11.1. Bayesian Procedures 66511.1.2.LetX 1 ,X 2 ,...,Xnbe a random sample from a distribution that isb(1,θ).
Let the prior of Θ be a beta one with parametersαandβ. Show that the posterior
pdfk(θ|x 1 ,x 2 ,...,xn)isexactlythesameask(θ|y) given in Example 11.1.2.11.1.3. LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution that is
N(θ, σ^2 ),where−∞<θ<∞andσ^2 is a given positive number. LetY=Xdenote
the mean of the random sample. Take the loss function to beL[θ, δ(y)] =|θ−δ(y)|.
Ifθis an observed value of the random variable Θ that isN(μ, τ^2 ), whereτ^2 > 0
andμare known numbers, find the Bayes solutionδ(y) for a point estimateθ.
11.1.4.LetX 1 ,X 2 ,...,Xndenote a random sample from a Poisson distribution
with meanθ, 0 <θ<∞.LetY =
∑n
1 Xi. Use the loss functionL[θ, δ(y)] =
[θ−δ(y)]^2 .Letθbe an observed value of the random variable Θ. If Θ has the prior
pdfh(θ)=θα−^1 e−θ/β/Γ(α)βα, for 0<θ<∞, zero elsewhere, whereα> 0 ,β> 0
are known numbers, find the Bayes solutionδ(y) for a point estimate forθ.
11.1.5.LetYn be thenth order statistic of a random sample of sizenfrom a
distribution with pdff(x|θ)=1/θ, 0 <x<θ, zero elsewhere. Take the loss
function to beL[θ, δ(y)] = [θ−δ(yn)]^2 .Letθbe an observed value of the random
variable Θ, which has the prior pdfh(θ)=βαβ/θβ+1,α<θ<∞, zero elsewhere,
withα> 0 ,β>0. Find the Bayes solutionδ(yn) for a point estimate ofθ.
11.1.6.LetY 1 andY 2 be statistics that have a trinomial distribution with param-
etersn, θ 1 ,andθ 2 .Hereθ 1 andθ 2 are observed values of the random variables Θ 1
and Θ 2 , which have a Dirichlet distribution with known parametersα 1 ,α 2 ,and
α 3 ; see expression (3.3.10). Show that the conditional distribution of Θ 1 and Θ 2 is
Dirichlet and determine the conditional meansE(Θ 1 |y 1 ,y 2 )andE(Θ 2 |y 1 ,y 2 ).
11.1.7.For Example 11.1.6, obtain the 95% credible interval forθ. Next obtain the
value of the mle forθand the 95% confidence interval forθdiscussed in Chapter 6.
11.1.8.In Example 11.1.2, letn=30,α= 10, andβ=5,sothatδ(y)=(10+y)/ 45
is the Bayes estimate ofθ.(a)IfY has a binomial distributionb(30,θ), compute the riskE{[θ−δ(Y)]^2 }.(b)Find values ofθfor which the risk of part (a) is less thanθ(1−θ)/30, the risk
associated with the maximum likelihood estimatorY/nofθ.11.1.9.LetY 4 be the largest order statistic of a sample of sizen=4froma
distribution with uniform pdff(x;θ)=1/θ, 0 <x<θ, zero elsewhere. If the prior
pdfoftheparameterg(θ)=2/θ^3 , 1 <θ<∞, zero elsewhere, find the Bayesian
estimatorδ(Y 4 )ofθ, based upon the sufficient statisticY 4 , using the loss function
|δ(y 4 )−θ|.
11.1.10.Refer to Example 11.2.3; suppose we selectσ^20 =dσ^2 ,whereσ^2 was known
in that example. What value do we assign todso that the variance of posterior is
two-thirds the variance ofY=X,namely,σ^2 /n?