# Bayes Estimation of Two-Phase Linear Regression Model

Authors: Mayuri Pandya; Krishnam Bhatt; Paresh Andharia
Journal: International Journal of Quality, Statistics, and Reliability
 (2011)

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## Abstract

Let the regression model beYi=β1Xi+εi, where εi are i. i. d. N (0,σ2) random errors with variance σ2>0 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  slope, regression parameter  β2. The problem of study is when and where this change has started occurring.  This is called change point inference problem. The estimators of m, β1,β2 are derived under asymmetric loss functions, namely, Linex loss & General Entropy loss functions.  The effects of correct and wrong prior information on the Bayes estimates are studied.


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## Body

## 1. Introduction

Regression analysis is an important statistical technique to analyze data in social, medical, and engineering sciences. Quite often, in practice, the regression coefficients are assumed constant. In many real-life problems, however, theoretical or empirical deliberations suggest models with occasionally changing one or more of its parameters. The main parameter of interest in such regression analyses is the shift point parameter, which indexes when or where the unknown change occurred.A variety of problems, such as switching straight lines [1], shifts of level or slope in linear time series models [2], detection of ovulation time in women   [3], and many others, have been studied during the last two decades. Holbert [4], while reviewing Bayesian developments in structural change from  1968 onward, gives a variety of interesting examples from economics and biology. The monograph by Broemeling and Tsurumi [5] provides a complete study of structural change in the linear model from the Bayesian viewpoint.Bayesian inference of the shift point parameter assumes availability of the prior distribution of the changing model parameters.  Bansal and Chakravarty [6] had proposed to study the effect of an ESD prior for the changed slope parameter of the two-phase linear regression (TPLR) model on the Bayes estimates of the shift point and also on the posterior odds ratio (POR) to detect a change in the simple regression model.In this paper, we studied a TPLR model. In Section2, we have given a change point model TPLR. In Sections 3.1 and 3.2, we obtained posterior densities of m considering σ2 unknown and of β1,β2 and m considering σ2 known, respectively. We derive Bayes estimators of β1,β2, and m under symmetric loss functions in Section 4 and asymmetric loss functions in Section 5. We have studied the sensitivity of the Bayes estimators of m when prior specifications deviate from the true values in Section 6. In Section 7, we have presented a numerical study to illustrate the above technique on generated observations. In this study, we have generated observations from the proposed model and have computed the Bayes estimates of m and of other parameters. Section 8 concludes the research paper.


## 2. Two-Phase Linear Regression Model

The TPLR model is one of the many models, which exhibits structural change.  Holbert [4] used a Bayesian approach, based on TPLR model, to reexamine the McGee and Kotz [7] data for stock market sales volume and reached the same conclusion that the abolition of splitups did hurt the regional exchanges.The TPLR model is defined as(1)yt={α1+β1xt+εt,t=1,2,…,m,α2+β2xt+εt,t=m+1,…,n,
where  εt’s are i. i. d. N (0,σ2) random errors with variance σ2>0, xt is a nonstochastic explanatory variable, and the regression parameters (α1, β1) ≠ (α2, β2). The shift point m is such that if m=n there is no shift, but when m=1,2,…,n-1 exactly one shift has occurred.


## 3. Bayes Estimation

The ML method, as well as other classical approaches are 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, say,g(β1,β2,σ-2,m∣z), which is proportional to the product of the likelihood function L(β1,β2,σ-2,m∣z), with a prior joint density, say, g(β1,β2,σ-2,m) representing uncertainty on the parameters values(2)g(β1,β2,θ1,θ2,m∣z)=L(β1,β2,σ-2,m∣z)⋅g(β1,β2,σ-2,m)∑m=1n-1∫β1∫β2∫σ-2L(β1,β2,σ-2,m∣z)⋅Z,
where ℨ denotes g(β1,β2,σ-2,m)dβ1dβ2dσ-2.The likelihood function ofβ1, β2, σ-2 and m, given the sample information  Zt=(xt,yt),t=1,2,…,m,m+1,…,n, is(3)L(β1,β2,σ-2,m∣z)=1(2π)n/2exp⁡[-12β12sm1σ2+β1(sm3σ2)]×exp⁡⁡[-12(sn1-sm1σ2)β22+β2[sm4σ2]]×e-A/2σ2σ-n,
where(4)sk1=∑i=1kxi2,sk2=∑i=1kxiyi,A=∑i=1nyi2+nα2-2α∑i=1nyi,Sm3=Sm2+2αSm1,Sm4=Sn2-Sm2+2α(Sn1-Sm1).


