# Bayes  Estimation of Change Point in Discrete Maxwell Distribution

Authors: Mayuri Pandya; Hardik Pandya
Journal: International Journal of Quality, Statistics, and Reliability
 (2011)

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## Abstract

A sequence of independent lifetimesX1,…,Xm,Xm+1,…,Xn
was observed from Maxwell distribution with reliability r1(t) at time t but later, it was found that there was a change in the system at some point of time m and it is reflected in the sequence after Xm by change in reliability r2(t) at time t. The Bayes estimators of m, θ1, θ2 are derived under different asymmetric loss functions. The effects of correct and wrong prior information on the Bayes estimates are studied.


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## Body

## 1. Introduction

Maxwell distribution plays an important role in physics and other allied sciences. This paper introduces a discrete analogue of the Maxwell distribution, called discrete Maxwell distribution (dMax distribution). This distribution is suggested as a suitable reliability model to fit a range of discrete lifetime data.In reliability theory many continuous lifetime models have been suggested and studied. However, it is sometimes impossible or inconvenient to measure the life length of a device, on a continuous scale. In practice, we come across situations, where lifetime of a device is considered to be a discrete random variable. For example, in an on/off switching device, the lifetime of the switch is a discrete random variable. Also, the number of voltage fluctuations which an electrical or electronic item can withstand before its failure is a discrete rv.If the lifetimes of individuals in some population are grouped or when lifetime refers to an integral number of cycles of some sort, it may be desirable to treat it as a discrete rv. When a discrete model is used with lifetime data, it is usually a multinomial distribution, which arises because effectively continuous data have been grouped. Some situations may demand for another discrete distribution, usually over the non negative integers. Such situations are best treated individually, but generally one tries to adopt one of the standard discrete distributions.In the last two decades, standard discrete distributions like geometric and negative binomial have been employed to model lifetime data. However, there is a need to find more plausible discrete lifetime distributions to fit to various types of lifetime data. For this purpose, popular continuous lifetime distributions can be helpful in the following manner.The Maxwell distribution defines the speed of molecules in thermal equilibrium under some conditions as defined in statistical mechanics. For example, this distribution explains many fundamental gas properties in kinetic theory of gases; distribution of energies and moments, and so forth.Tyagi and Bhattachary ([1], (1989b)) considered Maxwell distribution as a lifetime model for the first time. They obtained minimum variance unbiased estimator (MVUE) and Bayes estimator of the parameter and reliability function of this distribution.Chaturvedi and Rani [2] studied generalized Maxwell distribution by introducing one more parameter and obtained classical and Bayesian estimation procedures for it. A lifetime model is specified to represent the distribution of lifetimes, and statistical inferences are made on the basis of this model. Physical systems manufacturing the items are often subject to random fluctuations. It may happen that at some point of time instability in the sequence of lifetimes and reliability is observed. The problem of study is when and where this change has started occurring. This is called change point inference problem. Bayesian ideas may play an important role in the study of such change point problem and has been often proposed as a valid alternative to classical estimation procedure. The monograph of Broemeling and Tsurumi [3] on structural changes, Jani and Pandya [4], Ebrahimi and Ghosh [5] and a survey by Zacks [6], Pandya and Jani [7], Pandya and Bhatt [8], Mayuri Pandya and prbha Jadav [9], Pandya and Jadav [10] are useful references. In this paper we have proposed a discrete Maxwell model to represent the distribution of lifetimes with change point m and have obtained Bayes estimators of m,θ1,θ2.


## 2. Proposed Change Point Model

In this section we propose change point model on discrete Maxwell distribution. We also derived the Bayes estimates for the model. LetX1,X2,…,Xn(n≥3) be a sequence of random lifetimes. First m of them are coming from discrete Maxwell, dMax(θ1). So the probability mass function is given by(1)p(xi)=4π1θ1Q(xi,2,θ1),x=0,1,…,θ1>0i=1,2,…,m.
With reliability r1(t),(2)r1(t)=4πθ1-3/2J(t,2,θ1),x=0,1,…;θ1>0.Latern-m observations are coming from the discrete Maxwell, dMax(θ2). So the probability mass function is given by(3)p(xi)=4π1θ2Q(xi,2,θ2),x=0,1,…,θ2>0,i=m+1,…,n.
With reliability r2(t),(4)r2(t)=4πθ2-3/2J(t,2,θ2),x=0,1…;θ2>0,
where(5a)Q(x,k,θ1)=∫xx+1ukⅇ-u2/θ1du=J(x,k,θ1)-J(x+1,k,θ1),(5b)J(x,k,θ1)=∫x∞ukⅇ-u2/θ1du,
(5c)Q(x,k,θ2)=∫xx+1ukⅇ-u2/θ2du=J(x,k,θ2)-J(x+1,k,θ2),(5d)J(x,k,θ2)=∫x∞ukⅇ-u2/θ2du.The likelihood function given the sample information X̲=(X̲1,X̲2,…,X̲m,X̲m+1,…,X̲n)(6)L(θ1,θ2,m∣X̲)=(4π)nθ1-mG1(m,θ1)θ2-(n-m)G2(m,θ2)
where(7)G1(m,θ1)=∏i=0mQ(xi,2,θ1),G2(m,θ2)=∏i=m+1nQ(xi,2,θ2),Q(xi,2,θ1)and Q(xi,2,θ2) are explained in (5a) and (5c).


