Question

Let \(X_{1}, X_{2}, \ldots, X_{10}\) be a random sample of size \(n=10\) from a gamma distribution with \(\alpha=3\) and \(\beta=1 / \theta\). Suppose we believe that \(\theta\) has a gamma distribution with \(\alpha=10\) and \(\beta=2\). (a) Find the posterior distribution of \(\theta\). (b) If the observed \(\bar{x}=18.2\), 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 for \(\theta\). Would it be easier to find one having equal tail probabilities? Hint: Can the posterior distribution be related to a chi-square distribution?

Short answer

The posterior distribution of \(\theta\) is a gamma distribution with parameters \(\alpha=40\) and \(\beta= 2+18.2*10\). The Bayes point estimate associated with square-error loss function is \(40 / (2+18.2*10)\) and using the mode of the posterior distribution is \((40-1) / (2+18.2*10)\). The HDR interval would not likely have equal tail probabilities due to the skewness of the gamma distribution.

Step-by-step solution

Find the Posterior Distribution

Given: prior distribution of \(\theta\) is a gamma distribution = \(Gamma(10,2)\), likelihood function based on the sample is \(Gamma(3,1/\theta)\). \n\n Using Bayes’ rule, the posterior distribution is proportional to the product of the likelihood and the prior. Hence, \n posterior \(\propto\) likelihood \(\times\) prior \n \n This simplifies to: \(\Gamma(10+30, 1/ (2+18.2*10))\). Therefore, the posterior distribution of \(\theta\) is a gamma distribution with parameters \(\alpha=40\) and \(\beta= 2+18.2*10\).

Bayes Point Estimate for Square-Error Loss Function

For a Gamma distribution, the expected value E(\(\theta\)) is given by \(\alpha / \beta \). Therefore, with the square-error loss function, the Bayes estimate of \(\theta\) is the mean of the posterior distribution. So, the Bayes point estimate associated with square-error loss function is \(40 / (2+18.2*10)\).

Bayes Point Estimate Using the Mode of Posterior Distribution

The mode (peak) of a gamma distribution is given by \((\alpha-1)/\beta\) if \(\alpha > 1\). Here, since \(\alpha = 40\), which is greater than 1, we can calculate the mode of the posterior distribution, which gives the Bayes point estimate. So, the Bayes estimate of \(\theta\) is equal to \((40-1) / (2+18.2*10)\).

HDR Interval Estimate for \(\theta\)

Finding an HDR (Highest Density Region) interval requires finding the region under the posterior density with highest probability. The easiest way to accomplish this is by finding an interval with equal tail probabilities. Because the gamma distribution is skewed, not symmetric, intervals with equal tail probabilities are not the same as highest-density intervals. Since gamma distribution is a special case of chi-square distribution, one can relate this problem to chi-square distribution to determine such intervals.