EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 753-773 ISSN 1307-5543 – ejpam.com Published by New York Business Global E-Bayesian Estimation under Loss Functions in Competing Risks Didier Alain Njamen Njomen1,∗, Thiery Donfack1, Joseph Ngatchou-Wandji2, Georges Nguefack-Tsague3 1 Department of Mathematics and Computer’s Science, Faculty of Science, University of Maroua, Maroua, Cameroon 2 EHESP Sorbonne Paris City & Institut Elie Cartan de Nancy, Université de Lorraine, France 3 Department of Public Health, Faculty of Medicine and Biomedical Sciences, University of Yaounde I - Cameroon Abstract. Using gamma prior distribution of which shape hyperparameter has beta distribution and rate parameter has three different distributions over a finite interval, we studied the E-Bayesian estimation of one scale parameter of Gompertz distribution based on progressively type I censored sample from the competing risks model subject to K independent causes. The estimators obtained generalize those issued from the quadratic loss, entropy loss and DeGroot loss functions. 2020 Mathematics Subject Classifications: 62C10, 91G70, 33B15, 62P20 Key Words and Phrases: Bayesian estimation, E-Bayesian estimation, Competing risks, Type-I progressively censoring, Loss function 1. Introduction Several estimations of the parameters of the Gompertz distribution in a competing risks context have been studied in the literature. The maximum likelihood estimation has been studied by [17], while the Bayesian estimation and the hierarchical Bayesian estimation have been investigated by [19], [3, 27, 28], [30], [33] and [23]. All these methods involve integrals whose computation is not easy and may require numerical methods. The progress in computational mathematics and statistics during the past two decades has contributed to the development of a new method called E-Bayesian estimation intro- duced by [10]. By considering the quadratic loss function, [10] proved by means of simula- tions, that the E-Bayesian estimator is more efficient and easier to implement than others ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4351 Email addresses: didiernjamen1@gmail.com (D. A. N. Njamen), thierytd@yahoo.fr (T. Donfack), joseph.ngatchou-wandji@univ-lorraine.fr (J. Ngatchou-Wandji), nguefacktsague@gmail.com (G. Nguefack-Tsague) https://www.ejpam.com 753 © 2022 EJPAM All rights reserved. D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 754 authors. Since then, there has been a growing interest in studying E-Bayesian estimation. The studies are done under different distributions such as the binomial distribution [19], the exponential distribution ([30]), the Pareto distribution [26] and the distribution of Lomax ([6]). All these papers came to the same conclusion that the E-Bayesian estimate is better than the Bayesian estimate. [32] studied E-Bayesian estimation of a parameter in the context of competing risks model under the quadratic and Linex loss functions. They also concluded that the new method is more efficient under these loss functions. In this article, we assume that the survival time X is a positive and absolutely con- tinuous random variable. Instead of observing independent and identically distributed realizations (i.i.d.) of duration X, we observe the realization of the variable X subjected to various perturbations independent or not of the phenomenon studied. In the presence of right random censorship, the lifetimes are not all observed. For some of them, one only knows that they are greater than a certain known value. There are several types of censorship: Type I, II, and III censorship. The reader interested in the notion of censorship can refer to [1] or [7]. Type I censorship describes the situation where a test ends at a certain period and one knows that the remaining individuals have not yet been observed. In this case, the censorship time is fixed in advance and the number of individuals not observed is a random variable. Let C be a fixed value, instead of observing the complete