## 3. Bayes Estimation

### 3.1. The Conjugate Analysis Using Inverted Gamma Prior Distribution

The ML method, as well as other classical approaches, is based only on the empirical information provided by the data. However, when there is some technical knowledge on the parameters of the distribution available, a Bayes procedure seems to be an attractive inferential method. The Bayes procedure is based on a posterior density, sayg(θ1,θ2,m∣X̲), which is proportional to the product of the likelihood function L(θ1,θ2,m∣X̲) with a prior joint density, say g(θ1,θ2,m)| representing the uncertainty on the parameters values.We also suppose that some information onθ1 and θ2 is available and that this technical knowledge can be given in terms of prior mean values μ1, μ2 and variance σ1 and σ2, respectively. Suppose that the marginal prior density of θ1 and θ2is the inverted Gamma with respective mean μ1andμ2:(8)g(θ1)=a1b1Γb1(θ1)-(b1+1)ⅇ-a1/θ1,g(θ2)=a2b2Γb2(θ2)-(b2+1)ⅇ-a2/θ2,aibi>0,θi>0,i=1,2,
where the parameters ai,bii=1,2 are obtained by solving (9)bi=2+μi2σi2,ai=μi(bi-1),i=1,2.Following Calabria and Pulcini [13] we assume the prior information to be correct if the true value of θ1(θ2)is close to prior mean μ1(μ2)and is assumed to be wrong if θ1(θ2)is far from μ1(μ2).As in the study by Broemeling and Tsurumi [3], we suppose the marginal prior distribution of m to be discrete uniform over the set {1,2,…,n-1}(10)g(m)=1(n-1).
The joint prior density of θ1,θ2,m is(11)g1(θ1,θ2,m)=1n-1a1b1Γb1a2b2Γb2θ1-(b1+1)ⅇ-a1/θ1θ2-(b2+1)ⅇ-a2/θ2=kθ1-(b1+1)ⅇ-a1/θ1θ2-(b2+1)ⅇ-a2/θ2,
where(12)k=1n-1a1b1Γb1a2b2Γb2.
The joint posterior density of θ1,θ2,m is(13)g1(θ1,θ2,m∣x̲)=L(θ1,θ2,m∣x̲)g1(θ1,θ2,m)h1(x̲),g1(θ1,θ2,m∣x̲)=k2θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)θ2-(n-m+b2+1)×ⅇ-a2/θ2G2(m,θ2)⋅h1-1(x̲),(14)k2=k⋅k1=k(4π)n,G1(m,θ1) and G2(m,θ2) are explained in (7).Andh1(x̲) is the marginal posterior density of X̲:(15)h1(x̲)=∑m=1n-1∬0∞L(θ1,θ2,m∣x̲)g1(θ1,θ2,m)dθ1dθ2=k2∑m=1n-1I1(m)I2(m),
where (16)I1(m)=∫0∞θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)dθ1,(17)I2(m)=∫0∞θ2-(n-m+b2+1)ⅇ-a2/θ2G2(m,θ2)dθ2,G1(m,θ1) and G2(m,θ2) are explained in (7).Marginal posterior density ofθ1 and of θ2 is obtained by integrating the joint posterior density of θ1,θ2,m given in (13) with respect to θ2and with respect to θ1, respectively, and summing over m,(18)g1(θ1∣x̲)=k2∑m=1n-1θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)×∫0∞θ2-(n-m+b2+1)ⅇ-a2/θ2G2(m,θ2)dθ2h1-1(x̲),(19)g1(θ2∣x̲)=k2∑m=1n-1θ2-(n-m+b2+1)ⅇ-a2/θ2G2(m,θ2)×∫0∞θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)dθ1h1-1(x̲),G1(m,θ1) and G2(m,θ2) are explained in (7), andh1(x̲) is the same as in (15).The marginal posterior density of change pointm, say g1(m∣x̲), is obtained by integrating the joint posterior density of θ1,θ2,m (13) with respect to θ1andθ2(20)g1(m∣x̲)=∬0∞g1(θ1,θ2,m∣x̲)dθ1dθ2=∬0∞k2θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)θ2-(n-m+b2+1)×ⅇ-a2/θ2G2(m,θ2)dθ1dθ2h1-1(x̲)=I1(m)I2(m)∑m=1n-1I(m)1I2(m).
The Bayes estimator of a generic parameter (or function thereof) α, based on a squared error loss (SEL) function:(21)L1(α,d)=(α-d)2,
where d is a decision rule to estimate and α, is the posterior mean.Bayes estimator ofm under SEL and the inverted Gamma prior is(22)m1*=∑m=1n-1mI1(m)I2(m)∑m=1n-1I1(m)I2(m).I1(m) and I2(m) are the same as in (16).Other Bayes estimators ofα based on the loss functions
(23)L2(α,d)=|α-d|,L3(α,d)={0,if|α-d|<ϵ,ϵ>0,1,otherwise,
are the posterior median and posterior mode, respectively.


