Question

Note that if τ ∼ Gamma (a, b) then σ² ∼ Inverse Gamma(a,b). The PDF of the Inverse Gamma(a,b) for σ² is

f(σ²)= ((σ²)⁽⁻ᵃ⁻¹⁾e⁽−ᵇ/σ²⁾)/(bᵃΓ(a))
where a and b are called hyper-parameters The prior mean and variance are
E(σ²) = b/a − 1 for a > 1
V (σ²) = b²/(a − 1)²(a − 2) for a > 2

(a) Find the posterior pdf for σ²is

(b) Use the data in the class example, and assume known mean = 1.8. Use the Inverse Gamma(1,1) as the prior of σ²
Find the posterior distribution of σ²
Plot the prior and the posterior pdf Find the point estimate of σ²
Find the plot of the 95% HPD of σ²
Test H0 : σ² ≥ 0.2 vs H1 : σ² < 0.2

(c) Compare the results with those in the class example.
Hint: use library(pscl) for inverse gamma distribution

Answer 1

To find the posterior pdf for σ², apply Bayes' theorem with the given prior and likelihood. Plot the prior and posterior pdfs, determine the point estimate of σ², and test the hypothesis. Finally, compare the results with the class example.

Step-by-step explanation:

To find the posterior pdf for σ², we need to apply Bayes' theorem. Given that the prior distribution is Inverse Gamma(a=1, b=1), the likelihood function is based on the class example and assumes a known mean of 1.8. Using the given formula for the posterior pdf of Inverse Gamma(a,b), we can calculate the values for the posterior distribution. By plotting the prior and posterior pdfs, we can visualize the change in the distribution. The point estimate of σ² can be determined by taking the mean of the posterior distribution. To test the hypothesis H0: σ² ≥ 0.2 vs H1: σ² < 0.2, we can compute the 95% Highest Posterior Density (HPD) interval of σ² and compare it with the hypothesis. By comparing the results with those in the class example, we can analyze any similarities or differences.