life time variables X1, · · · , Xn, one observes Xi when Xi ≤ Ci if not, one knows that Xi > Ci. We use the following notation Ti = Xi ∧ Ci = min(Xi, Ci), with i = 1, · · · , n. In the case of simple type I censorship, all the individuals are censored after the same length of time while in the case of progressive type I censorship which is used in this article, all the individuals are censored at the same date whatever the time span they were followed. Progressively censored type I data were first proposed by [15]. Indeed, in the context of our article, progressively censored type I data are described as follows : assume that n units are wagered in a progressive life-test censorship scheme: (R1, R2, . . . , Rr), 1 ≤ r ≤ n. The experiment is over on the date τ ∈ (0,∞), Ri (i = 1, 2, . . . , r) and r is fixed in advance. At the time of the first failure t1, R1 of the remaining units are randomly removed, at the time of the second failure t2, R2 of the remaining units are randomly removed and so on. If the rth failure time tr occurs before time τ , all the remaining units Rr = n− r− (R1 + . . .+Rr−1) are removed and the terminal time of the experiment is tr. On the other hand, if the rth failure time tr does not occur before time τ and only J failures occur before time τ , where 0 ≤ J ≤ r, then at the time τ , all the remaining R∗ τ = n− J − (R1 + . . .+RJ) units are removed, and the terminal time of the experiment is τ . We denote the two cases as: Case 1 t1 < t2 < . . . < tr, tr < τ ; Case 2 t1 < t2 < . . . < tJ < τ < tJ+1 < . . . < tr, tr > τ. In the reminder of this article, we briefly present the notion of competing risks for the D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 755 Gompertz modeling, and the E-Bayesian estimation of the scale parameter of the Gom- pertz distribution under several loss functions. 2. Competing risks 2.1. Background In survival analysis, a competing risks situation is that where the event of interest is subject to several causes. In this context, the event of interest has been modelled by various distributions such as Gompertz distribution ([29], [32]), exponential distribution ([20]) and Lindley distribution ([21]), stochastic process ([22]). In this paper, we adopt the same concepts as Wu et al. [32] and Njamen et al. [23]. An examination of the literature has shown that for a given subject, at most one event denoted by δk (k ∈ {1, · · · ,K}) among K events will be observed. If no event occurs, then the subject is censored at the end of its tracking (δk = 0). In practice, one fixes a single event of interest (δk = 1) among the possible K. We assumed that : - there are K competing independent failure modes; - the system failure only occurs in one of the competingK failure modes with durations T1, · · · , TK ; - the system failure time is T = min{T1, · · · , TK} which is a latent time ; - the lifetime of the concurrent failure mode k (k = 1, · · · ,K) denoted by Tk, follows a Gompertz distribution with parameters αk and βk: Gompertz(αk, βk). 2.2. Gompertz distribution in competing risks [14] used the [9] distribution to model the cumulative incidence function (CIF) asso- ciated with an event. This CIF associated with an event of type k is denoted here by Fk(t,Ψk), defined by Fk(t; Ψk) = 1− exp { −βk αk [exp(αkt)− 1] } , (1) with Ψk = (αk;βk) ∈ R∗ × R; αk is the shape parameter and βk the scale parameter. The curve below is that of CIF. It is obtained by the R software version R i386 3.1.3 downloadable online. The CIF is a function defined on the interval [0; 1]. The curves of the distribution function of the Gompertz distribution inform us that, for any values of the parameters αk and βk, the curves take their origin in 0 and increase until they reach the value 1. The graph illustrates this perfectly. The CIF is used in survival data in aging biology where αk is called the coefficient of the age-dependent mortality rate, and βk is called the coefficient of the death-age-independent rate (see [31]). These models are