Bayes Estimation of Two-Phase Linear Regression Model

Abstract

Let the regression model be 𝑌_𝑖=𝛽₁𝑋_𝑖+𝜀_𝑖, where 𝜀_𝑖 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 𝑋_𝑚 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.

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 Section 2, 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 as𝑦_𝑡=𝛼¹𝛼+𝛽₁𝑥_𝑡+𝜀_𝑡,𝑡=1,2,…,𝑚,₂+𝛽₂𝑥_𝑡+𝜀_𝑡,𝑡=𝑚+1,…,𝑛,(1) where 𝜀_𝑡’s are i. i. d. N (0,𝜎²) random errors with variance 𝜎²>0, 𝑥_𝑡 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 values𝑔=𝛽₁,𝐿𝛽₂,𝛽𝜃₁,𝜃₂,𝑚∣𝑧₁,𝛽₂,𝜎⁻²,𝑚∣𝑧⋅𝑔𝛽₁,𝛽₂,𝜎⁻²,𝑚∑^{𝑛−1𝑚=1}∫_𝛽1∫_𝛽2∫_𝜎−2𝐿𝛽₁,𝛽₂,𝜎⁻²,𝑚∣𝑧⋅ℨ,(2) where ℨ denotes 𝑔(𝛽₁,𝛽₂,𝜎⁻²,𝑚)𝑑𝛽₁𝑑𝛽₂𝑑𝜎⁻².

The likelihood function of β₁, β₂, 𝜎⁻² and 𝑚, given the sample information 𝑍_𝑡=(𝑥_𝑡,𝑦_𝑡),𝑡=1,2,…,𝑚,𝑚+1,…,𝑛, is𝐿=𝛽₁,𝛽1₂,𝜎⁻²,𝑚∣𝑧(2𝜋)^𝑛/2exp−12𝛽²¹𝑠_𝑚1𝜎²+𝛽₁𝑠^𝑚3×exp−1𝜎²2𝑠^𝑛1−𝑠_𝑚1𝜎²𝛽²²+𝛽₂𝑠^𝑚4×𝑒_{−𝐴/2𝜎2}𝜎²𝜎^−𝑛,(3) where𝑠_𝑘1=^𝑘_𝑖=1𝑥^2𝑖,𝑠_𝑘2=^𝑘_𝑖=1𝐴=_𝑛𝑥_𝑖𝑦_𝑖,_𝑖=1𝑦^2𝑖+𝑛𝛼²−2𝛼^𝑛_𝑖=1𝑆_𝑚3𝑦_𝑖,𝑆_𝑚4=𝑆_𝑚2+2𝛼𝑆_𝑚1,=𝑆_𝑛2−𝑆_𝑚2+2𝛼𝑆_𝑛1−𝑆_𝑚1.(4)

3.1. Using Gamma Prior on 1/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 on β₁ and β₂ given 𝜎², with 𝑁(0,𝜎²) as𝑔₁𝛽₁∣𝜎²=1√2𝜋𝜎exp−21𝜎𝛽²¹²,−∞<𝛽¹𝑔<∞,₁𝛽₂∣𝜎²=1√2𝜋𝜎exp−21𝜎𝛽²²²,−∞<𝛽²<∞.(5) 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 𝜇𝑔₁1𝜎²=Γ𝑐𝑑^𝑑𝜎1²^𝑑−1𝑒^{−𝑐/𝜎2},𝜎²>0,𝑐,𝑑>0,(6) where Γ𝑑 is gamma function same as explained in (8).

The integral representation of Γ𝑧 is as below [Re𝑧>0,Re𝑥>0], Gradshteyn and Ryzhik [8, page 934]Γ𝑧=𝑥^𝑧^∞0𝑒^−𝑥𝑡𝑡^𝑧−1𝑑𝑡.(7)

The gamma function (Euler’s integral of the second kind) Γ(𝑧)

[Re𝑧>0], (Euler) Gradshteyn and Ryzhik [8, page 933], is defined asΓ(𝑧)=^∞0𝑒^−𝑡𝑡^𝑧−1𝑑𝑡.(8)

