# On Parameters Estimation of Lomax Distribution under General Progressive Censoring

Authors: Bander Al-Zahrani; Mashail Al-Sobhi
Journal: Journal of Quality and Reliability Engineering
 (2013)

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## Abstract

We consider the estimation problem of the probabilityS=P(Y<X) for Lomax distribution based on general progressive censored data. The maximum likelihood estimator and Bayes estimators are obtained using the symmetric and asymmetric balanced loss functions. The Markov chain Monte Carlo (MCMC) methods are used to accomplish some complex calculations. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation study.


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## Body

## 1. Introduction

The Lomax distribution, also called “Pareto type II” distribution is a particular case of the generalized Pareto distribution (GPD). The Lomax distribution has been used in the literature in a number of ways. For example, it has been extensively used for reliability modelling and life testing; see, for example, Balkema and de Haan [1]. It also has been used as an alternative to the exponential distribution when the data are heavy tailed; see Bryson [2]. Ahsanullah [3] studied the record values of Lomax distribution. Balakrishnan and Ahsanullah [4] introduced some recurrence relations between the moments of record values from Lomax distribution. The order statistics from nonidentical right-truncated Lomax random variables have been studied by Childs et al. [5]. Also, the Lomax model has been studied, from a Bayesian point of view, by many authors; see, for example, Arnold et al. [6] and El-Din et al. [7]. Howlader and Hossain [8] presented Bayesian estimation of the survival function of the Lomax distribution. Ghitany et al. [9] considered Marshall-Olkin approach and extended Lomax distribution. Cramer and Schmiedt [10] considered progressively type-II censored competing risks data from Lomax distribution. The Lomax distribution has applications in economics, actuarial modelling, queuing problems and biological sciences; for details, we refer to Johnson et al. [11].A positive random variableX is said to have the Lomax distribution, abbreviated as X~L(α,β), if it has the probability density function (pdf)
(1)f(x;α,β)=αβ(1+βx)-(α+1),x>0,α,β>0.
Here, α and β are the shape and the scale parameters, respectively. The survival function (sf) associated with (1) is
(2)F-(x;α,β)=(1+βx)-α,x>0.
Further probabilistic properties of this distribution are given, for example, in Arnold [12].This paper is concerned with the problem of estimatingS=P(Y<X) for Lomax based on general progressive censored data. The reliability of a component during a given period of time is defined as the probability that its strength X exceeds the stress Y, and symbolically we write S=P(Y<X). We assume X and Y to be independent, and each follows a Lomax distribution. A good overview on estimating S can be found in the monograph of Kotz et al. [13]. Later, the problem of estimating S attracted the attention of many authors; for example, see Baklizi [14], Raqab et al. [15], Kundu and Raqab [16], and Panahi and Asadi [17], and references cited therein.The rest of the paper is organized as follows. In Section2, we give a brief overview of the general progressive censoring. The maximum likelihood estimators (MLEs) are obtained in Section 3. In Section 4, we obtain Bayes estimators using the symmetric and asymmetric balanced loss functions. In Section 5, the MCMC methods are used to accomplish some complex calculations, and, therefore, comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation study.