### 3.2. Posterior Distribution Functions Using Noninformative Prior

A noninformative prior is a prior that adds no information to that contained in the empirical data. Thus, a Bayes inference based upon noninformative prior has generally a theoretical interest only, since, from an engineering view point, the Bayes approach is very attractive for it allows incorporating expert opinion or technical knowledge in the estimation procedure. Let the joint noninformative prior density ofθ1,θ2,andmbe given by,(24)g2(θ1,θ2,m)=1(n-1)θ1θ2.
The joint posterior density using noninformative prior g2(θ1,θ2,m), say g2(θ1,θ2,m∣x̲), is (25)g2(θ1,θ2,m∣x̲)=k2∑m=1n-1θ1-(m+1)ⅇ-1/θ1G1(m,θ1)×θ2-(n-m+1)ⅇ-1/θ2G2(m,θ2)⋅h2-1(x̲),
where G1(m,θ1),G2(m,θ2)and k2 are the same as in (7) and (14) and h2(x̲) is the marginal density of x̲ under the noninformative priors and is obtained by,(26)h2(x̲)=∑m=1n-1∬0∞L(θ1,θ2,m∣x̲)g2(θ1,θ2,m)dθ1dθ2=k2∑m=1n-1I3(m)I4(m),
where(27)I3(m)=∫0∞θ1-(m+1)ⅇ-1/θ1G1(m,θ1)θ2-(n-m+1)dθ1,I4(m)=∫0∞ⅇ-1/θ2G2(m,θ2)dθ2.
Marginal posterior density of θ1,θ2 and change point m under the noninformative prior (25) is obtained as(28)g2(θ1∣x̲)=k2∑m=1n-1θ1-(m+1)ⅇ-1/θ1G1(m,θ1)×∫0∞θ2-(n-m+1)ⅇ-1/θ2G2(m,θ2)dθ2⋅h2-1(x̲),(29)g2(θ2∣x̲)=k2∑m=1n-1ⅇ-1/θ2G2(m,θ2)θ2-(n-m+1)×∫0∞θ1-(m+1)ⅇ-1/θ1G1(m,θ1)dθ1⋅h2-1(x̲),
where G1(m,θ1), G2(m,θ2), and k2 are same as in (7) and (14). h2(x̲) is same as in (26).The marginal posterior density of change pointm is, say g2(m∣x̲) is obtained by integrating the joint posterior density of θ1,θ2,m (25) with respect to θ1andθ2,(30)g2(m∣x̲)=∬0∞g2(θ1,θ2,m∣x̲)dθ1dθ2=∬0∞k2∑m=1n-1θ1-(m+1)ⅇ-1/θ1G1(m,θ1)θ2-(n-m+1)×ⅇ-1/θ2G2(m,θ2)⋅h2-1(x̲)dθ1dθ2=I3(m)I4(m)∑m=1n-1I3(m)I4(m)
where I3(m) and I4(m) are the same as in (27).Bayes estimator ofm under SEL and noninformative prior is(31)m2*=∑m=1n-1mI3(m)I4(m)∑m=1n-1I3(m)I4(m).