If the prior information is given in terms of prior mean 𝜇 and coefficient of variation ∅, then the parameters 𝑐 and 𝑑 can be obtained by solving𝑑=1∅²,𝑐=1𝜇∅².(9)

Hence, joint prior pdf of 𝛽₁,𝛽₂,𝜎⁻² and 𝑚, say 𝑔₁(𝛽₁,𝛽₂,𝜎⁻²,𝑚) is𝑔₁𝛽₁,𝛽₂,𝜎⁻²,𝑚=𝑘₁𝑒^{−1/2𝜎2(𝛽21+𝛽22)}1𝜎²^𝑑𝑒^{−𝑐/𝜎2},(10) where𝑘₁=𝑐^𝑑2𝜋Γ𝑑(𝑛−1).(11)

Joint posterior density of 𝛽₁,𝛽₂,𝜎⁻², and 𝑚 say, 𝑔(β_1, β_2, 𝜎⁻², 𝑚∣𝑧) is𝑔₁=𝛽₁𝐿,𝛽₂𝛽,𝜎⁻²,𝑚∣𝑧₁,𝛽₂,𝜎²,𝑚∣𝑧𝑔𝛽₁,𝛽₂,𝑚𝑑𝛽₁𝑑𝛽₂𝑑𝜎⁻²=𝑘₂ℎ₁(𝑧)⋅exp−12⋅𝜎𝛽²¹²𝑆_𝑚1+1+𝛽₁𝑆_𝑚3×exp−1𝜎²2𝜎𝛽²²²𝑆_𝑛1−𝑆_𝑚1+1+𝛽₂𝑆_𝑚4×𝑒_{−((𝐴/2)+𝑐)(1/𝜎2)}𝜎²𝜎1₂^𝑑𝜎²^−𝑛/2ℎ₁(𝑧),(12) where 𝑘₂=𝑘₁1(2𝜋)^𝑛/2,(13)𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1, 𝑆_𝑚2and 𝐴 are as given in (4).

ℎ₁(𝑧) is the marginal density of 𝑧 given by ℎ₁(𝑧)=^𝑛−1_𝑚₌₁^∞0^∞−∞⋅𝑔𝛽₁𝐿𝛽1,𝛽2,𝜎⁻²,𝑚∣𝑧,𝛽₂,𝜎⁻²,𝑚𝑑𝛽₁𝑑𝛽₂𝑑𝜎⁻²=𝑘_{2𝑛−1𝑚}₌₁𝑇₁(𝑚),(14) where 𝑘₂ is as given in (13)𝑇₁(𝑚)=Γ((𝑛/2)+𝑑−1)(𝐴/2)+𝐶−ℎ_𝑚1−ℎ_𝑚2^{(𝑛/2)+𝑑−1}𝔵,(15) where 𝔵 denotes √𝑆_𝑚1+1⋅√𝑆_𝑛1−𝑆_𝑚1+1,ℎ_𝑚1=𝑆^2𝑚3ℎ2𝑆_𝑚1+1_𝑚2,=𝑆^2𝑚42𝑆_𝑛1−𝑆_𝑚1+1,(16)𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1, 𝑆_𝑚2, 𝑆_𝑛1, 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 point 𝑚, say 𝑔(𝑚∣𝑧) is 𝑔₁(𝑚∣𝑧)=𝑇¹(𝑚)∑^{𝑛−1𝑚=1}𝑇₁(𝑚),(17)𝑇₁(𝑚) is as given in (15).

3.2. Using Conditional Informative Priors on 𝛽₁,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 of β₁, β₂, and 𝑚, say 𝑔₂(β₁, β₂, 𝑚) is𝑔₂𝛽₁,𝛽₂,𝑚=𝑘₃⋅𝑒^{−1/2((𝛽21+𝛽22)/𝜎2)},(18) where𝑘₃=12𝜋(𝑛−1)𝜎².(19) Joint posterior density of β₁, β₂ and 𝑚 say, 𝑔₂(𝛽₁,𝛽₂,𝑚∣𝑧) is𝑔₂=𝛽𝑘₁₄,𝛽₂,𝑚∣𝑧exp−12𝛽²¹𝑆^𝑚1+1𝜎²+𝛽¹𝑆_𝑚3×exp−1𝜎²2𝛽²²𝑆^𝑛1−𝑆_𝑚1+1𝜎²+𝛽²𝑆_𝑚4𝜎²ℎ²(𝑧),(20) where𝑘₄=𝑘₃1(2𝜋)^𝑛/2𝑒^{−𝐴/2𝜎2}𝜎^−𝑛,(21)𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1, 𝑆_𝑚2, and 𝐴 are as given in (4).

