STA 2201 s 2011 Assignment One:

Your written answers and printouts are due at the beginning of class Friday on Jan 21. You may be asked to present your solution to the class before handing it in.

  1. Let X1, … Xn be a random sample from a normal distribution with mean θ and variance 1. The parameter space Θ consists of just two points: Θ = {0,1}. I know this is artificial, but it is intended to teach a lesson that will help with the next problem. Here are some data: 
     0.04  0.51 -0.34 -1.97  0.30 -0.48

    Calculate the maximum likelihood estimate of θ; your answer is a single number. Circle your answer. It is not the sample mean, because the MLE must be in the parameter space. I used R, but you don't absolutely have to.

  2. Let X1, … Xn be a random sample from a normal distribution with mean θ and variance θ2. Your job is to find the MLE of θ.
    1. It is possible to find the MLE with paper and pencil, and there is some benefit to giving it a try. But it's not the point of the assignment, and it's optional. So give it a try if you want and hand in your work if you do it, but the important thing is to get the answer numerically, as requested below.
    2. Using R, obtain the MLE of θ for the following data: 
       
-0.33  0.70  0.83  1.29  0.71  1.35  0.48  1.43 -0.22  1.46  1.07  2.90  1.02
-0.42  2.17 -0.91  1.84  0.98 -0.22  0.48  2.84  1.87  0.00  1.90  0.81  2.85
 2.74  2.31  0.31  2.82  1.19  1.52  0.01  0.59 -0.99  1.80  1.49  0.99  1.98
 0.85  1.79  0.21  2.10  0.39  0.97  0.53  0.60  0.17  0.15  2.45
    3. Using R, obtain the MLE of θ for the following data: 
       
-2.17 -2.43 -1.54 -1.09 -0.14  0.55 -1.34 -2.08 -0.36 -0.72 -0.98 -0.38 -0.68
-1.26 -0.17 -2.05 -2.22  0.31 -0.17 -1.01  1.23 -0.29 -1.51 -0.76  0.05 -3.51
-2.02 -1.54  0.17 -0.71 -0.99 -0.53 -0.10 -1.42 -0.92 -0.82 -0.54 -2.38 -0.56
 0.09 -0.41  0.95 -2.03 -0.99 -1.73  0.31 -1.27 -1.38 -1.10 -2.97

For the numerical work in Problem 2, please include the following in your printout.