D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 756 0.0 0.5 1.0 1.5 2.0 2.5 0 .0 0 .5 1 .0 1 .5 t α = 1,β = 1 α = 2,β = 2 α = 0.5,β = 0.5 α = 0.8,β = 2 α = 4.5,β = 0.1 Figure 1: Curve of the CIF of the Gompertz distribution also widely used in demography where they make it possible to estimate the lifespans of populations ([2]). The associated density function fk is obtained by deriving the CIF with respect to time. One has : fk(t; Ψk) = ∂Fk(t; Ψk) ∂t = βk exp(αkt) exp { −βk αk [exp(αkt)− 1] } . (2) Figure 2 below is the curve of the density function of the Gompertz distribution. It is obtained by the R software. We notice that when βk tends to 0, we obtain the curve of the Exponential distribution, which is a particular case of the Gompertz distribution (see the curve in blue). Thus, when βk → 0, the curve presents exponential distribution. Actually, through limit concept, lim αk→0 fk(t,Ψk) = βk exp(−tβk), for t > 0. If we fix the parameter βk, we obtain a family of distributions indexed by the parameter αk (> 0) which in fact constitutes a family of distributions at risk proportional. Thus, the other curves (green, black, red and pink) give us the basic Gompertz density under different parameters. Under progressive type I censorship, we consider a population of K competing risks. Let k be a fixed constant, k ∈ {1, 2, . . . ,K}. Let τ∗ = min{tr, τ} and R∗ = r, tr ≤ τ ; R∗ = J, tr > τ where τ∗ is the final time of the experiment, R∗ the number of failures before time τ∗. The couples (t1, α1), · · · , (tR∗ , αR∗) are the observed failure data, where t1, t2, . . . , tR∗ are the failure times in statistical order and αi takes any integer in the set D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 757 0.0 0.5 1.0 1.5 2.0 2.5 0 .0 0 .5 1 .0 1 .5 2 .0 t α = 1,β = 1 α = 2,β = 2 α = 0.5,β = 0.5 α = 0.8,β = 2 α = 4.5,β = 0.1 Figure 2: Curve of the density function of the Gompertz distribution {1, · · · ,K}. Note that αi = k (k = 1, 2, . . . ,K) indicates the failure mode caused by the kth event. Let δk(αi) = { 1 if αi = k 0 if αi ̸= k, and nk = ∑R∗ i=1 δk(αi) ≥ 0 the total number of failures caused by the kth event. Under type I progressive censorship as defined in the introduction, the likelihood function is given for all t = (t1, t2, . . . , tR∗) by : Lk(t|αk, βk) ∝ K∏ k=1 [ R∗∏ i=1 fk(ti) δk(αi)[1− Fk(ti)] 1−δk(αi)[1− Fk(ti)] Ri [1− Fk(τ ∗)]n−R∗− ∑R∗ i=1 Ri ] , (3) where Ri is the ith remainder in the random variables. After calculation (see [23]), the likelihood function is obtained from (2) as : Lk(t|αk, βk) = K∏ k=1 [ βnk k exp { αk R∗∑ i=1 δk(αi)ti − ( βk αk ) ×Ak }] , (4) with Ak = R∗∑ i=1 (Ri + 1) ( eαkti − 1 ) + ( n−R∗ − R∗∑ i=1 Ri )( eαkr ∗ − 1 ) and αk > 0. D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 758 In addition to the distribution of [9], we consider the gamma distribution with parameters a > 0 and b > 0, whose density function π is defined for all x ∈ R by : π(x; a, b) = ba Γ(k) xa−1 exp(−bx)I(x > 0), (5) where Γ is the Gamma function defined for all a > 0 by : Γ(a) = ∫ ∞ 0 e−xxa−1dx. (6) It is easy to check that for all a > 0, Γ(a+ 1) = aΓ(a), and in particular for an integer a, Γ(a) = (a− 1)!. By convention, Γ(12) = √ π. We also consider the Beta density function of parameters u > 0 and v > 0, defined for x ∈ R by: xu−1(1− x)v−1 B(u, v) I(0 ≤ x ≤ 1), where for u > 0 and v > 0, B(u, v) = ∫ 1 0 tu−1(1− t)v−1dt (i) One easily shows that for all x > 0 and y > 0, B(x, y + 1) = y x+ y B(x, y) (ii) and B(x, y) = Γ(x)Γ(y) Γ(x, y) = B(y, x). (iii) Bayesian estimation has received great attention by the researchers who have that Bayes estimators perform better than classical estimators. Expected Bayesian or E- Bayesian method as an extension to Bayesian estimation has been introduced by [11]. He obtained the E- Bayes estimate of failure probability by considering quadratic loss function and discussed the properties of E-Bayes estimate and showed that E-Bayes esti- mate is efficient and easy to operate. In this article, we examine E-Bayesian estimation of the