The question is related to Bayesian statistics, with Matlab coding involved. Please answer questions with specific details, thank you.
Question 8. [3.5+4+4+3+3=17.5 points] Uniform-Pareto Model
Suppose Xi|0~fo(xi|0), i = 1, ...,n, where fo(xi|0) is a PDF given by fo(xi|0)=0-1I(0xi0). Equivalently, we may write Xi|~unif([0,]), i = 1,...,n. The prior for is a Pareto distribution whose PDF is
where a > 0 and b > 0 are fixed parameters. This is denoted as ~Pareto(a,b). Note that information in this paragraph should be used to answer the sub-questions below unless stated otherwise. (a) Write down the likelihood and identify a sufficient statistic of dimension one (a real number summary of the data). Note: This means that you should write the likelihood in a
form that depends on the observed data x only through a sufficient statistic of dimension
one.
(b) Write down the posterior PDF of up to a proportionality constant and identify the posterior distribution of (what is the name of the posterior distribution and what are its parameters?). (c) Suppose a = 1, b = 1, and the observed data are given in Table 8.1. Calculate the posterior mean, posterior median, and posterior mode of (d) Suppose a = 1, b = 1, and the observed data are given in Table 8.1. Determine the shortest 95% credible interval for (note that it can be determined without using Matlab). (e) Suppose a = 1, b = 1, and the observed data are given in Table 8.1. Plot the prior PDF g(0) and posterior PDF g(0|x) over the interval [0.01,15] in the same figure using Matlab. Give your Matlab code.
Table 8.1: Observed data for Question 8. 1 2 3 4 3.4038 6.8605 2.6565 1.3732
i
5
Xi
5.2964