Where ℎ₂(𝑧) is the marginal density of 𝑧 given by ℎ₂(𝑧)=𝑘_{4𝑛−1𝑚}₌₁^∞−∞exp−12𝛽²¹𝑆_𝑚1+1𝜎²+𝛽₁𝑆_𝑚3𝜎²𝑑𝛽¹×^∞−∞exp−12𝜎𝛽²²²𝑆_𝑛1−𝑆_𝑚1+1+𝛽₂𝑆_𝑚4𝜎²𝑑𝛽².(22)

And the integrals,𝐺_1𝑚=^∞−∞exp−12𝜎𝛽²²²𝑆_𝑛1−𝑆_𝑚1+1+𝛽₂𝑆_𝑚4𝜎²𝑑𝛽²=√2𝜋⋅𝑒^{(𝑆2𝑚4/2𝜎2(𝑆𝑛1−𝑆𝑚1+1))}𝐺𝑆_𝑛1−𝑆_𝑚1+1/𝜎,(23)_2𝑚=^∞−∞exp−12𝛽²¹𝑆_𝑚1+1𝜎²+𝛽₁𝑆_𝑚3𝜎²𝑑𝛽¹=√2𝜋⋅𝑒^{𝑆2𝑚3/2𝜎2(𝑆𝑚1+1)}𝑆_𝑚1+1/𝜎.(24)

So using (23) and (24) results in (22), it reduces to ℎ₂(𝑧)=𝑘_{4𝑛−1𝑚}₌₁𝑇𝑇₂(𝑚),₂(𝑚)=𝐺_1𝑚𝐺_2𝑚,(25) where 𝑘₄ is as given in (21). 𝐺_1𝑚 and 𝐺_2𝑚 are as given in (23) and (24).

𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1, and 𝑆_𝑚2 are as given in (4).

Marginal posterior density of change point 𝑚, β₁, and β₂ is 𝑔₂(𝑚∣𝑧)=𝑇²(𝑚)∑^{𝑛−1𝑚=1}𝑇₂(𝑚),(26)𝑔₂𝛽₁∣𝑧=𝑘⁴ℎ₂(𝑧)^𝑛−1_𝑚=1𝑒^−1/2𝛽²¹𝑆^𝑚1+1𝜎²+𝛽¹𝑆𝜎^𝑚32𝐺^1𝑚,(27)𝑔₂=𝛽₂∣𝑘𝑧₄ℎ₂(𝑧)^𝑛−1_𝑚=1𝑒^−1/2𝛽²²𝑆^𝑛1−𝑆_𝑚1+1𝜎²+𝛽²𝑆_𝑚4𝜎²𝐺^2𝑚,(28)𝑘₄ and 𝐺_1𝑚,𝐺_2𝑚,ℎ₂(𝑧)are as given in (21), (23), (24), and (25), respectively.

𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1, 𝑆_𝑚2 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:𝐿₁(𝛼,𝑑)=(𝛼−𝑑)²,(29) where 𝑑 is a decision rule to estimate α, is the posterior mean. As a consequence, the SEL function relative to an integer parameter, 𝐿₁(𝑚,𝑣)=(𝑚−𝑣)²,𝑚,𝑣=0,1,2,….(30)

Hence, the Bayesian estimate of an integer-valued parameter under the SEL function 𝐿₁(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 of 𝑚 under SEL is𝑚^∗=∑^{𝑛−1𝑚=1}𝑚𝑇₁(𝑚)𝑚∑^{𝑛−1𝑚=1}𝑇₁(𝑚),(31)_∗∗=∑^{𝑛−1𝑚=1}𝑚𝑇₂(𝑚)∑^{𝑛−1𝑚=1}𝑇₂(𝑚),(32) where 𝑇₁(𝑚) and 𝑇₂(𝑚) are as given in (15) and (25).

