In this problem, we shall outline a form of robust linear regression. Assume throughout the exercise that the data consist of pairs (Yi, xi) for i = 1, . . . , n. Assume also that the xi €™s are all known and the Yi €™s are independent random variables. We shall only deal with simple regression here, but the method easily extends to multiple regression.
a. Let β0, β1, and σ stand for unknown parameters, and let a be a known positive constant. Prove that the following two models are equivalent. That is, prove that the joint distribution of (Y1, . . . , Yn) is the same in both models.
Model 1: For each i, [Yi ˆ’ (β0 + β1xi)]/σ has the t distribution with a degrees of freedom.
Model 2: For each i, Yi has the normal distribution with mean β0 + β1xi and variance 1/τi conditional on τi. Also, τ1, . . . , τn are i.i.d. having the gamma distribution with parameters a/2 and aσ2/2. Use the same argument that produced the marginal distribution of μ in Sec. 8.6 when μ and τ had a normal-gamma distribution.
b. Now consider Model 2 from part (a). Let η = σ2, and assume that η has a prior distribution that is the gamma distribution with parameters b/2 and f/2, where b and f are known constants. Assume that the parameters β0 and β1 have an improper prior with €œp.d.f.€ 1. Show that the product of likelihood and prior €œp.d.f.€ is a constant times

c. Consider (12.5.4) as a function of each parameter for fixed values of the others. Show that Table 12.7 specifies the appropriate conditional distribution for each parameter given all of the others.
Table 12.7 Parameters and conditional distributions for Exercise 8

(a+/2 1 ! n+-(1254) i=1 Parameter (12.5.4) looks like the p.d.f of this distribution gamma distribution with parameters (na + b)/2 and (f + a _1 %) /2 gamma distribution with parameters a + normal distribution with mean Bo precision Li ti normal distribution with mean precision 2 xf .1