Project 8

STAT 605: Advanced Statistical Computations
David B. Dahl
Spring 2010

Poisson Regression

Suppose that y_1,...y_n are indepedent and distributed Poisson(exp(beta) * x_i), for i=1,...,n. Interest lies in making inference regarding beta. Consider the following estimators for beta:

  1. Maximum likelihood estimation
  2. Mean of the posterior distribution where the prior distribution on beta is a normal distribution with mean 0.4 and standard deviation of 0.5. Note that the posterior mean is the Bayes estimate under squared-error loss.
  3. Mode of the posterior distribution where the prior distribution on beta is uniform on -2 to 2. Note that the posterior mode is the Bayes estimate under 0-1 loss.
Your task is to conduct a simulation study in a script named "simulation.R" that explores the long run relative-frequency performance of these estimators. Specifically, you are to estimate the bias and mean squared error (MSE) of the estimators of beta when the true value of beta varies from 0 to 0.8 in increments of 0.05 when x_1, ..., x_n are: 0.91, 2.34, 2.54, 1.57, 1.63, 1.08, 0.30, 0.90, 0.08, 2.01, 1.5, 1.2. In one plot, produce a publication-ready PDF file named "mse.pdf" showing the MSE vs. beta for each of the three estimators. Do the same for the bias and name it "bias.pdf". Write a one paragraph summary in "summary.txt" that gives your high-level conclusions from this simulation study.

Submission

Put the files "simulation.R", "mse.pdf", "bias.pdf", and "summary.txt" (with the correct names!) in your Project 8 directory.