## 2. General Progressive Censoring

We refer to the paper of Soliman et al. [18, page 452], for introducing the general progressive censoring as follows. Consider a general type-II progressive censoring scheme, proposed by Balakrishnan and Sandhu [19]. This scheme of censoring can be explained as follows: at time X0≡0, n randomly selected components were placed on a life test. The failure times of the first r components to fail, X1,…,Xr, were not observed. At the time of the (r+1)th failure, Xr+1:n, Rr+1 number of surviving components are removed from the test randomly and so on; at the time of the (r+i)th observed failure, Xr+i:n, Rr+i number of surviving components are removed from the test randomly; and finally, at the time of the mth failure, the remaining Rm=n-m-Rr+1-Rr+2-⋯-Rm-1 are removed from the test. Suppose that Xr+1:m:n≤Xr+2:m:n≤⋯≤Xm:m:n are the lifetimes of the completely observed components to fail and that Rr+1,Rr+2,…,Rm are the number of components removed from the test at these failure times, respectively. The Ri's, m, and r are prespecified integers such that 0≤r<m≤n,  0≤Ri≤n-i for i=r+1,…,m-1, and Rm=n-m-∑i=r+1m-1‍Ri. The resulting (m-r) ordered values Xr+1:m:n,Xr+2:m:n,…,Xm:m:n are appropriately referred to as general progressively type-II censored order statistics.Also referring to Soliman et al. [18], it should be noted that (i) if Ri=0, for i=r+1,…,m-1, and Rm=n-m, the general progressively type-II censoring scheme is reduced to the case of type-II doubly censored sample. (ii) If r=0, this scheme is reduced to the progressive type-II right censoring. (iii) If r=0 and Ri=0, for i=r+1,…,m-1 so that Rm=n-m, the general progressively type-II censoring scheme is reduced to conventional type-II one-stage right censoring, where just the first m usual order statistics are observed. (iv) If r=0 and Ri=0, for i=r+1,…,m, so that m=n, the general progressively type-II censoring scheme is reduced to the case of no censoring (complete sample case), where all n usual order statistics are observed. In this scheme R1,R2,…,Rm are all prefixed. Saraoglu et al. [20] discussed two examples showing the motivation behind the developments of the stress-strength models under censored samples. For more details, see Balakrishnan and Aggarwala [21].Suppose thatn randomly selected components from the L(α,β) are put in test. Further, let Xr+1:m:n,Xr+2:m:n,…,Xm:m:n denote a general progressively type-II censored sample from that population, with (Rr+1,Rr+2,…,Rm) being the progressive censoring scheme. For simplicity of notation, we will use xi instead of xi:m:n, and then x=(xr+1,…,xm) is the observed general progressive censored sample. The likelihood function for the parameters α and β is then
(3)ℒ(α,β∣x)=c*{1-F-(xr+1)}r∏i=r+1m‍f(xi)[F-(xi)]Ri,
where
(4)c*=(nr)(n-r)∏j=r+2m‍[n-∑i=r+1j-1‍Ri-j+1],
and the functions f(x) and F-(x) are given, respectively, by (1) and (2). Substituting (1) and (2) into (3), the likelihood function is
(5)ℒ(α,β∣x)=c*[1-(1+βxr+1)-α]r(αβ)m-r×∏i=r+1m‍(1+βxi)-α(1+Ri)-1.
Using the binomial expansion, r is a positive integer, one can rewrite the likelihood function as follows:
(6)ℒ(α,β∣x)=c*(αβ)m-re-u∑s=0r‍Cr,se-αTs,
where
(7)u=u(x;β)=∑i=r+1m‍ln(1+βxi),Ts=Ts(x;β)=sln(1+βxr+1)-∑i=r+1m‍(1+Ri)ln(1+βxi),Cr,s=(rs)(-1)s.
We focus our attention on the estimation of the probability S=P(Y<X), where X and Y are two independent random variables each is L(αj,βj), j=1,2 distributed, and the data obtained from both distributions are general progressively type-II censored. Here, X and Y are typically modeled as independent. The probability P(Y<X) has been widely studied under different approaches and distributional assumptions on X and Y. The case where X and Y are dependent has been considered by Nandi and Aich [22], Barbiero [23], and Rubio and Steel [24]. We investigate properties of S when the common scale parameter β1=β2=β is known. Then, it can be shown that
(8)S=P(Y<X)=∬0<y<xf(x,y)dxdy=α1α1+α2.
Here, f(x,y) is the joint pdf of X and Y, f(x,y)=f1(x)f2(y) by the independence. The general case, when β1≠β2, can be studied in a similar manner. We obtain that
(9)S=P(Y<X)=α1β1∫0∞‍(1+β1x)-(α1+1)(1+β2x)-α2dx.