### 3.1. Using  Gamma Prior on1/σ2 and  Conditional Informative Priors on β1, β2 with σ-2 Unknown

We consider the TPLR model (1) with unknown σ-2. As in Broemeling and Tsurumi [5], we suppose that the shift point m is a priori uniformly distributed over the set {1,2,…,n-1} and is independent of β1 and β2. We also suppose that some information on β1 and β2 are available that can be expressed in terms of conditional prior probability densities on β1 and β2.We have conditional prior density onβ1 and β2  given σ2, with N(0,σ2) as(5)g1(β1∣σ2)=12πσexp⁡[-12β12σ2],-∞<β1<∞,g1(β2∣σ2)=12πσexp⁡[-12β22σ2],-∞<β2<∞.
We also suppose that some information on 1/σ2 is available and that technical knowledge can be given in terms of prior mean  μ  and coefficient of variation ∅. We suppose marginal prior distribution of 1/σ2 to be gamma (c,d) distribution with mean μ(6)g1(1σ2)=cdΓd(1σ2)d-1e-c/σ2,σ2>0,c,d>0,
where Γd is gamma function  same as explained in (8).The integral representation ofΓz is as below [Rez>0,Rex>0], Gradshteyn and  Ryzhik  [8, page 934](7)Γz=xz∫0∞e-xttz-1dt.The gamma function (Euler’s integral of the second kind)Γ(z)[Rez>0], (Euler) Gradshteyn and Ryzhik [8, page 933], is  defined as(8)Γ(z)=∫0∞e-ttz-1dt.If the prior information is given in terms of prior meanμ and coefficient of variation ∅, then the parameters c and d can be obtained by solving(9)d=1∅2,c=1μ∅2.Hence, joint prior pdf ofβ1,β2,σ-2 and m, say g1(β1,β2,σ-2,m)  is(10)g1(β1,β2,σ-2,m)=k1e-1/2σ2(β12+β22)(1σ2)de-c/σ2,
where(11)k1=cd2πΓd(n-1).Joint posterior density ofβ1,β2,σ-2, and m say, g(β1,β2,σ-2, m∣z) is(12)g1(β1,β2,σ-2,m∣z)=L(β1,β2,σ2,m∣z)g(β1,β2,m)dβ1dβ2dσ-2h1(z)=k2⋅exp⁡[-12⋅β12σ2[Sm1+1]+β1Sm3σ2]×exp⁡[-12β22σ2[(Sn1-Sm1)+1]+β2Sm4σ2]×e-((A/2)+c)(1/σ2)(1σ2)d(σ2)-n/2h1(z),
where
(13)k2=k11(2π)n/2,Sm3,Sm4,Sm1, Sm2and A are as given in (4).h1(z) is the marginal density of z given by (14)h1(z)=∑m=1n-1∫0∞∬-∞∞L(β1,β2,σ-2,m∣z)⋅g(β1,β2,σ-2,m)dβ1dβ2dσ-2=k2∑m=1n-1T1(m),
where k2 is as given in (13)(15)T1(m)=Γ((n/2)+d-1)((A/2)+C-hm1-hm2)(n/2)+d-1x,
where 𝔵 denotes Sm1+1·Sn1-Sm1+1,(16)hm1=Sm322(Sm1+1),hm2=Sm422(Sn1-Sm1+1),Sm3,Sm4,Sm1, Sm2, Sn1, and A are as given in (4).Γ((n/2)+d-1) is the gamma function as explained in (8).Marginal posterior density of change pointm, say g(m∣z) is (17)g1(m∣z)=T1(m)∑m=1n-1T1(m),T1(m)  is as given in (15).


### 3.2. Using Conditional Informative Priors onβ1,β2 with σ-2 Known

We consider the TPLR model (1) with known σ2. We also assume same prior consideration of β1, β2,  and m as explained in Section 3.1.Hence, joint prior pdf ofβ1, β2, and m, say g2(β1, β2, m)  is(18)g2(β1,β2,m)=k3⋅e-1/2((β12+β22)/σ2),
where(19)k3=12π(n-1)σ2.
Joint posterior density of β1, β2 and m say, g2(β1,β2,m∣z) is(20)g2(β1,β2,m∣z)=k4exp⁡[-12β12[(Sm1+1)σ2]+β1Sm3σ2]×exp⁡[-12β22[(Sn1-Sm1+1)σ2]+β2Sm4σ2]/h2(z),
where(21)k4=k31(2π)n/2e-A/2σ2σ-n,Sm3,Sm4,Sm1, Sm2, and A are as given in (4).Whereh2(z) is the marginal density of z given by (22)h2(z)=k4∑m=1n-1∫-∞∞exp⁡[-12β12[Sm1+1]σ2+β1Sm3σ2]dβ1×∫-∞∞exp⁡⁡[-12β22σ2[(Sn1-Sm1)+1]+β2Sm4σ2]dβ2.And the integrals,(23)G1m=∫-∞∞exp⁡[-12β22σ2[(Sn1-Sm1)+1]+β2Sm4σ2]dβ2=2π⋅e(Sm42/2σ2(Sn1-Sm1+1))(Sn1-Sm1+1)/σ,(24)G2m=∫-∞∞exp⁡[-12β12[Sm1+1]σ2+β1Sm3σ2]dβ1=2π⋅eSm32/2σ2(Sm1+1)(Sm1+1)/σ.So using (23) and (24) results in (22), it reduces to                             (25)h2(z)=k4∑m=1n-1T2(m),T2(m)=G1mG2m,
where k4 is as given in (21). G1m and G2m are as given in (23) and (24).Sm3,Sm4,Sm1, and Sm2  are as given in (4).Marginal posterior density of change pointm, β1,  and β2 is  (26)g2(m∣z)=T2(m)∑m=1n-1T2(m),(27)g2(β1∣z)=k4h2(z)[∑m=1n-1e-1/2β12(Sm1+1σ2)+β1(Sm3σ2)]G1m,(28)g2(β2∣z)=k4h2(z)[∑m=1n-1e-1/2β22(Sn1-Sm1+1σ2)+β2Sm4σ2]G2m,k4 and  G1m,G2m,h2(z)are as given in (21), (23), (24), and (25), respectively.Sm3,Sm4,Sm1, Sm2 are as given in (4).