## 3.1. The Conjugate Analysis Using Inverted Gamma Prior Distribution

The ML method, as well as other classical approaches, is based only on the empirical information provided by the data. However, when there is some technical knowledge on the parameters of the distribution available, a Bayes procedure seems to be an attractive inferential method. The Bayes procedure is based on a posterior density, sayg(θ1,θ2,m∣X̲), which is proportional to the product of the likelihood function L(θ1,θ2,m∣X̲) with a prior joint density, say g(θ1,θ2,m)| representing the uncertainty on the parameters values.We also suppose that some information onθ1 and θ2 is available and that this technical knowledge can be given in terms of prior mean values μ1, μ2 and variance σ1 and σ2, respectively. Suppose that the marginal prior density of θ1 and θ2is the inverted Gamma with respective mean μ1andμ2:(8)g(θ1)=a1b1Γb1(θ1)-(b1+1)ⅇ-a1/θ1,g(θ2)=a2b2Γb2(θ2)-(b2+1)ⅇ-a2/θ2,aibi>0,θi>0,i=1,2,
where the parameters ai,bii=1,2 are obtained by solving (9)bi=2+μi2σi2,ai=μi(bi-1),i=1,2.Following Calabria and Pulcini [13] we assume the prior information to be correct if the true value of θ1(θ2)is close to prior mean μ1(μ2)and is assumed to be wrong if θ1(θ2)is far from μ1(μ2).As in the study by Broemeling and Tsurumi [3], we suppose the marginal prior distribution of m to be discrete uniform over the set {1,2,…,n-1}(10)g(m)=1(n-1).
The joint prior density of θ1,θ2,m is(11)g1(θ1,θ2,m)=1n-1a1b1Γb1a2b2Γb2θ1-(b1+1)ⅇ-a1/θ1θ2-(b2+1)ⅇ-a2/θ2=kθ1-(b1+1)ⅇ-a1/θ1θ2-(b2+1)ⅇ-a2/θ2,
where(12)k=1n-1a1b1Γb1a2b2Γb2.
The joint posterior density of θ1,θ2,m is(13)g1(θ1,θ2,m∣x̲)=L(θ1,θ2,m∣x̲)g1(θ1,θ2,m)h1(x̲),g1(θ1,θ2,m∣x̲)=k2θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)θ2-(n-m+b2+1)×ⅇ-a2/θ2G2(m,θ2)⋅h1-1(x̲),(14)k2=k⋅k1=k(4π)n,G1(m,θ1) and G2(m,θ2) are explained in (7).Andh1(x̲) is the marginal posterior density of X̲:(15)h1(x̲)=∑m=1n-1∬0∞L(θ1,θ2,m∣x̲)g1(θ1,θ2,m)dθ1dθ2=k2∑m=1n-1I1(m)I2(m),
where (16)I1(m)=∫0∞θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)dθ1,(17)I2(m)=∫0∞θ2-(n-m+b2+1)ⅇ-a2/θ2G2(m,θ2)dθ2,G1(m,θ1) and G2(m,θ2) are explained in (7).Marginal posterior density ofθ1 and of θ2 is obtained by integrating the joint posterior density of θ1,θ2,m given in (13) with respect to θ2and with respect to θ1, respectively, and summing over m,(18)g1(θ1∣x̲)=k2∑m=1n-1θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)×∫0∞θ2-(n-m+b2+1)ⅇ-a2/θ2G2(m,θ2)dθ2h1-1(x̲),(19)g1(θ2∣x̲)=k2∑m=1n-1θ2-(n-m+b2+1)ⅇ-a2/θ2G2(m,θ2)×∫0∞θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)dθ1h1-1(x̲),G1(m,θ1) and G2(m,θ2) are explained in (7), andh1(x̲) is the same as in (15).The marginal posterior density of change pointm, say g1(m∣x̲), is obtained by integrating the joint posterior density of θ1,θ2,m (13) with respect to θ1andθ2(20)g1(m∣x̲)=∬0∞g1(θ1,θ2,m∣x̲)dθ1dθ2=∬0∞k2θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)θ2-(n-m+b2+1)×ⅇ-a2/θ2G2(m,θ2)dθ1dθ2h1-1(x̲)=I1(m)I2(m)∑m=1n-1I(m)1I2(m).
The Bayes estimator of a generic parameter (or function thereof) α, based on a squared error loss (SEL) function:(21)L1(α,d)=(α-d)2,
where d is a decision rule to estimate and α, is the posterior mean.Bayes estimator ofm under SEL and the inverted Gamma prior is(22)m1*=∑m=1n-1mI1(m)I2(m)∑m=1n-1I1(m)I2(m).I1(m) and I2(m) are the same as in (16).Other Bayes estimators ofα based on the loss functions
(23)L2(α,d)=|α-d|,L3(α,d)={0,if|α-d|<ϵ,ϵ>0,1,otherwise,
are the posterior median and posterior mode, respectively.