Other Bayes estimators of α based on the loss functions𝐿𝐿₂(𝛼,𝑑)=||𝛼−𝑑₃||,(𝛼,𝑑)=01,,iofth||e𝛼rw−i𝑑se||,<𝜖,𝜖>0,(33) is the posterior median and posterior mode, respectively.

5. Asymmetric Loss Function

The Loss function 𝐿(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 𝐿₄(𝛼,𝑑)=exp𝑞₁(𝑑−𝛼)−𝑞₁(𝑑−𝛼)−𝐼,𝑞₁≠0.(34)

The sign of the shape parameter 𝑞₁ 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 is𝐸_𝛼𝐿₄(𝛼,𝑑)=−ex𝑞p₁𝑞₁𝑑𝐸_𝛼exp−𝑞₁𝛼𝑑−𝐸_𝛼{𝛼}−𝐼,(35) where 𝐸_𝛼{𝑓(𝛼)} denotes the expectation of 𝑓(𝛼) with respect to the posterior density 𝑔(𝛼∣𝑧). The Bayes estimate 𝛼^∗𝐿 is the value of 𝑑 that minimizes 𝐸_𝛼{𝐿₄(𝛼,𝑑)}𝛼^∗𝐿=−1𝑞₁ln𝐸_𝛼exp−𝑞₁𝛼(36) provided that 𝐸_𝛼{exp(−𝑞₁𝛼)} exists and is finite.

5.1. Assuming 𝜎⁻² Unknown

Minimizing the posterior expectation of the Linex loss function 𝐸_𝑚[𝐿₄(𝑚,𝑑)]. Where 𝐸_𝑚[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 𝑚^∗𝐿, as under. We get the Bayes estimators of 𝑚 using Linex loss function, respectively, as𝑚^∗𝐿=−1𝑞₁=−1𝑙_𝑛𝐸_𝑚exp−𝑞₁𝑚𝑞₁𝑙_𝑛∑^{𝑛−1𝑚=1}𝑒^−𝑞1𝑚𝑇₁(𝑚)∑^{𝑛−1𝑚=1}𝑇₁(𝑚).(37)

Another loss function, called general entropy (GE) loss function, proposed by Calabria and Pulcini [10], is given by𝐿₅(𝛼,𝑑)=𝑑𝛼^𝑞3−𝑞₃ln𝑑𝛼−𝐼.(38) The Bayes estimate 𝛼^∗𝐸 is the value of 𝑑 that minimizes 𝐸_𝛼[𝐿₅(𝛼,𝑑)]:𝛼^∗𝐸=[𝐸_𝛼(𝛼^−𝑞3)]^−1/𝑞3(39) provided that 𝐸_𝛼(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 𝑚^∗𝐸, as below. We get the Bayes estimates 𝑚^∗𝐸 of 𝑚 using General Entropy loss function as𝑚^∗𝐸=𝐸_𝑚𝑚^−𝑞3^−1/𝑞3=∑^{𝑛−1𝑚=1}𝑚^−𝑞3𝑇₁(𝑚)∑^{𝑛−1𝑚=1}𝑇₁(𝑚)^−1/𝑞3,(40) where 𝑇₁(𝑚) is as given in (15).

5.2. Assuming 𝜎⁻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 𝑚^𝐿∗∗ as below. 𝑚^𝐿∗∗=−1𝑞₁=−1ln𝐸_𝑚exp−𝑞₁𝑚𝑞₁ln∑^{𝑛−1𝑚=1}𝑒^−𝑞1𝑚𝑇₁(𝑚)∑^{𝑛−1𝑚=1}𝑇₁(𝑚).(41)