## 3.1. Using  Gamma Prior on1/σ2 and  Conditional Informative Priors on β1, β2 with σ-2 Unknown

We consider the TPLR model (1) with unknown σ-2. As in Broemeling and Tsurumi [5], we suppose that the shift point m is a priori uniformly distributed over the set {1,2,…,n-1} and is independent of β1 and β2. We also suppose that some information on β1 and β2 are available that can be expressed in terms of conditional prior probability densities on β1 and β2.We have conditional prior density onβ1 and β2  given σ2, with N(0,σ2) as(5)g1(β1∣σ2)=12πσexp⁡[-12β12σ2],-∞<β1<∞,g1(β2∣σ2)=12πσexp⁡[-12β22σ2],-∞<β2<∞.
We also suppose that some information on 1/σ2 is available and that technical knowledge can be given in terms of prior mean  μ  and coefficient of variation ∅. We suppose marginal prior distribution of 1/σ2 to be gamma (c,d) distribution with mean μ(6)g1(1σ2)=cdΓd(1σ2)d-1e-c/σ2,σ2>0,c,d>0,
where Γd is gamma function  same as explained in (8).The integral representation ofΓz is as below [Rez>0,Rex>0], Gradshteyn and  Ryzhik  [8, page 934](7)Γz=xz∫0∞e-xttz-1dt.The gamma function (Euler’s integral of the second kind)Γ(z)[Rez>0], (Euler) Gradshteyn and Ryzhik [8, page 933], is  defined as(8)Γ(z)=∫0∞e-ttz-1dt.If the prior information is given in terms of prior meanμ and coefficient of variation ∅, then the parameters c and d can be obtained by solving(9)d=1∅2,c=1μ∅2.Hence, joint prior pdf ofβ1,β2,σ-2 and m, say g1(β1,β2,σ-2,m)  is(10)g1(β1,β2,σ-2,m)=k1e-1/2σ2(β12+β22)(1σ2)de-c/σ2,
where(11)k1=cd2πΓd(n-1).Joint posterior density ofβ1,β2,σ-2, and m say, g(β1,β2,σ-2, m∣z) is(12)g1(β1,β2,σ-2,m∣z)=L(β1,β2,σ2,m∣z)g(β1,β2,m)dβ1dβ2dσ-2h1(z)=k2⋅exp⁡[-12⋅β12σ2[Sm1+1]+β1Sm3σ2]×exp⁡[-12β22σ2[(Sn1-Sm1)+1]+β2Sm4σ2]×e-((A/2)+c)(1/σ2)(1σ2)d(σ2)-n/2h1(z),
where
(13)k2=k11(2π)n/2,Sm3,Sm4,Sm1, Sm2and A are as given in (4).h1(z) is the marginal density of z given by (14)h1(z)=∑m=1n-1∫0∞∬-∞∞L(β1,β2,σ-2,m∣z)⋅g(β1,β2,σ-2,m)dβ1dβ2dσ-2=k2∑m=1n-1T1(m),
where k2 is as given in (13)(15)T1(m)=Γ((n/2)+d-1)((A/2)+C-hm1-hm2)(n/2)+d-1x,
where 𝔵 denotes Sm1+1·Sn1-Sm1+1,(16)hm1=Sm322(Sm1+1),hm2=Sm422(Sn1-Sm1+1),Sm3,Sm4,Sm1, Sm2, Sn1, and A are as given in (4).Γ((n/2)+d-1) is the gamma function as explained in (8).Marginal posterior density of change pointm, say g(m∣z) is (17)g1(m∣z)=T1(m)∑m=1n-1T1(m),T1(m)  is as given in (15).


## 3.2. Using Conditional Informative Priors onβ1,β2 with σ-2 Known

We consider the TPLR model (1) with known σ2. We also assume same prior consideration of β1, β2,  and m as explained in Section 3.1.Hence, joint prior pdf ofβ1, β2, and m, say g2(β1, β2, m)  is(18)g2(β1,β2,m)=k3⋅e-1/2((β12+β22)/σ2),
where(19)k3=12π(n-1)σ2.
Joint posterior density of β1, β2 and m say, g2(β1,β2,m∣z) is(20)g2(β1,β2,m∣z)=k4exp⁡[-12β12[(Sm1+1)σ2]+β1Sm3σ2]×exp⁡[-12β22[(Sn1-Sm1+1)σ2]+β2Sm4σ2]/h2(z),
where(21)k4=k31(2π)n/2e-A/2σ2σ-n,Sm3,Sm4,Sm1, Sm2, and A are as given in (4).Whereh2(z) is the marginal density of z given by (22)h2(z)=k4∑m=1n-1∫-∞∞exp⁡[-12β12[Sm1+1]σ2+β1Sm3σ2]dβ1×∫-∞∞exp⁡⁡[-12β22σ2[(Sn1-Sm1)+1]+β2Sm4σ2]dβ2.And the integrals,(23)G1m=∫-∞∞exp⁡[-12β22σ2[(Sn1-Sm1)+1]+β2Sm4σ2]dβ2=2π⋅e(Sm42/2σ2(Sn1-Sm1+1))(Sn1-Sm1+1)/σ,(24)G2m=∫-∞∞exp⁡[-12β12[Sm1+1]σ2+β1Sm3σ2]dβ1=2π⋅eSm32/2σ2(Sm1+1)(Sm1+1)/σ.So using (23) and (24) results in (22), it reduces to                             (25)h2(z)=k4∑m=1n-1T2(m),T2(m)=G1mG2m,
where k4 is as given in (21). G1m and G2m are as given in (23) and (24).Sm3,Sm4,Sm1, and Sm2  are as given in (4).Marginal posterior density of change pointm, β1,  and β2 is  (26)g2(m∣z)=T2(m)∑m=1n-1T2(m),(27)g2(β1∣z)=k4h2(z)[∑m=1n-1e-1/2β12(Sm1+1σ2)+β1(Sm3σ2)]G1m,(28)g2(β2∣z)=k4h2(z)[∑m=1n-1e-1/2β22(Sn1-Sm1+1σ2)+β2Sm4σ2]G2m,k4 and  G1m,G2m,h2(z)are as given in (21), (23), (24), and (25), respectively.Sm3,Sm4,Sm1, Sm2 are as given in (4).


## 4. Bayes Estimates under SymmetricLoss Function

The Bayes estimator of a generic parameter (or function there of)α based on a squared error loss (SEL) function:(29)L1(α,d)=(α-d)2,
where d is a decision rule to estimate α, is the posterior mean. As a consequence, the SEL function relative to an integer parameter, (30)L1(m,v)=(m-v)2,m,v=0,1,2,….Hence, the Bayesian estimate of an integer-valued parameter under the SEL functionL1(m,v) is no longer the posterior mean and can be obtained by numerically minimizing the corresponding posterior loss. Generally, such a Bayesian estimate is equal to the nearest integer value to the posterior mean. So, we consider the nearest value to the posterior mean as Bayes Estimate.The Bayes estimator ofm under SEL is(31)m*=∑m=1n-1mT1(m)∑m=1n-1T1(m),(32)m**=∑m=1n-1mT2(m)∑m=1n-1T2(m),
where T1(m) and T2(m) are as given in (15) and (25).Other Bayes estimators ofα based on the loss functions(33)L2(α,d)=|α-d|,L3(α,d)={0,if|α-d|<ϵ,ϵ>0,1,otherwise,
is the posterior median and posterior mode, respectively.


