1. In this question we will explore the connections between clustering and strati cation.
Suppose that we have test scores X, observed at the level of an individual student i.
Children belong to classrooms c, and the distribution of test scores across students
obeys
xic = c +"ic:
The marginal distribution of c has mean 0 and variance 2 . The marginal distribution
of "ic has mean 0 and variance 2". Furthermore, "ic??"jc for all i6= j and "ic??"jc0
for all i;j and c6= c0.
(a) Suppose that there are only k classrooms of equal size in the entire school system,
so the same classrooms are visited in each possible sample. Assume there are a
large number of students within each classroom, and m students are randomly
sampled within each classroom.
i. What is the variance of a single observation Xic? What is the variance of
Xic, conditional on the classroom c?
ii. What is the covariance between two observations within the same classroom?
iii. What is the variance of the sum of all m observations within a given classroom
c, Pmi=1Xic?
iv. Let x c be the mean within classroom c. Express xSTRAT as a weighted
average of the classroom-speci c means.
v. What is the variance xSTRAT?
(b) Now suppose there is a large set of classrooms and we randomly sample k of
them. There are exactly m students within each classroom, and if a classroom is
sampled, test scores for each of them are observed. Let the sample mean over all
km observations be xCLUST.
i. What is the variance of a single observation Xic? Why does it not make sense
to compute the conditional variance of c?
ii. What is the covariance between two observations within the same classroom?
iii. What is the variance of the sum of all m observations within a given classroom
c, Pmi=1Xic?
iv. Let x c be the mean within classroom c. Express xCLUST as an average of
the classroom-speci c means.
v. What is the variance xCLUST?
2. Consider a population with S > 1 strata. The mean and variance of a variable X
within each stratum are s and 2s, and each stratum constitutes a share s of the
population, where PSs=1 s = 1.
What is the optimal way to allocate a sample of size n to strata in order to minimize
the variance of the resulting sample mean? [Hint: you can pose this as a smooth
constrained optimization problem, if you ignore integer constraints. Let !s be the
fraction of the sample observations allocated to stratum s, and express the variance of
the strati ed mean in terms of the !s’s. The constraint is that they have to sum to 1.]
3. Recall that if X U(0; 0), then the ML estimate is b ML = maxfX1;:::Xng.
(a) Find the method of moments estimator, b MM.
(b) Which estimator has a lower mean square error?
4. Let X follow an exponential distribution with rate , i.e fX(x) = exp[ x] for x> 0.
We showed in class that the method of moments estimator, b MM, was biased. Write
a Monte Carlo simulation in STATA to estimate the bias of the MM estimate when
n = 100. Use B = 10000 replications and a seed of 11072017.
5. Wackerley, Ch. 8, ex. 8.17.
6. Wackerley, Ch. 9, ex. 9.25.
7. Let m be a positive integer, and consider an iid sample of size n = 2m + 1 from the
Laplace distribution, which has the following density:
f(xj ) = 12 exp[ jx j] (1)
(a) Let X(1) X(2) ::: X(n) be the order statistics from the sample. Express
the the sample median x in terms of these order statistics.
(b) Let 1 j m. Show that for any ,
jX(m+1 j) j+jX(m+1+j) j jX(m+1 j) x j+jX(m+1+j) x j(2)
(c) Use the above fact to show that x is the maximum likelihood estimate of .
This version: November 9, 2017