## 3. Maximum Likelihood Estimation ofS

Suppose thatX1,X2,…,Xn1 and Y1,Y2,…,Yn2 are two independent random samples of size n1 and n2 from L(α1,β1) and L(α2,β2) distributions, respectively. The log-likelihood function, with ignoring constants, is given by
(10)lnℒ(Θ∣x,y)=∑j=12‍(mj-rj)(lnαj+lnβj)+r1ln[1-(1+β1xr1+1)-α1]-∑i=r1+1m1‍(α1(1+R1i)+1)ln(1+β1xi)+r2ln[1-(1+β2yr2+1)-α2]-∑i=r2+1m2‍(α2(1+R2i)+1)ln(1+β2yi),
where Θ=(α1,α2,β1,β2). The MLEs of (α1,β1), say (α^1ML,β^1ML), are obtained as the solution of the system of equations
(11)∂lnℒ∂α1=m1-r1α1+r1ln(1+β1xr1+1)(1+β1xr1+1)α1-1-∑i=r1+1m1‍(1+R1i)ln(1+β1xi)=0,∂lnℒ∂β1=m1-r1β1+α1r1xr1+1(1+β1xr1+1)-1(1+β1xr1+1)α1-1-∑i=r1+1m1‍(α1(1+R1i)+1)xi1+β1xi=0.
In a similar way, we can obtain the MLEs of (α2,β2): say (α^2ML,β^2ML).The corresponding “ML plug-in estimation” ofS, when β1=β2=β, is obtained by replacing α1 and α2 by its MLEs, α^1ML, α^2ML, and substituting them into relation (8) which yields
(12)S^ML=α^1MLα^1ML+α^2ML.
For the general case, β1≠β2, the corresponding “ML plug-in estimation” of S is obtained by replacing α1, β1, α2, and β2 by its MLEs, α^1ML, β^1ML, α^2ML, and β^2ML and substituting them into relation (9) which results in obtaining S^ML as
(13)S^ML=α^1MLβ^1ML×∫0∞(1+β^1MLx)-(α^1ML+1)(1+β^2MLx)-α^2MLdx.


## 4. Bayes Estimation

In this section, Bayesian estimation for the probabilityS in the stress-strength model involving Lomax distribution is obtained. The estimation is based on balanced loss function (BLF) which is introduced by Zellner [25]. We will use an extended class of BLF introduced by Jozani et al. [26]. It is of the following form:
(14)Lρ,ω,δ0q(ζ(θ),δ)=ωq(θ)ρ(δ0,δ)+(1-ω)q(θ)ρ(ζ(θ),δ),
where q(·) is a suitable positive weight function and ρ(ζ(θ),δ) is an arbitrary loss function when estimating ζ(θ) by δ. The parameter δ0 is a chosen priori estimate of ζ(θ), obtained for instance from the criterion of maximum likelihood, least squares, or unbiasedness among others. An intuitive interpretation of the BLFs is given by Ahmadi et al. [27] who argue that they give a general Bayesian connection between the case of ω>0, and ω=0 where 0≤ω<1. By choosing ρ(ζ(θ),δ)=(δ-ζ(θ))2 and q(θ)=1, the BLF is reduced to the balanced squared error loss (BSEL) function, used by Ahmadi et al. [27], in the form
(15)Lω,δ0(ζ(θ),δ)=ω(δ-δ0)2+(1-ω)ρ(δ-ζ(θ))2.
The corresponding Bayes estimate of the function ζ(θ) is given by
(16)δω,ζ,δ0(x)=ωδ0+(1-ω)E(ζ(θ)∣x).
Also, by choosing ρ(ζ(θ),δ)=ea(δ-ζ(θ))-a(δ-ζ(θ))-1 and q(θ)=1, we get the balanced LINEX, abbreviated as BLINEX, loss function written as
(17)La,ω,δ0(ζ(θ),δ)=ω[ea(δ-δ0)-a(δ-δ0)-1]+(1-ω)[ea(δ-ζ(θ))-a(δ-ζ(θ))-1].
In this case, the Bayes estimate of ζ(θ) takes the form
(18)δa,ω,ζ,δ0(x)=-1aln[ωe-aδ0+(1-ω)E(e-aζ(θ)∣x)],
where a≠0 is the shape parameter of BLINEX loss function.


### 4.1. Bayes Estimation Whenβ1=β2=β

Assuming thatβ1=β2=β (β is known) and α1, α2 are random variable each having gamma prior with some parameters, we can write
(19)π(αj)=δjνjΓ(νj)αjνj-1e-δjαj,αj>0,(νj,δj>0),j=1,2.
Since αj, j=1,2 are independent, by combining the likelihood function with the priors pdf, the joint posterior density function of α1 and α2 is given by
(20)π*(α1,α2∣x,y)=J∏j=12‍αjAj*-1e-δjαj-uj*∑s=0rj‍Crj,se-αjTjs*,
where Aj*=mj-rj+νj, T1s*=Ts(x;β), T2s*=Ts(y;β), u1*=u(x;β), u2*=u(y;β), and
(21)J-1=∏j=12‍∫0∞‍αjAj*-1e-δjαj-uj*∑s=0rj‍Crj,se-αjTjs*dαj.
Under the BSEL function and using (12) and (20), the proposed “Bayesian estimators” of S are actually Bayesian “plug-in” estimators as they are obtained by replacing (α1,β1,α2,β2) with its Bayesian estimator (α^1,β^1,α^2,β^2)(22)S^BS=ωS^ML+(1-ω)∫0∞‍∫0∞‍Sπ*(α1,α2∣x,y)dα1dα2.
With the same argument, we can obtain Bayes estimator under the BLINEX loss function. It is obtained as follows:
(23)S^BL=-1aln[∫0∞‍ωe-aS^ML+(1-ω)-1aln×∫0∞‍∫0∞‍e-aSπ*(α1,α2∣x,y)dα1dα2],
where S^ML is the ML “plug-in” estimate of S as given by (12).


