tδ)∫01∫0kba+1Γ(a)Γ(n∗+a+1)[g1]n∗+a+1 dbda∫01∫0kba+1Γ(a)Γ(n∗+a)[g1]n∗+a dbda.(40)

4.2 H-Bayesian Estimation Based on ELF

Using ELF and the H-posterior distributions which are defined respectively in Eqs. (16), (36)–(38), the H-Bayesian estimates λ^HBE1, λ^HBE2, λ^HBE3 of λ can be defined as

λ^HBEj=E[(λ−1|z_)]−1;j=1,2,3.

Here,

λ^HBE1=E[(λ−1|z_)]=∫0∞P1(λ|z_)λ dλ=∫01∫0k(k−b)baΓ(a)Γ(n∗+a−1)[g1]n∗+a−1 dbda∫01∫0k(k−b)baΓ(a)Γ(n∗+a)[g1]n∗+a dbda,λ^HBE2=E[(λ−1|z_)]=∫0∞P2(λ|z_) λdλ=∫01∫0kbaΓ(a)Γ(n∗+a−1)[g1]n∗+a−1dbda∫01∫0kbaΓ(a)Γ(n∗+a)[g1]n∗+adbda,λ^HBE3=E[(λ−1|z_)]=∫0∞P3(λ|z_)λ dλ=∫01∫0kba+1Γ(a)Γ(n∗+a−1)[g1]n∗+a−1 dbda∫01∫0kba+1Γ(a)Γ(n∗+a)[g1]n∗+a dbda.(41)

For a time t, the H-Bayesian estimates of H(t) under ELF are given by

H^HBE1= δtδ−1(1−tδ)∫01∫0k(k−b)baΓ(a)Γ(n∗+a)[g1]n∗+a dbda∫01∫0k(k−b)baΓ(a)Γ(n∗+a−1)[g1]n∗+a−1 dbda,H^HBE2= δtδ−1(1−tδ)∫01∫0kbaΓ(a)Γ(n∗+a)[g1]n∗+a dbda∫01∫0kbaΓ(a)Γ(n∗+a−1)[g1]n∗+a−1 dbda,H^HBE3= δtδ−1(1−tδ)∫01∫0kba+1Γ(a)Γ(n∗+a)[g1]n∗+a dbda∫01∫0kba+1Γ(a)Γ(n∗+a−1)[g1]n∗+a−1 dbda.(42)