## 3.2. Posterior Distribution Functions Using Noninformative Prior

A noninformative prior is a prior that adds no information to that contained in the empirical data. Thus, a Bayes inference based upon noninformative prior has generally a theoretical interest only, since, from an engineering view point, the Bayes approach is very attractive for it allows incorporating expert opinion or technical knowledge in the estimation procedure. Let the joint noninformative prior density ofθ1,θ2,andmbe given by,(24)g2(θ1,θ2,m)=1(n-1)θ1θ2.
The joint posterior density using noninformative prior g2(θ1,θ2,m), say g2(θ1,θ2,m∣x̲), is (25)g2(θ1,θ2,m∣x̲)=k2∑m=1n-1θ1-(m+1)ⅇ-1/θ1G1(m,θ1)×θ2-(n-m+1)ⅇ-1/θ2G2(m,θ2)⋅h2-1(x̲),
where G1(m,θ1),G2(m,θ2)and k2 are the same as in (7) and (14) and h2(x̲) is the marginal density of x̲ under the noninformative priors and is obtained by,(26)h2(x̲)=∑m=1n-1∬0∞L(θ1,θ2,m∣x̲)g2(θ1,θ2,m)dθ1dθ2=k2∑m=1n-1I3(m)I4(m),
where(27)I3(m)=∫0∞θ1-(m+1)ⅇ-1/θ1G1(m,θ1)θ2-(n-m+1)dθ1,I4(m)=∫0∞ⅇ-1/θ2G2(m,θ2)dθ2.
Marginal posterior density of θ1,θ2 and change point m under the noninformative prior (25) is obtained as(28)g2(θ1∣x̲)=k2∑m=1n-1θ1-(m+1)ⅇ-1/θ1G1(m,θ1)×∫0∞θ2-(n-m+1)ⅇ-1/θ2G2(m,θ2)dθ2⋅h2-1(x̲),(29)g2(θ2∣x̲)=k2∑m=1n-1ⅇ-1/θ2G2(m,θ2)θ2-(n-m+1)×∫0∞θ1-(m+1)ⅇ-1/θ1G1(m,θ1)dθ1⋅h2-1(x̲),
where G1(m,θ1), G2(m,θ2), and k2 are same as in (7) and (14). h2(x̲) is same as in (26).The marginal posterior density of change pointm is, say g2(m∣x̲) is obtained by integrating the joint posterior density of θ1,θ2,m (25) with respect to θ1andθ2,(30)g2(m∣x̲)=∬0∞g2(θ1,θ2,m∣x̲)dθ1dθ2=∬0∞k2∑m=1n-1θ1-(m+1)ⅇ-1/θ1G1(m,θ1)θ2-(n-m+1)×ⅇ-1/θ2G2(m,θ2)⋅h2-1(x̲)dθ1dθ2=I3(m)I4(m)∑m=1n-1I3(m)I4(m)
where I3(m) and I4(m) are the same as in (27).Bayes estimator ofm under SEL and noninformative prior is(31)m2*=∑m=1n-1mI3(m)I4(m)∑m=1n-1I3(m)I4(m).