## 5. Asymmetric Loss Function

The Loss functionL(α,d) provides a measure of the financial consequences arising from a wrong decision rule d to estimate an unknown quantity (a generic parameter or function thereof) α. The choice of the appropriate loss function depends on financial considerations only and is independent from 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.A useful asymmetric loss, known as the Linex loss function was introduced by  Varian  [9]. Under the assumption that the minimal loss occurs at α, the Linex loss function can be expressed as (34)L4(α,d)=exp⁡[q1(d-α)]-q1(d-α)-I,q1≠0.The sign of the shape parameterq1 reflects the direction of the asymmetry, q1>0 if overestimation 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(35)Eα{L4(α,d)}=exp⁡(q1d)Eα{exp⁡(-q1α)}-q1(d-Eα{α})-I,
where Eα{f(α)} denotes the expectation of f(α) with respect to the posterior density g(α∣z). The Bayes estimate αL* is the value of d that minimizes Eα{L4(α,d)}(36)αL*=-1q1ln⁡⁡[Eα{exp⁡(-q1α)}]
provided that Eα{exp(-q1α)} exists and is finite.


### 5.1. Assumingσ-2 Unknown

Minimizing the posterior expectation of the Linex loss functionEm[L4(m,d)]. Where Em[L4(m,d)] denotes the expectation of L4(m,d) with respect to the posterior density g1(m∣z) given in (17), we get the Bayes estimate of m by means of the nearest integer value to (37), say mL*, as under. We get the Bayes estimators of m using Linex loss function, respectively, as(37)mL*=-1q1ln[Em{exp⁡(-q1m)}]=-1q1ln[∑m=1n-1e-q1mT1(m)∑m=1n-1T1(m)].Another loss function, called general entropy (GE) loss function, proposed by Calabria and Pulcini [10], 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.Combining the General Entropy Loss with the posterior density (17), we get the estimate m by means of the nearest integer value to (40), say mE*, as below. We get the Bayes estimates mE* of m using General Entropy loss function as(40)mE*=[Em[m-q3]]-1/q3=[∑m=1n-1m-q3T1(m)∑m=1n-1T1(m)]-1/q3,
where T1(m) is as given in (15).


