1. Chapter 4. Supplementary Questions

2. Question 1. Consider the systematic sample estimator based on the trapezoidal rule:
Discuss the bias and variance of this estimator.
In the case , how does it compare with other estimators such as crude Monte Carlo and antithetic random numbers requiring n function evaluations.
Are there any disadvantages to its use?

3. -
can not be calculated by the sample variance because ‘s are not independent.

4. The difference (ARN) is 2.6667e-004, almost 0.
Results
Actual value is
The difference (SS) is
The difference (MC) is
The difference (ARN) is e-004, almost 0.
Disadvantage : careful to calculate the variance of the estimator N
Crude MC
S. S
Antithetic 50
0.3217
0.3283
0.3336
100
0.3865
0.3300

5. Question 2. For any random variables , , prove that for all x, y.

6. Proof.
Case 1. X and Y are independent. Obvious!
Case 2. X and Y are not independent.

7. Question 4. Suppose we wish to generate the partial sum of independent identically distributed summands,
for
(a) is generated with having
(b) is generated with a student
What is the maximum possible correlation we can achieve between and ? What is the minimum correlation?

8. Theorem 40. (maximum/minimum covariance)
Suppose and are both non-decreasing (or both
non-increasing) functions. Subject to the constraint that
X, Y have cumulative distribution functions , respectively, the covariance
is maximized when and
and is minimized when and ,
where U~U[0,1].

9. Solution
Using common random number(CRN), the maximum correlation is obtained and
Using antithetic random number(ARN), the minimum correlation is obtained.
Let ,
So, ,

10. Algorithms (sigma=1, n=10, m=100000)
Generate
For CRN, x=norminv(u,0,1);
For ARN, x=norminv(1-u,0,1);
y=tinv(u,5);
Compute sums of x and y
Compute the covariance matrix
Compute Corr(sum(x),sum(y))

11. Results
Maximum of covariance is e+004 .
Minimum of covariance is e+005.
Thus, the maximum of correlation is and
the minimum of correlation is
When dof=2,
maximum = and minimum =