### 4.2. Bayes Estimation whenβ1≠β2

We assume thatαj and βj, j=1,2, are random variables each having gamma prior with some parameters; that is,
(24)π(αj∣βj)=bjajΓ(aj)αjaj-1e-bjαj,αj>0,(aj,bj>0),j=1,2,π(βj)=djcjΓ(cj)βcj-1e-djβj,βj>0,(cj,dj>0),j=1,2.
Since αj and βj are independent, then the joint density function of (αj,βj) is given by
(25)π(αj,βj)=bjajdjcjΓ(aj)Γ(cj)αjaj-1βjcj-1e-bjαj-djβj,j=1,2.
Combining the likelihood function with the priors pdf yields the posterior density function of all parameters Θ=(α1,α2,β1,β2) as follows:
(26)π*(Θ∣x,y)=K∏j=12‍αjAj-1βjBj-1e-uj-djβj×∑s=0rj‍Crj,se-αj(Tjs+bj),
where Aj=mj-rj+aj, Bj=mj-rj+cj, T1s=Ts(x;β1), T2s=Ts(y;β2), u1=u(x;β1), u2=u(y;β2), and
(27)K-1=∏j=12‍∫0∞‍∫0∞‍αjAj-1βjBj-1e-uj-djβj×∑s=0rj‍Crj,se-αj(Tjs+bj)dαjdβj.
Under the BSEL function, and by using (13) and (26), the Bayesian “plug-in” estimate of S is given by
(28)S^BS=ωS^ML+(1-ω)∫Θ‍Sπ*(Θ∣x,y)dΘ.
Also, based on the BLINEX loss function, the Bayes estimate of S is obtained by using (13) and is written as
(29)S^BL=-1aln[ωe-aS^ML+(1-ω)∫Θ‍e-aSπ*(Θ∣x,y)dΘ],
where S^ML is the ML “plug-in” estimate of S as given by (13).It may be noted, from (22), (23), (28), and (29), that the Bayes estimates of S contain integrals that cannot be obtained in simple closed form, and numerical techniques must be used for computations. We, therefore, propose to consider MCMC methods.


## 4.1. Bayes Estimation Whenβ1=β2=β

Assuming thatβ1=β2=β (β is known) and α1, α2 are random variable each having gamma prior with some parameters, we can write
(19)π(αj)=δjνjΓ(νj)αjνj-1e-δjαj,αj>0,(νj,δj>0),j=1,2.
Since αj, j=1,2 are independent, by combining the likelihood function with the priors pdf, the joint posterior density function of α1 and α2 is given by
(20)π*(α1,α2∣x,y)=J∏j=12‍αjAj*-1e-δjαj-uj*∑s=0rj‍Crj,se-αjTjs*,
where Aj*=mj-rj+νj, T1s*=Ts(x;β), T2s*=Ts(y;β), u1*=u(x;β), u2*=u(y;β), and
(21)J-1=∏j=12‍∫0∞‍αjAj*-1e-δjαj-uj*∑s=0rj‍Crj,se-αjTjs*dαj.
Under the BSEL function and using (12) and (20), the proposed “Bayesian estimators” of S are actually Bayesian “plug-in” estimators as they are obtained by replacing (α1,β1,α2,β2) with its Bayesian estimator (α^1,β^1,α^2,β^2)(22)S^BS=ωS^ML+(1-ω)∫0∞‍∫0∞‍Sπ*(α1,α2∣x,y)dα1dα2.
With the same argument, we can obtain Bayes estimator under the BLINEX loss function. It is obtained as follows:
(23)S^BL=-1aln[∫0∞‍ωe-aS^ML+(1-ω)-1aln×∫0∞‍∫0∞‍e-aSπ*(α1,α2∣x,y)dα1dα2],
where S^ML is the ML “plug-in” estimate of S as given by (12).