### 5.2. Assumingσ-2 Known

Combining the Linex loss with posterior density (26), we get the Bayes estimate of m by means of the nearest integer value to (41), say mL** as below. (41)mL**=-1q1ln⁡[Em{exp⁡(-q1m)}]=-1q1ln⁡[∑m=1n-1e-q1mT1(m)∑m=1n-1T1(m)].Combining the Linex loss with the posterior distributions (27) and (28),  respectively, we get the Bayes estimators of β1  and β2 using Linex loss function as(42)β1L**=-1q1ln⁡⁡[Eβ1e-q1β1]=-1q1ln⁡k4h2(z)∑m=1n-12π⋅e(sm3-q1σ2)2/2σ2(Sm1+1)(Sm1+1)/σG1m,(43)β2L**=-1q1ln⁡k4h2(z)×[∑m=1n-1∫-∞∞exp⁡{-12β12(Sn1-Sm1+1σ2)+β1(Sm4σ2)-q1}dβ1∑m=1n-1]G2m,
where k4,h2(z), G1m, and G2m are same as given in (21), (22), (23), and (24), respectively. Sm3,Sm4,Sm1,Sm2 are as given in (4).Minimizing expectationE[L5(m,d)] and then taking expectation with respect to posterior density g2(m∣z), we get the estimate m by means of the nearest integer value to (44) say mE*, as below.  We get the Bayes estimates mE* of m using General Entropy loss function as (44)mE**=[E1[m-q3]]-1/q3=[∑m=1n-1m-q3T2(m)∑m=1n-1T2(m)]-1/q3,
where T2(m)  is same as given in (25).Note 1.
The confluent hypergeometric function of the first kind1F1(a,b;x) [11] is a degenerate form of the hypergeometric function 2F1(a,b,c;x) which arises as a solution to the confluent hypergeometric differential equation. It is also known as Kummer's function of the first kind  and denoted  by 1F1, defined as follows:
(45)1F1(a,b;x)=∑m=0∞(a,m)xm(b,m)m!,for|x|<1.
With Pochhammer coefficients (a,m)=Γ(a+m)/Γ(a) for m≥1 and (a,0)=1 [12, page 755],  also has an integral representation
(46)1F1(a,b;x)=∫01exuua-1(1-u)b-a-1B(a,b-a)du.
The symbols Γ and B denoting the usual functions gamma and beta, respectively.
Whena and b are both integer, some special results are obtained. If a<0, and either b>0 or b<a, the series yields a polynomial with a finite number of terms. If integer b≤0, the function is undefined.Note 2.
pFq[{(a1),…,(ap)},{(b1),…,(bq)},z] is called a generalized hypergeometric series and defined as (Gradshteyn and Ryzhik [8, page 1045]).
pFq[{(a1),…,(ap)},{(b1),…,(bq)},z] has series expansion
(47)pFq[{(a1),…,(ap)},{(b1),…,(bq)},z]=∑m=0∞(a1)m,…,(ap)m,(b1)m,…,(bq)m,zm/m!.
In many special cases hypergeometric pFq is automatically converted to other functions.
Forp=q+1, hypergeometric pFq [a list, b list, z] has a branch cut discontinuity in the complex z plane running from 1 to ∞.
HypergeometricpFq (Regularized) is finite for all finite values of its argument so long as p≤q.Note 3.
β(x,y) is the beta function Euler’s integral of the first kind defined as  Gradshteyn and Ryzhik [8, pages 948, 950],
(48)β(x,y)=∫01tx-1(1-t)y-1dt,β(x,y)=ΓxΓyΓxy.
The gamma function is as explained in (8).
Minimizing expected loss functionE[L5(βi,d)] and using posterior distributions (27) and (28), we get the Bayes estimates of βi,i=1,2 using General Entropy  loss function, respectively, as
(49)βiE**={E(βi-q3)}-1/q3,i=1,2,β1E**={k4h2(z)∑m=1n-1J1mG1m}-1/q3,J1m=∫-∞∞β1-q3e-(1/2σ2)β12(Sm1+1)+β1Sm3/σ2dβ1=1((Sm1+1)/σ2)(-1+q3)×{k5mΓ(1-q32)1F1(q32,12,-sm32/σ22(Sm1+1))}+{({12,1},{1-q32,32-q32},-sm32/σ22(Sm1+1)σ2)G3mΓ(1-q32)Hypergeometric1F1×(1+q32,32,-sm32/σ22(Sm1+1)σ2)}+{({12,1},{1-q32,32-q32},-sm32/σ22(Sm1+1)σ2)G4mHypergeometricpFq,×(sm32/σ22(Sm1+1)σ2{12,1},{1-q32,32-q32},-sm32/σ22(Sm1+1)σ2)},
where
(50)k5m=(-1)-q32(1/2)(-1-q3){-(Sm1+1)2sm32}-q3×{-sm32(Sm1+1)2}-q3exp⁡⁡{sm32/σ22(Sm1+1)}×[Sm1+1σ2(Sm1+1σ2)q3/2(-1+q3)×{Sm1+1σ2((-1)q3(-Sm1+1Sm3)q3×(-Sm3Sm1+1)q3+1)Sm1+1σ2}],G3m=2Sm3σ2[(-1)q3(-Sm1+1Sm3)q3(-Sm3Sm1+1)q3-1],G4m=2(1+q3)/2[{-(Sm1+1)2sm32}q3⋅Sm3σ2×{(-1)q3(-Sm3Sm1+1)q3-(Sm3Sm1+1)q3}{-(Sm1+1)2sm32}].
Hypergeometric 1F1((1+q3)/2,3/2,-(sm32/σ2)/2(Sm1+1)σ2) and hypergeometric pFq({1/2,1},{(1-q3/2,3/2-q3/2},-(sm32/σ2)/2(Sm1+1)σ2) are hypergeometric functions same as explained in Notes 1 and  2, respectively. k5m,G3m,G4m, and G1mare as explained in (23) and (50), respectively. k4 is as given in (21). Γ((1-q3)/2) and Γ(1-q3/2) are gamma functions same as explained in (8)′. Sm3,Sm4,Sm1, and Sm2 are as given in (4)
(51)β2E**={k4h2(z)∑m=1n-1J2mG2m}-1/q3,J2m=∫-∞∞β2-q3e-(1/2σ2)β22(Sn1-Sm1+1)+β2(Sm4/σ2)dβ2=1((Sn1-Sm1+1)/σ2)(-1+q3)×{(q32,12,-sm42/σ22(Sn1-Sm1+1))k6m⋅Γ(1-q32)Hypergeometric1F1×(q32,12,-sm42/σ22(Sn1-Sm1+1))}+{(1+q32,32,-sm42/σ22(Sn1-Sm1+1))G5mΓ(1-q32)Hypergeometric1F1×(1+q32,32,-sm42/σ22(Sn1-Sm1+1))}+{({12,1},{1-q32,32-q32},-sm42/σ22(Sn1-Sm1+1))G6mHypergeometricpFq×(sm42/σ22(Sn1-Sm1+1){12,1},{1-q32,32-q32},-sm42/σ22(Sn1-Sm1+1))},
where
(52)k6m=(-1)-q32(1/2)(-1-q3)×{-(Sn1-Sm1+1)2sm42}-q3×{-sm42(Sn1-Sm1+1)2}-q3×exp⁡{sm42/σ22(Sn1-Sm1+1)}×[Sn1-Sm1+1σ2(Sn1-Sm1+1σ2)q3/2(-1+q3)×{Sn1-Sm1+1σ2×((-1)q3(-(Sn1-Sm1+1)Sm4)q3×(-Sm4(Sn1-Sm1+1))q3+1)Sn1-Sm1+1σ2}Sn1-Sm1+1σ2],G5m=2Sm4σ2[(-1)q3(-Sn1-Sm1+1Sm4)q3×(-Sm4Sn1-Sm1+1)q3-1],G6m=2(1+q3)/2[{-(Sn1-Sm1+1)2sm42}q3Sm4σ2×{(-1)q3(-Sm4Sn1-Sm1+1)q3-(Sm4Sn1-Sm1+1)q3}{-(Sn1-Sm1+1)2sm42}q3].
Hypergeometric 1F1((1+q3)/2,3/2,-(sm32/σ2)/2(Sm1+1)σ2) and hypergeometric pFq({1/2,1},{1-q3/2,(3/2)-q3/2},-(sm32/σ2)/2(Sm1+1)σ2) are hypergeometric functions same as explained in Notes 1 and  2, respectively.  k6m,G5m,G6m, and G2m are as explained in (23) and (50), respectively. k4 is as given in (21).
k4 is as given in (21). Γ((1-q3)/2) and Γ(1-q3/2) are gamma functions same as explained in (8)′. Sm3,Sm4,Sm1, and Sm2 are as given in (4).Remark 1.
Puttingq3=-1 in (40) and (44), we get the Bayes estimators of m, posterior means under the squared error loss as given in (31) and (32). Note that for q3=-1, the GE loss function reduces to the squared error loss function.


## 5.1. Assumingσ-2 Unknown

Minimizing the posterior expectation of the Linex loss functionEm[L4(m,d)]. Where Em[L4(m,d)] denotes the expectation of L4(m,d) with respect to the posterior density g1(m∣z) given in (17), we get the Bayes estimate of m by means of the nearest integer value to (37), say mL*, as under. We get the Bayes estimators of m using Linex loss function, respectively, as(37)mL*=-1q1ln[Em{exp⁡(-q1m)}]=-1q1ln[∑m=1n-1e-q1mT1(m)∑m=1n-1T1(m)].Another loss function, called general entropy (GE) loss function, proposed by Calabria and Pulcini [10], 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.Combining the General Entropy Loss with the posterior density (17), we get the estimate m by means of the nearest integer value to (40), say mE*, as below. We get the Bayes estimates mE* of m using General Entropy loss function as(40)mE*=[Em[m-q3]]-1/q3=[∑m=1n-1m-q3T1(m)∑m=1n-1T1(m)]-1/q3,
where T1(m) is as given in (15).