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 as𝛽^∗∗1𝐿=−1𝑞₁=−1ln𝐸_𝛽1𝑒^{−𝑞1𝛽1}𝑞₁ln𝑘⁴ℎ₂(𝑧)^𝑛−1_𝑚=1√2𝜋⋅𝑒^{(𝑠𝑚3−𝑞1𝜎2)2/2𝜎2(𝑆𝑚1+1)}𝑆_𝑚1+1/𝜎𝐺^1𝑚,(42)𝛽^∗∗2𝐿=−1𝑞₁ln𝑘⁴×ℎ₂(𝑧)_𝑛−1𝑚₌₁^∞−∞exp−12𝛽²¹𝑆^𝑛1−𝑆_𝑚1+1+𝛽₁𝜎²𝑆𝜎^𝑚42−𝑞¹𝑑𝛽¹𝐺^2𝑚,(43) where 𝑘₄,ℎ₂(𝑧), 𝐺_1𝑚, and 𝐺_2𝑚 are same as given in (21), (22), (23), and (24), respectively. 𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1,𝑆_𝑚2 are as given in (4).

Minimizing expectation 𝐸[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 𝑚^∗𝐸, as below. We get the Bayes estimates 𝑚^∗𝐸 of 𝑚 using General Entropy loss function as 𝑚^𝐸∗∗=𝐸₁𝑚^−𝑞3^−1/𝑞3=∑^{𝑛−1𝑚=1}𝑚^−𝑞3𝑇₂(𝑚)∑^{𝑛−1𝑚=1}𝑇₂(𝑚)^−1/𝑞3,(44) where 𝑇₂(𝑚) is same as given in (25).

