Description
Three problems:
4.8 (textbook) (Note: The reference to “simulation time” in part (c) refers to a value of T
potentially different than T in parts (a) and (b) (not to, say, computer CPU time)).
A. The Kantorovich inequality mentioned in class lecture (and handout slides) relates 1 () E X to
E X (1 ) for a positive random variable X. In particular, if 0 < m ≤ X ≤ M < ∞ for some positive
constants m ≤ M, the Kantorovich inequality states that
2 1( ) 1
4 ( ) m M EXE
X mM
+
≤ ≤ .
Suppose that for all i it is known that the samples τi in a regenerative process satisfy 0 < m ≤ τi ≤
M and that the samples Ri ≥ 0. Further, suppose that the τi and Ri are independent of each other
for all i (so the sample path for collecting the τi is distinct and independent of the sample path for
collecting the Ri). Do the following:
(a) Use the Kantorovich inequality to derive an upper bound to the magnitude of the bias in the
regenerative estimate l
ˆin terms of m and M.
(b) What are the numerical values of the upper bound in part (a) as a function of the ratio M m
for M m = 1.1, 1.5, and 2.0? Given the answers as a fraction of the true value l. Also give a
brief (~2 sentence) interpretation of the results relative to potential practical applications.
B. Reproduce and expand on the results in slide 17 of the online class handout,
Chap4_bootstrap_handout.pdf (use current version at the website). In particular do
parts (a) and (b):
(a) Generate an original sample of 5000 replicates of sample means (each the average of 25
points) and give a histogram (like the left plot on slide 17). Then resample from the original
sample (i.e., from the 5000 sample means) according to the EDF and produce a histogram of
the bootstrap sample (like the right plot on slide 17). Note that the bootstrap samples do not
need to be built from the individual points going into the sample means; rather, the bootstrap
samples should come directly from the sample means themselves.
(b) Compare the two-sided 95% uncertainty bounds (i.e., 2.5% in each tail) centered around 0 of:
(i) original sample, (ii) bootstrap sample, (iii) central limit theorem. Provide a brief
discussion on how the three bounds differ and why they are like they are. Use the true
variance, not the estimated variance, for part (iii). Note: The problem asks you to center
around 0, which is convenient when considering the central limit theorem. However, it is
trivial to modify the bounds to shift to the true mean 0.5. Please submit your answers
centered around 0.