parameter of the reliability function for the competing risk model from the Gompertz distribution developed in [23]. Under the type I progressive censorship, the new estimators obtained generalize not only the estimators of the generalized quadratic loss function proposed by [32], but also those of the DeGroot and Entropy loss functions. D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 759 3. The E-Bayesian notion Bayesian methods in the context of competing risks are studied for instance by [10–13], [32], [24], [33], [18] and [25]. E-Bayesian estimation is a new method for estimating the probability of failure in the case of two hyper-parameters. Introduced by [10–12], in the context of a single risk, it is based on the calculation of the posterior mean of the Bayes estimators. Here we consider E-Bayesian estimation of the shape parameter λ of the Gompertz dis- tribution in a competing risks context. Under the Entropy loss function, we assume that λ follows a Gamma prior distribution π(λ|a, b) where a > 0 and b > 0 are the hyper- parameters. Definition 1. ([10]) Then the E-Bayesian estimate of λ is given by : λ̂EB = ∫ ∫ D λ̂Bπ(λ|(a, b)dadb = Eπ[λ̂B(a, b)], (7) where D is the domain of space of the parameters a and b. Definition 2. ([10]) An E-Bayesian estimate of λ is Eπ[λ̂B(a, b)] the expectation of λ̂B(a, b) obtained with respect to any joint distribution π(a, b) of (a, b). In the context of competitive risks, [32] considered the Bayesian estimation of the parameter βk and assumed that βk follows an a-priori Gamma distribution π(βk|ak, bk), with hyper-parameters ak > 0 and bk > 0. These must be selected to guarantee that π(βk) be a decreasing function of βk. For this reason, they must be choosen such that ∂π(βk|ak, bk) ∂βk < 0. Since ∂π(βk|ak, bk) ∂βk = bakk Γ(ak) (ak − 1− bkβk)β ak−2 k exp(−bkβk), (8) from (8), ak and bk must achieve 0 < ak < 1 and bk > 0. We assume subsequently that ak and bk are independent random variables, with joint (a-priori) distribution of the form π̃(ak, bk) = π̃(ak)π̃(bk), where π̃ is a density function. In the sequel, we use three different a priori distributions for ak and bk as defined by [32]. The influence of each of these on the E-Bayesian estimation of βk is investigated. The three a priori joint distributions of the hyperparameters ak and bk are defined as Beta distributions by : π̃1(ak, bk) = 1 ckB(uk,vk) auk−1 k (1− ak) vk−1 π̃2(ak, bk) = 2 c2kB(uk,vk) (ck − bk)a uk−1 k (1− ak) vk−1 π̃3(ak, bk) = 2bk c2kB(uk,vk) auk−1 k (1− ak) vk−1, (9) where ak and bk are such that 0 < ak < 1, 0 < bk < ck, ck > 0. D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 760 4. E-Bayesian estimation under the generalized quadratic loss function 4.1. Generalized quadratic loss function The quadratic loss function proposed by [16] and [8] is defined by : L(θ, d) = (θ − d)2, where θ ∈ Θ the parameters space, d ∈ D the decision space. A variant of this loss function is the weighted squared loss function of the form L(θ, d) = ω(θ)(θ − d)2, where ω is a weight function. A cost function is any real-valued function L defined on Θ × D. In general, a loss function is a measurable positive function defined on Θ×D. The parameters space Θ is endowed with a probability π such that (Θ,A, π) is a probabilized space. We write θ ∼ π to mean that θ has distribution π called a priori law. This distribution determines what we know and what we don’t know before observing the the event under consideration. Under the assumption of a quadratic cost, the Bayes estimator δπ(x) of θ associated with the prior distribution π is the conditional mean a posteriori of θ defined for any observation x = (x1, x2, · · · , xn) by : δπ(x) = Eπ(·|x)(θ) = ∫ θ∈Θ L(θ, δ(x))π(θ|x)dθ. The E-Bayesian estimator of βk with hyper-parameters ak and bk is given by : β̂k(EBQGi) = ∫ ∫ D β̂k(BQG)(ak, bk)πi(ak, bk)dbkdak, i = 1, 2, 3, where D is the decision space, and β̂k(BQG) is the Bayesian estimator of βk defined in Theorem 4.1 of [23] and recalled below : β̂k(BQG)(αk, βk) = nk + ak + α− 1 bk + Ak αk , with αk > 0. The a priori distributions defined above will allow us in the following subsection to determine the estimators of the Bayesian expectation for the different loss functions con- sidered. D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 761 4.2. The E-Bayesian estimators Theorem 1. Under the generalized quadratic loss function, the E-Bayesian estimators of βk obtained with the priors πi(ak, bk), i ∈ {1, 2, 3} are given respectively by : β̂k(EBQG1) = c−1 k ln ( 1 + ck Ak αk )( nk + α− 1 + uk uk + vk ) β̂k(EBQG2) = 2c−2 k [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )]( nk + α− 1 + uk uk + vk ) β̂k(EBQG3) = 2c−2 k [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )]( nk + α− 1 + uk uk + vk ) , (10) with αk > 0, 0 < ak < 1 and 0 < bk < ck. Proof. For i = 1, we have, for the generalized quadratic loss function, and the prior π1(ak, bk), the E-Bayesian estimator of βk given by : β̂k(EBQG1) = ∫ 1 0 ∫ ck 0 nk + ak + α− 1 bk + Ak αk × 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 nk + ak + α− 1 bk + Ak αk dbk ) dak = ∫ 1 0 nk + ak + α− 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 1 bk + Ak αk dbk ) dak = ∫ 1 0 nk + ak + α− 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × ln ( 1 + ck Ak αk ) dak = 1 ckB(uk, vk) × ln ( 1 + ck Ak αk )∫ 1 0 (nk + ak + α− 1)× auk−1 k (1− ak) vk−1dak; = 1 ckB(uk, vk) × ln ( 1 + ck Ak αk ) × I, with I = ∫ 1 0 (nk + ak + α− 1)× auk−1 k (1− ak) vk−1dak. One has the following equalities : I = ∫ 1 0 (nk + ak + α− 1)× auk−1 k (1− ak) vk−1dak + ∫ 1 0 auk k (1− ak) vk−1dak = ∫ 1 0 [(nk + α− 1) + ak]a uk−1 k (1− ak) vk−1 D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 762 = (nk + α− 1) ∫ 1 0 (auk−1 k )dak + ∫ 1 0 aka uk−1 k (1− ak) vk−1dak = (nk + α− 1) ∫ 1 0 (auk−1 k )dak + ∫ 1 0 a (uk−1)+1 k (1− ak) vk−1dak = (nk + α− 1) ∫ 1 0 (auk−1 k )dak + ∫ 1 0 a (uk+1)−1 k (1− ak) vk−1dak = (nk + α− 1)B(uk, vk) +B(uk + 1, vk) by (i) = (nk + α− 1)B(uk, vk) +B(vk, uk + 1) since B(x, y) = B(y, x) = (nk + α− 1)B(uk, vk) + uk uk + vk B(uk, vk) by (ii) = ( nk + α− 1 + uk uk + vk ) B(uk, vk). (11) It results from above that β̂k(EBQG1) = 1 ckB(uk, vk) × ln ( 1 + ck Ak ck ) × [ nk + α− 1 + uk uk + vk ] B(uk, vk) = 1 ck ln ( 1 + ck Ak ck ) × [ nk + α− 1 + uk uk + vk ] . For i = 2, we have for the generalized quadratic loss function, and for the prior π2(ak, bk), the E-Bayesian estimator of βk given by : β̂k(EBQG2) = ∫ 1 0 ∫ ck 0 nk + ak + α− 1 bk + Ak αk × 2 c2kB(uk, vk) (ck − bk)a uk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 2(nk + ak + α− 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 (∫ ck 0 ck − bk bk + Ak αk dbk ) dak = ∫ 1 0 2(nk + ak + α− 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 [ −bk + ( ck + Ak αk ) ln ( bk + Ak αk )]ck 0 dak = ∫ 1 0 2(nk + ak + α− 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] dak = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2kB(uk, vk) ∫ 1 0 (nk + ak + α− 1)auk−1 k (1− ak) vk−1dak = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2kB(uk, vk) × I = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2kB(uk, vk) ×B(uk, vk) [ (nk + α− 1) + uk uk + vk ] by (14) D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 763 = 2c−2 k [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] × [ (nk + α− 1) + uk uk + vk ] . For i = 3, we have for the generalized quadratic loss function, and for the prior π3(ak, bk), the E-Bayesian estimator of βk given by : β̂k(EBQG3) = ∫ 1 0 ∫ ck 0 nk + ak + α− 1 bk + Ak αk × 2 c2kB(uk, vk) × (bk)a uk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 2(nk + ak + α− 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 bk bk + Ak αk dbk ) dak = ∫ 1 0 2(nk + ak + α− 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ bk − ( Ak αk ) ln ( bk + Ak αk )]ck 0 dak = ∫ 1 0 2(nk + ak + α− 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] dak = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )]∫ 1 0 (nk + ak + α− 1)auk−1 k (1− ak) vk−1dak = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] × I = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] ×B(uk, vk) [ (nk + α− 1) + uk uk + vk ] by (14) = 2c−2 k [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] × [ (nk + α− 1) + uk uk + vk ] . Remark 1. From the decompositions resulting from the above theorem, one observes that : • For α = 1, one obtains the estimator of βk under the quadratic loss function as in [32]; • For α = 2, one gets the estimator of βk under the DeGroot loss function; • For α = 0, one gets the estimator of βk under the Entropy loss function. Thus, our estimators generalize not only the one associated with the quadratic loss function of [32], but also those associated with the DeGroot and the Entropy loss functions. 5. E-Bayesian Estimation for the DeGroot loss function 5.1. The DeGroot loss function [5] introduced several types of loss functions and then obtained the Bayes estimators under them. An example of a symmetric loss function is defined by : D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 764 L(θ, δ(x)) = ( θ − δ(x) δ(x) )2 . Under this loss function, the Bayes estimator is defined by : δπ(x) = Eπ(θ 2|x) Eπ(θ|x) . The E-Bayesian estimator of βk with hyper-parameters ak and bk is given by the formula: β̂k(EBDi) = ∫ ∫ D β̂k(BD)(ak, bk)πi(ak, bk)dbkdak, i = 1, 2, 3, where D is the decision space and where β̂k(BD) is the Bayesian estimator of βk defined in Theorem 4.2 of [23] and given below: β̂k(BD)(αk, βk) = nk + ak + 1 bk + Ak αk , with αk > 0. The a priori distributions defined above will allow us in the following subsection to determine the E-Bayesian estimators for the different loss functions of DeGroot. 5.2. The E-Bayesian Estimators Theorem 2. Under DeGroot’s loss function, the E- Bayesian estimators of βk with the priors πi(ak, bk), i ∈ {1, 2, 3} are given by : β̂k(EBD1) = c−1 k ln ( 1 + ck Ak αk )( nk + 1 + uk uk + vk ) β̂k(EBD2) = 2c−2 k [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )]( nk + 1 + uk uk + vk ) β̂k(EBD3) = 2c−2 k [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )]( nk + 1 + uk uk + vk ) , (12) where αk > 0, 0 < ak < 1 and 0 < bk < ck. Proof. For i = 1, under DeGroot’s loss function, and for the prior π1(ak, bk), the E-Bayesian estimator of βk is given by : β̂k(EBD1) = ∫ 1 0 ∫ ck 0 nk + ak + 1 bk + Ak αk × 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 nk + ak + 1 bk + Ak αk dbk ) dak D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 765 = ∫ 1 0 nk + ak + 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 1 bk + Ak αk dbk ) dak = ∫ 1 0 nk + ak + 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × ln ( 1 + ck Ak αk ) dak = 1 ckB(uk, vk) × ln ( 1 + ck Ak αk )∫ 1 0 (nk + ak + 1)× auk−1 k (1− ak) vk−1dak; = 1 ckB(uk, vk) × ln ( 1 + ck Ak αk ) × I1, with I1 = ∫ 1 0 (nk + ak + 1)× auk−1 k (1− ak) vk−1dak. One has the following developments: I1 = (nk + 1) ∫ 1 0 auk−1 k (1− avk−1 k )dak + ∫ 1 0 aka uk−1 k (1− avk−1 k )dak = (nk + 1)B(uk, vk) + ∫ 1 0 a (uk−1)−1 k (1− ak)dak = (nk + 1)B(uk, vk) +B(uk + 1, vk) = (nk + 1)B(uk, vk) +B(vk, uk + 1) because B(x, y) = B(y, x) = (nk + 1)B(uk, vk) + uk uk + vk B(uk, vk) = [ (nk + 1) + uk uk + vk ] B(uk, vk), (13) From which one has: β̂k(EBD1) = 1 ckB(uk, vk) × ln ( 1 + ck Ak ck ) × ( nk + 1 + uk uk + vk ) B(uk, vk) = 1 ck ln ( 1 + ck Ak ck ) × ( nk + 1 + uk uk + vk ) . For i = 2, under DeGroot’s loss function, and for the a priori π2(ak, bk), the E-Bayesian estimator of βk is given by : β̂k(EBD2) = ∫ 1 0 ∫ ck 0 nk + ak + 1 bk + Ak αk × 2 c2kB(uk, vk) × (ck − bk)a uk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 2(nk + ak + 1) c1kB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 ck − bk bk + Ak αk dbk ) dak D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 766 = ∫ 1 0 2(nk + ak + 1) c1kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ −bk + ( ck + Ak αk ) ln(bk + Ak αk ) ]ck 0 dak = ∫ 1 0 2(nk + ak + 1) c1kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ −ck + ( ck + Ak αk ) ln(1 + ck Ak αk ) ] dak = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2B(uk, vk) × ∫ 1 0 (nk + ak + 1)auk+1 k (1− ak) vk−1dak = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2B(uk, vk) × I1, where I1 is defined above. Thus, β̂k(EBD2) = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2B(uk, vk) × [ (nk + 1) + uk uk + vk ] B(uk, vk) = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2 × [ (nk + 1) + uk uk + vk ] = 2c−2 k [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] × ( nk + 1 + uk uk + vk ) . For i = 3, under DeGroot’s loss function, and for the a priori distribution π3(ak, bk), the E-Bayesian estimator of βk is given by : β̂k(EBD3) = ∫ 1 0 ∫ ck 0 β̂k(BD)(ak, bk)π3(ak, bk)dbkdak = ∫ 1 0 ∫ ck 0 nk + ak + 1 bk + Ak αk × 2bk c2kB(uk, vk) × auk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 2(nk + ak + 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 bk bk + Ak αk dbk ) dak = ∫ 1 0 2(nk + ak + 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ bk − ( Ak αk ) ln ( bk + Ak αk )]ck 0 dak = ∫ 1 0 2(nk + ak + 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] dak = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )]∫ 1 0 (nk + ak + 1)auk−1 k (1− ak) vk−1dak = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] × I1 D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 767 = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] ×B(uk, vk) [ (nk + 1) + uk uk + vk ] from (18) = 2 c2k [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] × [ (nk + 1) + uk uk + vk ] . This expression gives the E-Bayesian estimator of βk for the a priori π3(ak, bk). 6. E-Bayesian estimation for the entropy loss function 6.1. Entropy loss function [4] proposed a loss function which results from the Linex loss function called the entropy loss function, defined by : LE(θ, d) ∝ ( d θ )p − p ln ( d θ ) − 1. LE(θ, d) is minimal at d = θ. The Bayes estimator of the parameter θ under this loss function is defined for all p ∈ R by : δ(x) = ( Eθ(θ) −p )−1 p . • When p = 1, the Bayes estimator coincides with the Bayes estimator under the weighted squared loss function : (d− θ)2 θ . • When p = −1, the Bayes estimator coincides with the Bayes estimator under the quadratic loss function. The E-Bayesian estimator of βk for the hyper-parameters ak and bk is given by the formula : β̂k(EBEi) = ∫ ∫ D β̂k(BE)(ak, bk)πi(ak, bk)dbkdak, i = 1, 2, 3, where D is the decision space and β̂k(BE) is the Bayesian estimator of βk defined by (see [23]) : β̂k(BE)(αk, βk) =  1( bk + Ak αk )−1 × Γ(nk + ak − p) Γ(nk + ak)  −1/p , with αk > 0. D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 768 6.2. E-Bayesian estimators of βk Theorem 3. Under the Entropy loss function for p = 1, the E-Bayesian estimators of βk for the priors πi(ak, bk), i ∈ {1, 2, 3} are given by :  β̂k(EBE1) = c−1 k ln ( 1 + ck Ak αk )( nk − 1 + uk uk + vk ) β̂k(EBE2) = 2c−2 k [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )]( nk − 1 + uk uk + vk ) β̂k(EBE3) = 2c−2 k [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )]( nk − 1 + uk uk + vk ) (14) where αk > 0, 0 < ak < 1 and 0 < bk < ck. Proof. For i = 1, under the entropy loss function for p = 1 and for the prior π1(ak, bk), the E-Bayesian estimator of βk is given by : β̂k(EBE1) = ∫ 1 0 ∫ ck 0  1( bk + Ak αk )−1 × Γ(nk + ak − p) Γ(nk + ak)  −1/1 × 1 ckB(uk, vk) auk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 ∫ ck 0  1( bk + Ak αk )−1 × Γ(nk + ak − p) (nk + ak − 1)Γ(nk + ak − 1)  −1 × 1 ckB(uk, vk) auk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 ∫ ck 0 nk + ak − 1 bk + Ak αk × 1 ckB(uk, vk) auk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 nk + ak − 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 1 bk + Ak αk dbk ) dak = ∫ 1 0 nk + ak − 1 ckB(uk, vk) × auk−1 k (1− ak) vk−1 × ln ( 1 + ck Ak αk ) dak = 1 ckB(uk, vk) × ln ( 1 + ck Ak αk )∫ 1 0 (nk + ak − 1)× auk−1 k (1− ak) vk−1dak = 1 ckB(uk, vk) × ln ( 1 + ck Ak αk ) × I2, with I2 = ∫ 1 0 (nk + ak − 1)× auk−1 k (1− ak) vk−1dak. D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 769 As for I and I1, one has: I2 = ∫ 1 0 [(nk − 1) + ak]× auk−1 k (1− ak) vk−1 = [ (nk − 