Note 1. The confluent hypergeometric function of the first kind ₁𝐹₁(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: ₁𝐹₁(𝑎,𝑏;𝑥)=^∞_𝑚₌₀(𝑎,𝑚)𝑥^𝑚(𝑏,𝑚)𝑚!,for|𝑥|<1.(45) 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 ₁𝐹₁(𝑎,𝑏;𝑥)=¹⁰𝑒^𝑥𝑢𝑢^𝑎−1(1−𝑢)^{𝑏−𝑎−1}𝐵(𝑎,𝑏−𝑎)𝑑𝑢.(46) The symbols Γ and 𝐵 denoting the usual functions gamma and beta, respectively.
When 𝑎 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. _𝑝𝐹_𝑞[{(𝑎₁),…,(𝑎_𝑝)},{(𝑏₁),…,(𝑏_𝑞)},𝑧] is called a generalized hypergeometric series and defined as (Gradshteyn and Ryzhik [8, page 1045]).
_𝑝𝐹_𝑞[{(𝑎₁),…,(𝑎_𝑝)},{(𝑏₁),…,(𝑏_𝑞)},𝑧] has series expansion _𝑝𝐹_𝑞=𝑎₁,…,𝑎_𝑝,𝑏₁,…,𝑏_𝑞,𝑧_∞_𝑚₌₀𝑎₁_𝑚,…,𝑎_𝑝_𝑚,𝑏₁_𝑚,…,𝑏_𝑞_𝑚,𝑧^𝑚𝑚!.(47) In many special cases hypergeometric _𝑝𝐹_𝑞 is automatically converted to other functions.
For 𝑝=𝑞+1, hypergeometric _𝑝𝐹_𝑞 [𝑎 list, 𝑏 list, 𝑧] has a branch cut discontinuity in the complex 𝑧 plane running from 1 to ∞.
Hypergeometric _𝑝𝐹_𝑞 (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], 𝛽(𝑥,𝑦𝛽)=(𝑥,𝑦¹⁰)𝑡=^𝑥−1Γ(1𝑥Γ−𝑦𝑡)^𝑦−1𝑑𝑡,Γ𝑥𝑦.(48) The gamma function is as explained in (8).
Minimizing expected loss function 𝐸[𝐿₅(𝛽_𝑖,𝑑)] and using posterior distributions (27) and (28), we get the Bayes estimates of 𝛽_𝑖,𝑖=1,2 using General Entropy loss function, respectively, as 𝛽𝛽^{∗∗𝑖𝐸}=𝐸𝛽^−𝑞3_𝑖^−1/𝑞3,𝑖=1,2,_∗∗1𝐸=𝑘⁴ℎ₂(𝑧)^𝑛−1_𝑚=1𝐽_1𝑚𝐺_1𝑚^−1/𝑞3𝐽_1𝑚,=^∞−∞𝛽^−𝑞3₁𝑒^{−(1/2𝜎2)𝛽12(𝑆𝑚1+1)+𝛽1𝑆𝑚3/𝜎2}𝑑𝛽₁=1×𝑆𝑘_𝑚15𝑚+1/𝜎²−1+𝑞₃Γ1−𝑞³21^𝐹1𝑞³2,12,−𝑠^2𝑚3/𝜎²+𝐺2𝑆_𝑚1+1_3𝑚Γ1−𝑞³2Hypergeometric1^𝐹×1+𝑞₃12,32,−𝑠^2𝑚3/𝜎²+𝐺2𝑆_𝑚1+1𝜎²_4𝑚×H1ypergeometric𝑝^𝐹𝑞,−2,1𝑠,_2𝑚31−𝑞2³,23−𝑞2³,/𝜎²2𝑆_𝑚1+1𝜎²,(49) where 𝑘_5𝑚=(−1)^−𝑞32^{(1/2)(−1−𝑞3)}−𝑆^𝑚1+1²𝑠^2𝑚3^−𝑞3×−𝑠^2𝑚3𝑆_𝑚1+1²^−𝑞3exp𝑠^2𝑚3/𝜎²×𝑆2𝑆_𝑚1+1_𝑚1+1𝜎²^𝑞3/2×−1+𝑞₃𝑆_𝑚1+1𝜎²(−1)^𝑞3−𝑆^𝑚1+1𝑆_𝑚3^𝑞3×−𝑆^𝑚3𝑆_𝑚1+1^𝑞3𝐺+1_3𝑚,=√2𝑆^𝑚3𝜎²(−1)^𝑞3−𝑆^𝑚1+1𝑆_𝑚3^𝑞3−𝑆𝑆_𝑚1+^𝑚31^𝑞3𝐺−1_4𝑚,=2^(1+𝑞3)/2⎡⎢⎢⎣−𝑆_𝑚1+1²𝑠^2𝑚3^𝑞3⋅𝑆^𝑚3𝜎²×(−1)^𝑞3−𝑆𝑆_𝑚1+^𝑚31^𝑞3−𝑆^𝑚3𝑆_𝑚1+1^𝑞3.(50) Hypergeometric 1^𝐹1((1+𝑞₃)/2,3/2,−(𝑠^2𝑚₃/𝜎²)/2(𝑆_𝑚1+1)𝜎²) and hypergeometric 𝑝^𝐹𝑞({1/2,1},{(1−𝑞₃/2,3/2−𝑞₃/2},−(𝑠^2𝑚₃/𝜎²)/2(𝑆_𝑚1+1)𝜎²) are hypergeometric functions same as explained in Notes 1 and 2, respectively. 𝑘_5𝑚,𝐺_3𝑚,𝐺_4𝑚, and 𝐺_1𝑚are 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). 𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1, and 𝑆_𝑚2 are as given in (4) 𝛽^∗∗2𝐸=𝑘⁴ℎ₂(𝑧)^𝑛−1_𝑚=1𝐽_2𝑚𝐺_2𝑚^−1/𝑞3𝐽_2𝑚,=^∞−∞𝛽^−𝑞3₂𝑒^{−(1/2𝜎2)𝛽22(𝑆𝑛1−𝑆𝑚1+1)+𝛽2(𝑆𝑚4/𝜎2)}𝑑𝛽₂=1×𝑆𝑘_𝑛1−_6𝑚𝑆_𝑚1+1/𝜎²−1+𝑞₃⋅Γ1−𝑞³2Hypergeometric1^𝐹×𝑞₃12,21,−𝑠^2𝑚4/𝜎²+𝐺2𝑆_𝑛1−𝑆_𝑚1+1_5𝑚Γ1−𝑞³2Hypergeometric1^𝐹×1+𝑞₃12,32,−𝑠^2𝑚4/𝜎²+𝐺2𝑆_𝑛1−𝑆_𝑚1+1_6𝑚×H1ypergeometric𝑝^𝐹𝑞−2,1,𝑠_2𝑚41−𝑞2³,23−𝑞2³,/𝜎²2𝑆_𝑛1−𝑆_𝑚1+1,(51) where 𝑘_6𝑚=(−1)^−𝑞32^{(1/2)(−1−𝑞3)}×−𝑆^𝑛1−𝑆_𝑚1+1²𝑠^2𝑚4^−𝑞3×−𝑠^2𝑚4𝑆_𝑛1−𝑆_𝑚1+1²^−𝑞3×exp𝑠^2𝑚4/𝜎²×𝑆2𝑆_𝑛1−𝑆_𝑚1+1_𝑛1−𝑆_𝑚1+1𝜎²^𝑞3/2×−1+𝑞₃𝑆_𝑛1−𝑆_𝑚1+1𝜎²×(−1)^𝑞3−𝑆^𝑛1−𝑆_𝑚1+1𝑆_𝑚4^𝑞3×−𝑆^𝑚4𝑆_𝑛1−𝑆_𝑚1+1^𝑞3𝐺+1_5𝑚,=√2𝑆^𝑚4𝜎²(−1)^𝑞3−𝑆^𝑛1−𝑆_𝑚1+1𝑆_𝑚4^𝑞3×−𝑆^𝑚4𝑆_𝑛1−𝑆_𝑚1+1^𝑞3𝐺−1_6𝑚,=2^(1+𝑞3)/2⎡⎢⎢⎣−𝑆_𝑛1−𝑆_𝑚1+1²𝑠^2𝑚4^𝑞3𝑆_𝑚4𝜎²×(−1)^𝑞3−𝑆_𝑛1−𝑆𝑆^𝑚4_𝑚1+1^𝑞3−𝑆^𝑚4𝑆_𝑛1−𝑆_𝑚1+1^𝑞3⎤⎥⎥⎦.(52) Hypergeometric 1^𝐹1((1+𝑞₃)/2,3/2,−(𝑠^2𝑚₃/𝜎²)/2(𝑆_𝑚1+1)𝜎²) and hypergeometric 𝑝^𝐹𝑞({1/2,1},{1−𝑞₃/2,(3/2)−𝑞₃/2},−(𝑠^2𝑚₃/𝜎²)/2(𝑆_𝑚1+1)𝜎²) are hypergeometric functions same as explained in Notes 1 and 2, respectively. 𝑘_6𝑚,𝐺_5𝑚,𝐺_6𝑚, and 𝐺_2𝑚 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). 𝑆_𝑚3,𝑆_𝑚4,𝑆_𝑚1, and 𝑆_𝑚2 are as given in (4).