## 4.2. Bayes Estimation whenβ1≠β2

We assume thatαj and βj, j=1,2, are random variables each having gamma prior with some parameters; that is,
(24)π(αj∣βj)=bjajΓ(aj)αjaj-1e-bjαj,αj>0,(aj,bj>0),j=1,2,π(βj)=djcjΓ(cj)βcj-1e-djβj,βj>0,(cj,dj>0),j=1,2.
Since αj and βj are independent, then the joint density function of (αj,βj) is given by
(25)π(αj,βj)=bjajdjcjΓ(aj)Γ(cj)αjaj-1βjcj-1e-bjαj-djβj,j=1,2.
Combining the likelihood function with the priors pdf yields the posterior density function of all parameters Θ=(α1,α2,β1,β2) as follows:
(26)π*(Θ∣x,y)=K∏j=12‍αjAj-1βjBj-1e-uj-djβj×∑s=0rj‍Crj,se-αj(Tjs+bj),
where Aj=mj-rj+aj, Bj=mj-rj+cj, T1s=Ts(x;β1), T2s=Ts(y;β2), u1=u(x;β1), u2=u(y;β2), and
(27)K-1=∏j=12‍∫0∞‍∫0∞‍αjAj-1βjBj-1e-uj-djβj×∑s=0rj‍Crj,se-αj(Tjs+bj)dαjdβj.
Under the BSEL function, and by using (13) and (26), the Bayesian “plug-in” estimate of S is given by
(28)S^BS=ωS^ML+(1-ω)∫Θ‍Sπ*(Θ∣x,y)dΘ.
Also, based on the BLINEX loss function, the Bayes estimate of S is obtained by using (13) and is written as
(29)S^BL=-1aln[ωe-aS^ML+(1-ω)∫Θ‍e-aSπ*(Θ∣x,y)dΘ],
where S^ML is the ML “plug-in” estimate of S as given by (13).It may be noted, from (22), (23), (28), and (29), that the Bayes estimates of S contain integrals that cannot be obtained in simple closed form, and numerical techniques must be used for computations. We, therefore, propose to consider MCMC methods.


## 5. MCMC Algorithm for Bayesian Estimation

The MCMC algorithm is conducted to compare the Bayes estimates ofS. We consider the Metropolis-Hastings algorithm to generate samples from the conditional posterior distributions, and then we compute the Bayes estimates. For more details about the MCMC methods, see, for example, Robert and Casella [28], Upadhyaya and Gupta [29], and Jaheen and Al Harbi [30]. The Metropolis-Hastings algorithm generates samples from an arbitrary proposal distribution.


### 5.1. The Case Whenβ1=β2=β

The conditional posteriors distributions of the parametersαj, j=1,2, can be computed and written, respectively, as
(30)π*(α1α2,x,y)∝Jα1A1*-1e-δ1α1∑s=0r1‍Cr1,se-α1T1s*,π*(α2∣α1,x,y)∝Jα2A2*-1e-δ2α2∑s=0r2‍Cr2,se-α2T2s*.


### 5.2. The Case Whenβ1≠β2

The conditional posteriors distributions of the parametersαj and βj, j=1,2, can be computed and written, respectively, by
(31)π*(α1Θ-α1,x,y)∝α1A1-1∑s=0r1‍Cr1,se-α1(T1s+b1),π*(β1Θ-β1,x,y)∝β1B1-1e-u1-d1β1∑s=0r1‍Cr1,se-α1(T1s+b1),π*(α2Θ-α2,x,y)∝α2A2-1∑s=0r2‍Cr2,se-α2(T2s+b2),π*(β2∣Θ-β2,x,y)∝β2B2-1e-u2-d2β2∑s=0r2‍Cr2,se-α2(T2s+b2).
The following MCMC procedure is proposed to compute Bayes estimators for S=P(Y<X).Step 1.
Start with initial guess ofαj and βj; say α0j and β0j,j=1,2.Step 2.
Seti=1.Step 3.
Generateα1i from π*(α1∣Θ-α1,x,y) and β1i from π*(β1∣Θ-β1,x,y).Step 4.
Generateα2i from π*(α2∣Θ-α2,x,y) and β2i from π*(β2∣Θ-β2,x,y).Step 5.
ComputeS(αi,βi)=α1iβ1i∫0∞‍(1+β1ix)-(α1i+1)(1+β2ix)-α2idx.Step 6.
Repeat Steps 2–5,N times.Now, the approximate means of S(αi,βi) with respect to the posterior distribution are given, respectively, by
(32)E(S∣x,y)=1N-M∑i=M+1N‍S(αi,βi),E[exp(-aS)∣x,y]=1N-M∑i=M+1N‍exp(-aS(αi,βi)),
where M is the burn-in period. Therefore, the Bayes estimators of S=S(αi,βi) based on BSEL and BLINEX loss functions are given, respectively, by
(33)S^BS=ωS^ML+(1-ω)E(Sx,y),S^BL=-1aln[ωe-aS^ML+(1-ω)E[exp(-aS)∣x,y]].


