Homeworks

Homework 1, Due April 9th, 2004: to hand in before class- Hints are here!

From the text:
Page 15, 2.4.
Page 28-30 : 3.1, 3.5, 3.10.

Hints for Efron & Tibshirani, Problem 2.4

We want to calculate

$$P(\mbox{ median is equal to a certain} x_{(i)}).$$

We do this by calculating the difference between probabilities of getting a median that is greater than or equal to $x_{(i-1)}$ and getting a median that is greater than or equal to $x_{(i)}$. That is,

$$P(X^*_{(4)} = x_{(i)}) = P(x^*_{(4)} \geq x_{(i-1)})-P(x^*_{(4)} \geq x_{(i)})$$

.

To find

$P(x^*_{(4)} \geq x_{(i)}),$ (1)

1. Think what is the ``minimum requirements'' for $x^*$ such that its median is larger than a number.
2. Calculate the probability in (1). How many ways you could get a bootstrap sample that fulfill the ``minmum requirements'' This should be
   
   $$\sum^3_{j=0} (Bi(j; n, \frac{i}{n}))$$
   
   

Homework 2, Due April 21st, 2004: to hand in before class From text: 5.5,6.3,6.4,6.8,6.10. New hints for Numbers 6.3 and 6.10

Hint for 6.3:
For the variance question, if you can't do it analytically, explain in words why you believe this inequality to be true. (Remember about variance reduction in Monte Carlo techniques in general, and that a good idea is often to condition on the sample.)

Be careful we corrected the values in 6.10 so that there are only 8 of them, and they are: 1.2, 3.5, 4.7,7.3, 8.6,12.4,13.8,18.1.

Hint: If you want to try the gray code idea here, there is the full gray code for 8 values here

Homework 4, Due May 26th, 2004: to hand in before class

1. Turn in the code of 4 functions that you have written yourself for your final project. The functions can be in R or Matlab, but must be clearly commented so an outsider can read them and understand what they do.

   The grade will be assigned according to clarity and modularity of the programming, not necessarily speed of execution, unless you can show that you have encountered a intractibility issue in your computing.

2. Give 3 websites that you found useful in compiling programs in matlab or R for running Monte Carlo simulations (parametric bootstrapping) or randomized hypothesis testing or bootstrapping.