Remark 1. Putting 𝑞₃=−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 model𝑦𝑦_𝑡=3𝑥_𝑡+𝜀_𝑡,𝑡=1,2,3,4,_𝑡=3.5𝑥_𝑡+𝜀_𝑡,𝑡=5,6,…,15,(53) where 𝜀_𝑡's are i.i.d. 𝑁(0,1) random errors. We take the first 15 values of 𝑥_𝑡 and 𝜀_𝑡from Table 4.1 of Zellner [13] to generate 15 sample values (𝑥_𝑡,𝑦_𝑡)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.

We have calculated posterior mean, posterior median and posterior mode of m. The results are shown in Table 2.

Table 2 

The values of Bayes estimates of shift point, regression coefficients and precision.

We also compute the Bayes estimators m_E of 𝑚 using (40) for unknown 𝜎² and (44) for known 𝜎² and 𝑚_𝐿 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.

Table 3 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 Table 3 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.

For 𝑞₁=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 estimators 𝛽^1∗∗_L,𝛽^∗∗2𝐿,𝛽^∗∗1𝐸, and 𝛽^∗∗2𝐸 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.

Tables 3 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 Sections 4 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.

Table 5 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 Sections 4 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,𝑚^∗𝐿,𝑚^∗𝐸 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 each β₁, 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.

Table 6 leads to conclusion that performance of 𝑚^∗𝐿,𝑚^∗𝐹, 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 𝑚^∗𝐿 are closed to actual values of 𝑚. 66% values of 𝑚^∗𝐸 are closed to actual values of 𝑚.

Table 6 

Frequency distributions of the Bayes estimates of the change point.

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 of 𝑚 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.

Acknowledgments

The authors would like to thank the editor and the referee for their valuable suggestions which improved the earlier version of the paper.