## 5.1. The Case Whenβ1=β2=β

The conditional posteriors distributions of the parametersαj, j=1,2, can be computed and written, respectively, as
(30)π*(α1α2,x,y)∝Jα1A1*-1e-δ1α1∑s=0r1‍Cr1,se-α1T1s*,π*(α2∣α1,x,y)∝Jα2A2*-1e-δ2α2∑s=0r2‍Cr2,se-α2T2s*.


## 5.2. The Case Whenβ1≠β2

The conditional posteriors distributions of the parametersαj and βj, j=1,2, can be computed and written, respectively, by
(31)π*(α1Θ-α1,x,y)∝α1A1-1∑s=0r1‍Cr1,se-α1(T1s+b1),π*(β1Θ-β1,x,y)∝β1B1-1e-u1-d1β1∑s=0r1‍Cr1,se-α1(T1s+b1),π*(α2Θ-α2,x,y)∝α2A2-1∑s=0r2‍Cr2,se-α2(T2s+b2),π*(β2∣Θ-β2,x,y)∝β2B2-1e-u2-d2β2∑s=0r2‍Cr2,se-α2(T2s+b2).
The following MCMC procedure is proposed to compute Bayes estimators for S=P(Y<X).Step 1.
Start with initial guess ofαj and βj; say α0j and β0j,j=1,2.Step 2.
Seti=1.Step 3.
Generateα1i from π*(α1∣Θ-α1,x,y) and β1i from π*(β1∣Θ-β1,x,y).Step 4.
Generateα2i from π*(α2∣Θ-α2,x,y) and β2i from π*(β2∣Θ-β2,x,y).Step 5.
ComputeS(αi,βi)=α1iβ1i∫0∞‍(1+β1ix)-(α1i+1)(1+β2ix)-α2idx.Step 6.
Repeat Steps 2–5,N times.Now, the approximate means of S(αi,βi) with respect to the posterior distribution are given, respectively, by
(32)E(S∣x,y)=1N-M∑i=M+1N‍S(αi,βi),E[exp(-aS)∣x,y]=1N-M∑i=M+1N‍exp(-aS(αi,βi)),
where M is the burn-in period. Therefore, the Bayes estimators of S=S(αi,βi) based on BSEL and BLINEX loss functions are given, respectively, by
(33)S^BS=ωS^ML+(1-ω)E(Sx,y),S^BL=-1aln[ωe-aS^ML+(1-ω)E[exp(-aS)∣x,y]].


