Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

670 Bayesian Statistics

has at-distribution with 2α+ndegrees of freedom. Of course, using these degrees

of freedom, we can findtγ/ 2 so that

xk±tγ/ 2

sk

√

nk

is an HDR credible interval estimate forθ 1 with probability 1−γ. Naturally, it falls
upon the Bayesian to assign appropriate values toα, β, n 0 ,andθ 0. Small values of
αandn 0 with a large value ofβwould create a prior, so that this interval estimate
would differ very little from the usual one.

Finally, it should be noted that when dealing with symmetric, unimodal pos-
terior distributions, it was extremely easy to find the HDR interval estimate. If,
however, that posterior distribution is not symmetric, it is more difficult and often
the Bayesian would find the interval that has equal probabilities on each tail.

EXERCISES

11.2.1.LetX 1 ,X 2 be a random sample from a Cauchy distribution with pdf

f(x;θ 1 ,θ 2 )=

(

1

π

)

θ 2

θ 22 +(x−θ 1 )^2

, −∞<x<∞,

where−∞<θ 1 <∞, 0 <θ 2. Use the noninformative priorh(θ 1 ,θ 2 )∝1.

(a)Find the posterior pdf ofθ 1 ,θ 2 , other than the constant of proportionality.

(b)Evaluate this posterior pdf ifx 1 =1,x 2 =4forθ 1 =1, 2 , 3 ,4andθ 2 =

0. 5 , 1. 0 , 1. 5 , 2 .0.

(c)From the 16 values in part (b), where does the maximum of the posterior pdf

seem to be?

(d)Do you know a computer program that can find the point (θ 1 ,θ 2 )ofmaximum?

11.2.2. LetX 1 ,X 2 ,...,X 10 be a random sample of sizen= 10 from a gamma
distribution withα=3andβ=1/θ. Suppose we believe thatθhas a gamma
distribution withα=10andβ=2.

(a)Find the posterior distribution ofθ.

(b)If the observedx=18.2, what is the Bayes point estimate associated with

square-error loss function?

(c)What is the Bayes point estimate using the mode of the posterior distribution?

(d)Comment on an HDR interval estimate forθ. Would it be easier to find one

having equal tail probabilities?

Hint:Can the posterior distribution be related to a chi-square distribution?