## 5.2. Assumingσ-2 Known

Combining the Linex loss with posterior density (26), we get the Bayes estimate of m by means of the nearest integer value to (41), say mL** as below. (41)mL**=-1q1ln⁡[Em{exp⁡(-q1m)}]=-1q1ln⁡[∑m=1n-1e-q1mT1(m)∑m=1n-1T1(m)].Combining the Linex loss with the posterior distributions (27) and (28),  respectively, we get the Bayes estimators of β1  and β2 using Linex loss function as(42)β1L**=-1q1ln⁡⁡[Eβ1e-q1β1]=-1q1ln⁡k4h2(z)∑m=1n-12π⋅e(sm3-q1σ2)2/2σ2(Sm1+1)(Sm1+1)/σG1m,(43)β2L**=-1q1ln⁡k4h2(z)×[∑m=1n-1∫-∞∞exp⁡{-12β12(Sn1-Sm1+1σ2)+β1(Sm4σ2)-q1}dβ1∑m=1n-1]G2m,
where k4,h2(z), G1m, and G2m are same as given in (21), (22), (23), and (24), respectively. Sm3,Sm4,Sm1,Sm2 are as given in (4).Minimizing expectationE[L5(m,d)] and then taking expectation with respect to posterior density g2(m∣z), we get the estimate m by means of the nearest integer value to (44) say mE*, as below.  We get the Bayes estimates mE* of m using General Entropy loss function as (44)mE**=[E1[m-q3]]-1/q3=[∑m=1n-1m-q3T2(m)∑m=1n-1T2(m)]-1/q3,
where T2(m)  is same as given in (25).Note 1.
The confluent hypergeometric function of the first kind1F1(a,b;x) [11] is a degenerate form of the hypergeometric function 2F1(a,b,c;x) which arises as a solution to the confluent hypergeometric differential equation. It is also known as Kummer's function of the first kind  and denoted  by 1F1, defined as follows:
(45)1F1(a,b;x)=∑m=0∞(a,m)xm(b,m)m!,for|x|<1.
With Pochhammer coefficients (a,m)=Γ(a+m)/Γ(a) for m≥1 and (a,0)=1 [12, page 755],  also has an integral representation
(46)1F1(a,b;x)=∫01exuua-1(1-u)b-a-1B(a,b-a)du.
The symbols Γ and B denoting the usual functions gamma and beta, respectively.
Whena and b are both integer, some special results are obtained. If a<0, and either b>0 or b<a, the series yields a polynomial with a finite number of terms. If integer b≤0, the function is undefined.Note 2.
pFq[{(a1),…,(ap)},{(b1),…,(bq)},z] is called a generalized hypergeometric series and defined as (Gradshteyn and Ryzhik [8, page 1045]).
pFq[{(a1),…,(ap)},{(b1),…,(bq)},z] has series expansion
(47)pFq[{(a1),…,(ap)},{(b1),…,(bq)},z]=∑m=0∞(a1)m,…,(ap)m,(b1)m,…,(bq)m,zm/m!.
In many special cases hypergeometric pFq is automatically converted to other functions.
Forp=q+1, hypergeometric pFq [a list, b list, z] has a branch cut discontinuity in the complex z plane running from 1 to ∞.
HypergeometricpFq (Regularized) is finite for all finite values of its argument so long as p≤q.Note 3.
β(x,y) is the beta function Euler’s integral of the first kind defined as  Gradshteyn and Ryzhik [8, pages 948, 950],
(48)β(x,y)=∫01tx-1(1-t)y-1dt,β(x,y)=ΓxΓyΓxy.
The gamma function is as explained in (8).
Minimizing expected loss functionE[L5(βi,d)] and using posterior distributions (27) and (28), we get the Bayes estimates of βi,i=1,2 using General Entropy  loss function, respectively, as
(49)βiE**={E(βi-q3)}-1/q3,i=1,2,β1E**={k4h2(z)∑m=1n-1J1mG1m}-1/q3,J1m=∫-∞∞β1-q3e-(1/2σ2)β12(Sm1+1)+β1Sm3/σ2dβ1=1((Sm1+1)/σ2)(-1+q3)×{k5mΓ(1-q32)1F1(q32,12,-sm32/σ22(Sm1+1))}+{({12,1},{1-q32,32-q32},-sm32/σ22(Sm1+1)σ2)G3mΓ(1-q32)Hypergeometric1F1×(1+q32,32,-sm32/σ22(Sm1+1)σ2)}+{({12,1},{1-q32,32-q32},-sm32/σ22(Sm1+1)σ2)G4mHypergeometricpFq,×(sm32/σ22(Sm1+1)σ2{12,1},{1-q32,32-q32},-sm32/σ22(Sm1+1)σ2)},
where
(50)k5m=(-1)-q32(1/2)(-1-q3){-(Sm1+1)2sm32}-q3×{-sm32(Sm1+1)2}-q3exp⁡⁡{sm32/σ22(Sm1+1)}×[Sm1+1σ2(Sm1+1σ2)q3/2(-1+q3)×{Sm1+1σ2((-1)q3(-Sm1+1Sm3)q3×(-Sm3Sm1+1)q3+1)Sm1+1σ2}],G3m=2Sm3σ2[(-1)q3(-Sm1+1Sm3)q3(-Sm3Sm1+1)q3-1],G4m=2(1+q3)/2[{-(Sm1+1)2sm32}q3⋅Sm3σ2×{(-1)q3(-Sm3Sm1+1)q3-(Sm3Sm1+1)q3}{-(Sm1+1)2sm32}].
Hypergeometric 1F1((1+q3)/2,3/2,-(sm32/σ2)/2(Sm1+1)σ2) and hypergeometric pFq({1/2,1},{(1-q3/2,3/2-q3/2},-(sm32/σ2)/2(Sm1+1)σ2) are hypergeometric functions same as explained in Notes 1 and  2, respectively. k5m,G3m,G4m, and G1mare as explained in (23) and (50), respectively. k4 is as given in (21). Γ((1-q3)/2) and Γ(1-q3/2) are gamma functions same as explained in (8)′. Sm3,Sm4,Sm1, and Sm2 are as given in (4)
(51)β2E**={k4h2(z)∑m=1n-1J2mG2m}-1/q3,J2m=∫-∞∞β2-q3e-(1/2σ2)β22(Sn1-Sm1+1)+β2(Sm4/σ2)dβ2=1((Sn1-Sm1+1)/σ2)(-1+q3)×{(q32,12,-sm42/σ22(Sn1-Sm1+1))k6m⋅Γ(1-q32)Hypergeometric1F1×(q32,12,-sm42/σ22(Sn1-Sm1+1))}+{(1+q32,32,-sm42/σ22(Sn1-Sm1+1))G5mΓ(1-q32)Hypergeometric1F1×(1+q32,32,-sm42/σ22(Sn1-Sm1+1))}+{({12,1},{1-q32,32-q32},-sm42/σ22(Sn1-Sm1+1))G6mHypergeometricpFq×(sm42/σ22(Sn1-Sm1+1){12,1},{1-q32,32-q32},-sm42/σ22(Sn1-Sm1+1))},
where
(52)k6m=(-1)-q32(1/2)(-1-q3)×{-(Sn1-Sm1+1)2sm42}-q3×{-sm42(Sn1-Sm1+1)2}-q3×exp⁡{sm42/σ22(Sn1-Sm1+1)}×[Sn1-Sm1+1σ2(Sn1-Sm1+1σ2)q3/2(-1+q3)×{Sn1-Sm1+1σ2×((-1)q3(-(Sn1-Sm1+1)Sm4)q3×(-Sm4(Sn1-Sm1+1))q3+1)Sn1-Sm1+1σ2}Sn1-Sm1+1σ2],G5m=2Sm4σ2[(-1)q3(-Sn1-Sm1+1Sm4)q3×(-Sm4Sn1-Sm1+1)q3-1],G6m=2(1+q3)/2[{-(Sn1-Sm1+1)2sm42}q3Sm4σ2×{(-1)q3(-Sm4Sn1-Sm1+1)q3-(Sm4Sn1-Sm1+1)q3}{-(Sn1-Sm1+1)2sm42}q3].
Hypergeometric 1F1((1+q3)/2,3/2,-(sm32/σ2)/2(Sm1+1)σ2) and hypergeometric pFq({1/2,1},{1-q3/2,(3/2)-q3/2},-(sm32/σ2)/2(Sm1+1)σ2) are hypergeometric functions same as explained in Notes 1 and  2, respectively.  k6m,G5m,G6m, and G2m are as explained in (23) and (50), respectively. k4 is as given in (21).
k4 is as given in (21). Γ((1-q3)/2) and Γ(1-q3/2) are gamma functions same as explained in (8)′. Sm3,Sm4,Sm1, and Sm2 are as given in (4).Remark 1.
Puttingq3=-1 in (40) and (44), we get the Bayes estimators of m, posterior means under the squared error loss as given in (31) and (32). Note that for q3=-1, the GE loss function reduces to the squared error loss function.