## 6. Simulation Study

In order to find the Bayes and likelihood estimates of the parameterS, a Monte Carlo study is performed following the algorithms as follows.(1)
For particular values ofαj and βj, j=1,2, Lomax observations of various sizes are generated for different general progressive censored schemes.(2)
The ML estimates ofαj and βj,j=1,2, are computed from the ML equations. The ML estimate of S is computed from (13) after replacing αj and βj, j=1,2, by their ML estimates.(3)
ForN=20000, M=2000, the Bayes estimates of S are computed from (22), (23), (28), and (29) for BSEL and BLINEX loss functions based on MCMC algorithm.(4)
The squared deviations(S*-S)2 are calculated for different sample sizes and different schemes, where S* is ML or Bayes estimates of S.(5)
The above steps are repeated 1000 times, and the estimated risk (ER) is computed by averaging the squared deviations over the 1000 repetitions.The computational results are displayed in Tables1, 2, and 3. Table 1 shows different censoring schemes used in the simulation study. In the case of β1=β2=β, we take ν1=4, δ1=3, ν2=3, δ2=5, ω=0.5, α1=1.1339, β=3, α2=0.7216, and the true value of S=0.4516. For this case, Table 2 presents simulation results and the MLE, Bayes estimate, and the corresponding mean squared error is reported within bracket. In the general case, we take a1=4, b1=3, a2=3, b2=2, ω=0.5, α1=1.5788, β1=1.6899, α2=1.5690, β2=2.2002, and the true value of S=0.6111. The obtained simulation results for this case are shown in Table 3.Table 1
Censoring schemes used in the simulation study.
CS
n
m
r
R
(i)
50
35
5
(
6
*
0,5
,
4
*
0,2
,
3
*
0,3
,
1
3
*
0,5
)
(ii)
30
20
2
(
6
*
0,5
,
1
0
*
0,5
)
(iii)
25
20
2
(
8
*
0,5
,
9
*
0
)
(iv)
35
30
5
(
5
*
0,2
,
3
*
0,3
,
1
5
*
0
)
(v)
40
35
3
(
1
2
*
0,5
,
1
9
*
0
)
(vi)
30
25
2
(
8
*
0,3
,
1,1
,
1
2
*
0
)
(vii)
80
75
5
(
3
0
*
0,1
,
1
0
*
0,2
,
5
*
0,2
,
2
2
*
0
)
(viii)
70
60
3
(
2
*
0,2
,
1
5
*
0,3
,
1
0
*
0,3
,
2
1
*
0,2
,
5
*
0
)Table 2
The simulation results and estimates ofS when β1=β2=β.
CS(1)
CS(2)
ML
Bayes  (MCMC)
S
ML
S
BS
S
BL
a
=
-
2
a
=
1
a
=
3
(i)
(ii)
0.4871
0.4867
0.4871
0.4865
0.4860
(
0.0017
)
(
0.0016
)
(
0.0016
)
(
0.0016
)
(
0.0015
)
(iii)
(iv)
0.5471
0.5389
0.5393
0.5387
0.5383
(
0.0092
)
(
0.0079
)
(
0.0079
)
(
0.0078
)
(
0.0078
)
(v)
(vi)
0.5408
0.5386
0.5388
0.5384
0.5382
(
0.0080
)
(
0.0079
)
(
0.0079
)
(
0.0078
)
(
0.0078
)
(vii)
(viii)
0.5548
0.5436
0.5440
0.5434
0.5430
(
0.0107
)
(
0.0088
)
(
0.0089
)
(
0.0088
)
(
0.0087
)Table 3
The simulation results and estimates ofS when β1≠β2.
CS(1)
CS(2)
ML
Bayes (MCMC)
S
ML
S
BS
S
BL
a
=
-
2
a
=
1
a
=
3
(i)
(ii)
0.7217
0.6968
0.6977
0.6964
0.6954
(
0.0124
)
(
0.0077
)
(
0.0078
)
(
0.0076
)
(
0.0074
)
(iii)
(iv)
0.6440
0.6085
0.6102
0.6077
0.6059
(
0.0013
)
(
0.0004
)
(
0.0004
)
(
0.0004
)
(
0.0005
)
(v)
(vi)
0.7001
0.6731
0.6741
0.6726
0.6716
(
0.0080
)
(
0.0041
)
(
0.0042
)
(
0.0040
)
(
0.0039
)
(vii)
(viii)
0.6771
0.6418
0.6434
0.6410
0.6395
(
0.0044
)
(
0.0010
)
(
0.0011
)
(
0.0010
)
(
0.0009
)


## 7. Conclusions

In this paper, the estimation of the stress-strength parameter,S, for two Lomax distributions under general progressive type-II censoring has been considered. The maximum likelihood and Bayes estimators of the stress-strength parameter have been derived. The MCMC method is used for computing Bayes estimates. It is observed that Bayes estimators outperform the ML estimators in small samples, while the estimators are almost equally efficient in large samples. It may be noted, from Tables 2 and 3, that the Bayes estimates have the smallest mean squared errors as compared with their corresponding maximum likelihood estimates. Based on the obtained results in this study and because of the need to deal with small samples in life testing, we recommend to use Bayes estimators in place of ML estimators.



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*Source: 431541-2013-06-05.xml*