1) + uk uk + vk ] B(uk, vk). (15) For i = 2, under the Entropy loss function for p = 1 and for the prior π2(ak, bk), the E-Bayesian estimator of βk is given by : β̂k(EBE2) = ∫ 1 0 ∫ ck 0 β̂k(BE)(ak, bk)π2(ak, bk)dbkdak = ∫ 1 0 ∫ ck 0 nk + ak − 1 bk + Ak αk × 2 c2kB(uk, vk) (ck − bk)a uk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 2(nk + ak − 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 (∫ ck 0 ck − bk bk + Ak αk dbk ) dak = ∫ 1 0 2(nk + ak − 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 [ −bk + ( ck + Ak αk ) ln ( bk + Ak αk )]ck 0 dak = ∫ 1 0 2(nk + ak − 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] dak = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2kB(uk, vk) ∫ 1 0 (nk + ak − 1)auk−1 k (1− ak) vk−1dak = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2kB(uk, vk) × I2 By replacing I2 in the expression for β̂k(EBE2) above, we have : β̂k(EBE2) = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2kB(uk, vk) × [ (nk − 1) + uk uk + vk ] B(uk, vk) = 2 [ −ck + ( ck + Ak αk ) ln ( 1 + ck Ak αk )] c2k × [ (nk − 1) + uk uk + vk ] . For i = 3, under the entropy loss function for i = 3 and for the prior π3(ak, bk), the E-Bayesian estimator of βk is given by: β̂k(EBE3) = ∫ 1 0 ∫ ck 0 β̂k(BE)(ak, bk)π3(ak, bk)dbkdak D. A. N. Njamen et al. / Eur. J. Pure Appl. Math, 15 (2) (2022), 753-773 770 = ∫ 1 0 ∫ ck 0 nk + ak − 1 bk + Ak αk × 2 c2kB(uk, vk) × (bk)a uk−1 k (1− ak) vk−1dbkdak = ∫ 1 0 2(nk + ak − 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × (∫ ck 0 bk bk + Ak αk dbk ) dak = ∫ 1 0 2(nk + ak − 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ bk − ( Ak αk ) ln ( bk + Ak αk )]ck 0 dak = ∫ 1 0 2(nk + ak − 1) c2kB(uk, vk) × auk−1 k (1− ak) vk−1 × [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] dak = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )]∫ 1 0 (nk + ak − 1)auk−1 k (1− ak) vk−1dak = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] × I2 = 2 c2kB(uk, vk) [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] ×B(uk, vk) [ (nk − 1) + vk uk + vk ] = 2 c−2 k [ ck − ( Ak αk ) ln ( 1 + ck Ak αk )] × [ (nk − 1) + vk uk + vk ] . This expression gives the E-Bayesian estimator of βk for the a-priori distribution of the density π3(ak, bk) of the hyper-parameters ak and bk under this loss function. 7. Conclusion and Perspectives In this paper, we have studied the E-Bayesian estimation of the scaling parameter in the context of competing risks of the Gompertz distribution under several loss functions. Under progressive type I censoring, we have determined the new estimators which gen- eralize not only the generalized quadratic loss function estimators proposed by [32] but also those of the DeGroot and Entropy loss functions. We also determine the E-Bayesian estimators of the scale parameter for the prior distributions of the hyper-parameters under the entropy loss function for p = 1 and under the DeGroot loss function. As perspectives, we plan to work on the relations existing among the estimators β̂k(EBQGi) (i = 1, 2, 3), β̂k(EBDi) (i = 1, 2, 3) and the relations among β̂k(EBEi) (i = 1, 2, 3). In a second step, we will do simulation experiments to evaluate the performance of Bayesian and E-Bayesian estimation of the reliability functions based on the generalized quadratic loss, DeGroot and Entropy functions in terms of estimated risks by Monte Carlo methods. We will also make applications in survival data in biology of aging where αk is called the coefficient of the age-dependent mortality rate, and βk is called the coefficient of the death-age-independent rate. The curves that we will obtain will be compared with those REFERENCES 771 obtained by [31] and we will allow us to judge the robustness and/or efficiency of our estimators obtained. Finally, we will consider the case of the 0-1 loss function. Acknowledgements The authors thank the editor and the reviewers for their criticisms and suggestions which have considerably raised the standard of this article. 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