## 6. Numerical Study

### 6.1. Illustration

Let us consider the two-phase regression model(53)yt=3xt+εt,t=1,2,3,4,yt=3.5xt+εt,t=5,6,…,15,
where  εt's are i.i.d. N(0,1) random errors. We take the first 15 values of  xt and εtfrom Table 4.1 of Zellner [13] to generate 15 sample values (xt,yt)t=1,2,…,15. The generated sample values are given in Table 1. The β1,β2, and σ2 themselves were random observations. β1 and  β2 were from standard normal distribution and precision 1/σ2  was from gamma distribution with μ =1 and coefficient of variation  ∅=1.4, respectively, in c=0.5,d=0.5.Table 1
Zellener data for two-phase linear regression model.
TxtytTxtytTxtyt11.74.664.111.27115.116.9022.26.474.213.90125.217.3232.76.984.514.18136.017.9943.38.094.715.45146.218.9253.913.75104.816.40156.519.36We have calculated posterior mean, posterior median and posterior mode of m. The results are shown in Table2.Table 2
The values of Bayes estimates of shift point, regression coefficients and precision.
PriorBayes estimates ofmPosterior MeanPosterior MedianPosterior ModeUnknownσ-2454Knownσ-2454We also compute the Bayes estimators mE of m using (40) for unknown σ2 and (44) for known σ2 and mL using (37) for unknown σ2 and (41) for known σ2 for data given in Table 1. The results are shown in Table 3.Table 3
The Bayes estimates of change point using asymmetric loss functions.
PriorShape parameterBayes estimates of   change pointq1q3mL*mE*Unknownσ-21.51.5221.21.2330.90.9440.50.5440.20.2440.10.144−1.0−1.064−2.0−2.076q1q3mL**mE**Knownσ-21.51.5221.21.2330.90.9440.50.5440.20.2440.10.144−1.0−1.054−2.0−2.065Table3 shows that for small values of |q|,q=0.9, 0.5, 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=q3=1.5, 1.2, Bayes estimates are less than actual value  of m=4.It can be seen from Table3 that positive sign of shape parameter of loss functions reflects overestimation is more serious than underestimation. Thus, problem of overestimation can be solved by taking the value of shape parameter of  Linex and General Entropy loss functions positive and high.Forq1=q3=-1, −2, Bayes estimates are quite large than actual value m=4. It can be seen from Table 3 that the negative sign of shape parameter of loss functions reflects 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.We get Bayes estimatorsβ1L**,β2L**,β1E**, and β2E** of β1 and β2 using (42), (43), (49), and (51), respectively, for the data given in Table 1 and for different value  of shape parameter q1 and q3. The results are shown in Table 4.Table 4
The Bayes estimates of Bayes estimates of  regression coefficients using asymmetric loss functions.
Shape parameterBayes estimates of  regression co-efficientsβ1β2q1β1L**β2L**Knownσ-21.51.32.21.21.82.40.92.53.20.53.03.50.23.23.70.13.43.9q3β**1Eβ**2EKnownσ-21.51.02.61.21.52.80.92.93.30.53.13.40.23.33.60.13.53.8Tables3 and 4 show that the values of the shape parameters of  Linex and General Entropy loss functions increase, the values of Bayes estimates decrease.