## 4. Bayes Estimates of Change Point and Other Parameters under Asymmetric Loss Functions

The loss functionL(α,d) provides a measure of the financial consequences arising from a wrong decision rule d to estimate an unknown quantity α. The choice of the appropriate loss function depends on financial considerations only and is independent of the estimation procedure used. The use of symmetric loss function was found to be generally inappropriate, since, for example, an overestimation of the reliability function is usually much more serious than an underestimation.In this section, we derive Bayes estimators of change pointm under different asymmetric loss functions using both prior considerations explained in Sections 3.1 and 3.2. A useful asymmetric loss, known as the Linex loss function was introduced by Varian [12]. Under the assumption that the minimal loss occurs at d, the Linex loss function can be expressed as(32)L4(α,d)=exp⁡⁡[q1(d-α)]-q1(d-α)-1,q1≠0.
The sign of the shape parameter q1 reflects the deviation of the asymmetry, q1>0, if over, estimation is more serious than underestimation, and vice versa, and the magnitude of q1 reflects the degree of asymmetry.The posterior expectation of the Linex loss function is(33)Eα{L4(α,d)}=exp⁡⁡(q1d)Eα{exp⁡⁡(-q1α)}-q1(d-E{α})-I,
where Eα{f(a)}denotes the expectation of f(a) with respect to the posterior density g(α∣x̲). The Bayes estimate al* is the value of d that minimizes Eα{l4(α,d)}(34)al*=-1q1ln⁡⁡[Eα{exp⁡⁡(-q1α)}]
provided that Eα{exp(-q1α)} exists and is finite.Minimizing expected loss functionEm[L4(m,d)] and using posterior distributions (20) and (30), we get the Bayes estimators of m using Linex loss function, respectively, as
(35)mL*=-1q1ln⁡[∑m=1n-1ⅇ-mq1I1(m)I2(m)∑m=1n-1I1(m)I2(m)],mL**=-1q1ln⁡[∑m=1n-1ⅇ-mq1I3(m)I4(m)∑m=1n-1I3(m)I4(m)],
where I1(m)I2(m) and I3(m)I4(m) are the same as in (16) and (27).Minimizing expected loss functionEθ1[L4(θ1,d)] and using posterior distributions (18) and (28), we get the Bayes estimators of θ1 using Linex loss function, respectively, as(36)θ1L*=-1q1ln⁡[k2∑m=1n-1∫0∞θ2-(n-m+b2+1)ⅇ-a2/θ2G2(m,θ2)dθ2×∫0∞θ1-(m+b1+1)ⅇ-a1/θ1-θ1q1×G1(m,θ1)dθ1h1-1(x̲)∑m=1n-1],θ1L**=1q1ln⁡⁡[∫0∞k2∑m=1n-1θ1-(m+1)ⅇ-1/θ1-θ1q1G1(m,θ1)dθ1×∫0∞θ2-(n-m+1)ⅇ-1/θ2G2(m,θ2)dθ2⋅h2-1(x̲)],
where G1(m,θ1), G2(m,θ2), and k2 are the same as in (7) and (14). and h1(x̲) and h2(x̲) are the same as in (15) and (26).Minimizing expected loss functionEθ2[L4(θ2,d)] and using posterior distribution (19) and (29), we get the Bayes estimators of θ2 using Linex loss function, respectively, as(37)θ2L*=1q1ln⁡[k2∑m=1n-1∫0∞θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)dθ1×∫0∞θ2-(n-m+b2+1)ⅇ-a2/θ2-θ2q1×G2(m,θ2)dθ2h1-1(x̲)∑m=1n-1],θ2L**=-1q1ln⁡[k2∫0∞ⅇ-1/θ2-θ2q1G2(m,θ2)θ2-(n-m+1)dθ2×∫0∞θ1-(m+1)ⅇ-1/θ1G1(m,θ1)dθ1⋅h2-1(x̲)],
where G1(m,θ1), G2(m,θ2), and k2 are the same as in (7) and (14) and h1(x̲) and h2(x̲) are the same as in (15) and (26).Another loss function, called General Entropy (GE) loss function, proposed by Calabria and Pulcini [11], is given by,(38)L5(α,d)=(dα)q3-q3ln⁡⁡(dα)-I.
The Bayes estimate αE* is the value of d that minimizes Eα[L5(α,d)]:(39)αE*=[Eα(α-q3)]-1/q3,
provided that Eα(α-q3) exists and is finite.Minimizing expectationEm[L5(m,d)] and using posterior distributions (20) and (30), we get the Bayes estimators mE*,mE** of m, respectively, as(40)mE*=[Em[m-q3]]-1/q3=[∑m=1n-1m-q3I1(m)I2(m)∑m=1n-1I1(m)I2(m)]-1/q3,mE**=[∑m=1n-1m-q3I3(m)I4(m)∑m=1n-1I3(m)I4(m)]-1/q3,
where I1(m)I2(m) and I3(m)I4(m) are the same as in (16) and (27).Minimizing expected loss functionEθ1[L5(θ1,d)] and using posterior distributions (18) and (28), we get the Bayes estimates of θ1 using General Entropy loss function and informative and noninformative prior, respectively, as(41)θ1E*=[k2∑m=1n-1∫0∞θ2-(n-m+b2+q3+1)ⅇ-a2/θ2G2(m,θ2)dθ2×∫0∞θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)dθ1h1-1(x̲)]-1/q3,θ1E**=[∫0∞k2∑m=1n-1θ1-(m+q3+1)ⅇ-1/θ1G1(m,θ1)dθ1×∫0∞θ2-(n-m+1)ⅇ-1/θ2G2(m,θ2)dθ2⋅h2-1(x̲)]-1/q3,
where G1(m,θ1), G2(m,θ2) and k2 are same as in (7) and (14). h1(x̲) and h2(x̲) are same as in (15) and (26).Minimizing expected loss functionEθ2[L5(θ2,d)] and using posterior distributions (19) and (29), we get the Bayes estimates of θ2 using General Entropy loss function as(42)θ2E*=[k2∑m=1n-1∫0∞θ1-(m+b1+1)ⅇ-a1/θ1G1(m,θ1)dθ1×∫0∞θ2-(n-m+b2+q3+1)ⅇ-a2/θ2×G2(m,θ2)dθ2h1-1(x̲)∑m=1n-1]-1/q3,(43)θ2E**=[k2∫0∞ⅇ-1/θ2G2(m,θ2)θ2-(n-m+q3+1)dθ2×∫0∞θ1-(m+1)ⅇ-1/θ1G1(m,θ1)dθ1⋅h2-1(x̲)]-1/q3,
where, G1(m,θ1), G2(m,θ2)and k2 are same as in (7) and (14) and h1(x̲) and h2(x̲)are the same as in (15) and (26).Remark 1.
Puttingq3=-1 in (40), (41), and (42), we get the Bayes estimators of m, θ1, and θ2, posterior means under the squared error loss.Note that, forq3=-1, the GE loss function reduces to the squared error loss function.


