Homework 7 in STAT630: Advanced Statistical Data Analysis @ CSU
[3pts] Consider a random variable \(X\) that is distributed with a gamma distribution with density \[ f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha -1} \exp(-\beta x). \] for \(x > 0\) and \(\alpha, \beta > 0\). Use the fact that \(E[X] = \alpha/\beta\) and \(Var[X] = \alpha/\beta^2\) to obtain method-of-moments estimators for \(\alpha\) and \(\beta\) in terms of the sample mean and sample variance.
[8pts] Assume the data in the file dat.csv are from a gamma distribution with unknown \(\alpha\) and \(\beta\). Use your method-of-moments estimators to obtain estimates \(\hat \alpha\) and \(\hat \beta\). Then, draw 1000 bootstrap samples of these estimators. Create 95% bootstrap-based confidence intervals for both \(\alpha\) and \(\beta\) using both the percentile method and the basic method. Report these confidence intervals.
[8pts] Create a 95% bootstrap based confidence interval using the studentized method (write your own function to perform this).
[8pts] Create a 95% bootstrap based confidence interval using the BC (not BCa) method (write your own function to perform this).
[8pts] Create 95% bootstrap based confidence interval using the BCa method.
[10pts] Design a simulation experiment to assess the coverage rates of the percentile method, the basic method, the studentized, the BC method, and the BCa method for a gamma distribution with \(\alpha = 5\) and \(\beta = 1\). Report the coverage rates of these methods.
[5pts] On homework 5, you investigated both asymptotic confidence intervals for maximum likelihood and profile likelihood based confidence intervals for a gamma distribution. Of all the methods you’ve investigated (both on homework 5 and here on homework 9) which method would you recommend for producing confidence intervals for the parameters of the gamma distribution? Explain your choice.