## 6.1. Illustration

Let us consider the two-phase regression model(53)yt=3xt+εt,t=1,2,3,4,yt=3.5xt+εt,t=5,6,…,15,
where  εt's are i.i.d. N(0,1) random errors. We take the first 15 values of  xt and εtfrom Table 4.1 of Zellner [13] to generate 15 sample values (xt,yt)t=1,2,…,15. The generated sample values are given in Table 1. The β1,β2, and σ2 themselves were random observations. β1 and  β2 were from standard normal distribution and precision 1/σ2  was from gamma distribution with μ =1 and coefficient of variation  ∅=1.4, respectively, in c=0.5,d=0.5.Table 1
Zellener data for two-phase linear regression model.
TxtytTxtytTxtyt11.74.664.111.27115.116.9022.26.474.213.90125.217.3232.76.984.514.18136.017.9943.38.094.715.45146.218.9253.913.75104.816.40156.519.36We have calculated posterior mean, posterior median and posterior mode of m. The results are shown in Table2.Table 2
The values of Bayes estimates of shift point, regression coefficients and precision.
PriorBayes estimates ofmPosterior MeanPosterior MedianPosterior ModeUnknownσ-2454Knownσ-2454We also compute the Bayes estimators mE of m using (40) for unknown σ2 and (44) for known σ2 and mL using (37) for unknown σ2 and (41) for known σ2 for data given in Table 1. The results are shown in Table 3.Table 3
The Bayes estimates of change point using asymmetric loss functions.
PriorShape parameterBayes estimates of   change pointq1q3mL*mE*Unknownσ-21.51.5221.21.2330.90.9440.50.5440.20.2440.10.144−1.0−1.064−2.0−2.076q1q3mL**mE**Knownσ-21.51.5221.21.2330.90.9440.50.5440.20.2440.10.144−1.0−1.054−2.0−2.065Table3 shows that for small values of |q|,q=0.9, 0.5, 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=q3=1.5, 1.2, Bayes estimates are less than actual value  of m=4.It can be seen from Table3 that positive sign of shape parameter of loss functions reflects overestimation is more serious than underestimation. Thus, problem of overestimation can be solved by taking the value of shape parameter of  Linex and General Entropy loss functions positive and high.Forq1=q3=-1, −2, Bayes estimates are quite large than actual value m=4. It can be seen from Table 3 that the negative sign of shape parameter of loss functions reflects 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.We get Bayes estimatorsβ1L**,β2L**,β1E**, and β2E** of β1 and β2 using (42), (43), (49), and (51), respectively, for the data given in Table 1 and for different value  of shape parameter q1 and q3. The results are shown in Table 4.Table 4
The Bayes estimates of Bayes estimates of  regression coefficients using asymmetric loss functions.
Shape parameterBayes estimates of  regression co-efficientsβ1β2q1β1L**β2L**Knownσ-21.51.32.21.21.82.40.92.53.20.53.03.50.23.23.70.13.43.9q3β**1Eβ**2EKnownσ-21.51.02.61.21.52.80.92.93.30.53.13.40.23.33.60.13.53.8Tables3 and 4 show that the values of the shape parameters of  Linex and General Entropy loss functions increase, the values of Bayes estimates decrease.


## 7. Sensitivity of Bayes Estimates

In this section, we study the sensitivity of the Bayes estimator, obtained in Sections4 and 5 with respect to change in the prior of parameters. The mean μ  of  gamma prior on σ-2 has been used as prior information in computing the parameters c,d of the prior. We have computed posterior mean m* using (31) and m** using (32) for the data given in Table 1 considering  different sets of values of (μ).  Following Calabria and Pulcini [10], we also assume the prior information to be correct if the true value of σ-2 is closed to prior mean μ and is assumed to be wrong if σ-2 is far from μ. We observed that the posterior mode m* appears to be robust with respect to the correct choice of the prior density of σ-2and also with a wrong choice of the prior density of σ-2. This can be seen from Table 5.Table 5
The values of Posterior mean of change point for different values of prior means.
Values of prior meanμBayes estimates of change pointPosterior meanPosterior medianPosterior mode0.556414541.5664Table5 shows that when prior mean μ = 1 = actual value of  σ-2, it means correct choice of prior of   σ-2, The values of Bayes estimator posterior mode is 4. It gives  correct estimation of change point. Now, when μ = 0.5 and 1.5 (far from true value of σ-2 = 1), it means wrong choice of prior of σ-2. The value of Bayes estimator of posterior mode remains 4. But, posterior mean and posterior median do not remain same for wrong choice of prior of σ-2. Thus, posterior mode is not sensitive with wrong choice of prior density of σ-2. While, posterior mean and posterior median are sensitive with wrong choice of prior density of σ-2.


## 8. Simulation Study

In Sections4 and 5 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=4,n=15,β1=3.2, 3.3, 3.4, β2=3.5, 3.6, 3.7 and obtained the frequency distributions of posterior mean, median of m,mL*,mE* with the correct prior consideration. The result is shown in Tables 2 and 3. The value of shape parameter of the general entropy loss and Linex loss used in simulation study for shift point is taken as 0.1.We have also simulated several standard normal samples. For eachβ1, β2,  and m and n, 1000 pseudorandom samples from two-phase linear regression model discussed in Section 2 have been simulated and Bayes estimators of change point m has been computed using q3=0.9  and for different prior mean μ.Table6 leads to conclusion that performance of mL*,mF*, posterior mode’s and posterior median’s has better performance than that of posterior mean of change point explained in Sections 4 and 5. 46% values of posterior mean are closed to actual value of change point with correct choice of prior. 62% values of posterior median are closed to actual value of change point with correct choice of prior. 70% values of posterior mode are close to correct values of change point with correct prior considerations. 65% values of mL*  are closed to actual values of m. 66% values of  mE* are closed to actual values of m.Table 6
Frequency distributions of the Bayes estimates of the change point.
Bayes estimate% Frequency for01-0203–0506–15Posterior mean354619Posterior median236215Posterior mode127018mL*206515mF*226612


## 9. Conclusions

In this study, we are discussing the Bayes estimator of shift point, the integer parameter, posterior mean is less appealing. Posterior median and posterior mode  appear as better estimators as they would be always integer. Our numerical study showed that the Bayes estimators posterior mode ofm is robust with respect to the correct choice of the prior specifications on σ-2 and wrong choice of the prior specifications on σ-2, posterior median and posterior mode are sensitive in case prior specifications on  1/σ2 deviate simultaneously from the true values. Here, we discussed regression model with one change point, in practice it may have two or more change point models. One can apply these models to econometric data such as poverty and irrigation.



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*Source: 357814-2011-07-26.xml*