## 5. Numerical Study

We have generated 30 random observations from dmax distribution involving change point discussed in Section2. The first 15 observations from dMax with θ1=1.0 and at t=5.0, R1t=0.0460 and next 15 observations from dMax distribution with θ2=0.5 and R2t=0.0011, θ1 and θ2 themselves were random observation from inverted Gamma prior distributions with prior means μ1=1.0, μ2=0.5 and variance σ1=1.0 and σ2=1.0, respectively, resulting in a1=2.0,b1=3.0, and a2=.5, b2=.75. These observations are given in Table 1 first raw.Table 1
Generated samples from dMax distribution.
Sample no.NMSampleActual reliabilityθ1=1.0(first m observation)θ2=0.5(last (n-m) observation)R1(t)R2(t)130150×5, 1×100×11,1×4250250×9, 1×160×18, 1×70.04600.0011350350×11, 1×240×25, 1×10θ1=5.0θ2=2.0430151×6, 2×4, 3×2, 4×20,1×10, 2×4550250,1×9,2×6, 3×6,4×30×3, 1×16, 2×60.84950.6594650350×3,1×9,2×11,3×8,4×40×5, 1×21, 2×8, 3We also compute the Bayes estimatorsmL*,mE* of m, θ1E*, θ1L* and θ2E*, θ2L* of θ1 and θ2 using the results given in Section 4 for the data given in Table 1 and for different values of shape parameter q1 and q3. The results are shown in Tables 3 and 4.We have generated 6 random samples from discrete Maxwell distribution involving change point discussed in Section2 with n=30,50,50, m=15,25,35, and θ1=1.0,5.0 and θ2=0.5 and 2.0. As explained in Section 3.1, θ1 and θ2 themselves were random observations from inverted Gamma prior distributions with prior means μ1, μ2, respectively. These observations are given in Table 1. We have calculated posterior means of m, θ1, and θ2 under both the prior for all samples, and the results are shown in Table 2.Table 2
Bayes estimate ofm, θ1, and θ2 under SEL.
Bayes estimates ofm (posterior mean)Bayes Estimates ofθ1andθ2 (posterior mean)Sample no.Nθ1*θ2*130151.00.5250251.00.5350351.00.5430155.02.0550255.02.0650355.02.0Table 3
The Bayes estimates using Linex loss function. (n=30, m=15, θ1=1.0, θ2=0.5).
q1mL*θ1L*θ2L*Informative prior0.09151.00.530.10151.00.520.20151.00.511.2140.970.481.5130.900.44−1.0161.50.57−2.0171.70.59Noninformative Prior0.09140.930.540.10140.920.510.20140.910.501.2130.850.461.5120.820.41−1.0161.50.56−2.0171.70.57Table 4
The Bayes estimates using General Entropy loss function. (n=30, m=15, θ1=1.0, θ2=0.5).
q3θ1E*mE*θ2E*Informative Prior0.091.3150.530.101.3150.510.201.2150.501.201.0130.471.500.93120.45−1.01.5160.55−2.01.7170.58Noninformative Prior0.091.2140.530.101.2140.530.201.0140.521.20.90120.471.50.84110.44−1.01.4170.58−2.01.8190.61Table3 shows that for small values of |q|, q1=0.09, 0.2, 0.1 Linex loss function is almost symmetric and nearly quadratic and the values of the Bayes estimate under such a loss is not far from the posterior mean. Table 3 also shows that, for q1=1.5,1.2, Bayes estimates are less than actual value of m=15.Forq1=q3=-1, −2, Bayes estimates are quite large than actual value m=15. It can be seen from Tables 3 and 4 that the negative sign of shape parameter of loss functions reflecting underestimation is more serious than overestimation. Thus, problem of underestimation can be solved by taking the value of shape parameters of Linex and General Entropy loss functions negative.Table4 shows that, for small values of |q|, q3=0.09, 0.2, 0.1 General Entropy loss function, the values of the Bayes estimate under such a loss is not far from the posterior mean. Table 4 also shows that, for q3=1.5,1.2, Bayes estimates are less than actual value of m=15.It can be seen from Tables3 and 4 that positive sign of shape parameter of loss functions reflecting overestimation is more serious than under estimation. Thus, problem of over estimation can be solved by taking the value of shape parameter of Linex and General Entropy loss functions positive and high.


### 5.1. Sensitivity of Bayes Estimates

In this section, we study the sensitivity of the Bayes estimator, obtained in Section3 with respect to change in the prior of parameters. The means μ1 and μ2 of inverted Gamma prior on θ1 and θ2 have been used as prior information in computing the parameters a1, a2, b1, and b2 of the prior. We have computed posterior mean m* for the data given in Table 1, considering different sets of values of (μ1, μ2). Following Calabria and Pulcini [13], we also assume the prior information to be correct if the true value of θ1 and θ2 is closed to prior mean μ1(μ2) and is assumed to be wrong if θ1 and θ2 are far from μ1(μ2). We observed that the posterior mean m* appears to be robust with respect to the correct choice of the prior density of θ1(θ2) and a wrong choice of the prior density of θ2(θ1).This can be seen from Tables5, 6, and 7.Table 5
Bayes estimate ofm for Sample 1.
μ1μ2m*1.00.3151.00.5151.00.7150.70.5150.90.5151.40.515Table 6
Bayes estimate ofm for Sample 5.
μ1μ2m*5.01.0255.03.0255.04.0254.02.0256.02.0256.52.025Table 7
Bayes Estimate ofm for Sample 6.
μ1μ2m*1.00.3351.00.5351.00.7350.70.5350.90.5351.40.535Table5 shows that, when prior mean μ1=1 = actual value of θ1, μ2=0.3 and 0.7 (far from true value of θ2=0.5), it means correct choice of prior of θ1 and wrong choice of prior of θ2, the values of Bayes estimator posterior mean remains the same and is 15. It gives correct estimation of m. Thus, posterior mean is not sensitive with wrong choice of prior density of θ2(θ1) and correct choice of prior of θ1(θ2).


## 5.1. Sensitivity of Bayes Estimates

In this section, we study the sensitivity of the Bayes estimator, obtained in Section3 with respect to change in the prior of parameters. The means μ1 and μ2 of inverted Gamma prior on θ1 and θ2 have been used as prior information in computing the parameters a1, a2, b1, and b2 of the prior. We have computed posterior mean m* for the data given in Table 1, considering different sets of values of (μ1, μ2). Following Calabria and Pulcini [13], we also assume the prior information to be correct if the true value of θ1 and θ2 is closed to prior mean μ1(μ2) and is assumed to be wrong if θ1 and θ2 are far from μ1(μ2). We observed that the posterior mean m* appears to be robust with respect to the correct choice of the prior density of θ1(θ2) and a wrong choice of the prior density of θ2(θ1).This can be seen from Tables5, 6, and 7.Table 5
Bayes estimate ofm for Sample 1.
μ1μ2m*1.00.3151.00.5151.00.7150.70.5150.90.5151.40.515Table 6
Bayes estimate ofm for Sample 5.
μ1μ2m*5.01.0255.03.0255.04.0254.02.0256.02.0256.52.025Table 7
Bayes Estimate ofm for Sample 6.
μ1μ2m*1.00.3351.00.5351.00.7350.70.5350.90.5351.40.535Table5 shows that, when prior mean μ1=1 = actual value of θ1, μ2=0.3 and 0.7 (far from true value of θ2=0.5), it means correct choice of prior of θ1 and wrong choice of prior of θ2, the values of Bayes estimator posterior mean remains the same and is 15. It gives correct estimation of m. Thus, posterior mean is not sensitive with wrong choice of prior density of θ2(θ1) and correct choice of prior of θ1(θ2).


## 6. Simulation Study

In Section5, we have obtained Bayes estimates of m on the basis of the generated data given in Table 1 for given values of parameters. To justify the results, we have generated 10,000 different random samples with m=10, n=30,50, θ1=1, θ2=0.5 and obtained the frequency distributions of posterior mean mL*, mE* of m with the correct prior consideration. The result is shown in Table 8. The value of shape parameter of the General Entropy loss and Linex loss used in simulation study for change point is taken as 0.1. We have also simulated several dMax samples with m=15,25,35; n=30,50, and θ1=0.15,0.11,0.10; θ2=0.55,0.45,0.35. For each m,n,θ1, and θ2 1000 pseudorandom samples have been simulated and Bayes estimators of change point m using q1=q3=0.9 have been computed for same value of a1, a2 and for different prior means μ1 and μ2. We observed that the posterior mean m* appears to be robust with respect to the correct choice of the prior density of θ1(θ2) and a wrong choice of the prior density of θ2(θ1).for each combination of prior means μ1 and μ2.Table 8
Frequency distributions of the Bayes estimates of the change point.
Bayes estimate% frequency for01–0809–1112–20Posterior mean127810mL*206515mE*226612The value of Bayes estimator of change pointm, based on Linex loss and general Entropy loss, using q1=q3=0.9 is 10.Table8 leads to conclusion that performance of posterior means has better performance than that of mL*, mE*, of change point. 78% values of posterior mean are closed to actual value of change point with correct choice of prior. 65% values of mL* are closed to actual value of change point with correct choice of prior. 66% values of mE* are close to correct values of change point with correct prior considerations.



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*Source: 395034-2011-08-04.xml*