Feel free to ask me any questions or use your text book as a reference. You can also
use wolfram alpha to do any calculations, in addition to R of course. If you
want to use any other resource, run it by me rst.
1. Consider again a shifted exponential distribution but with > 0 unknown.
f(x) = 1 e (x ) 1(x )
Suppose we observe n observations from this distribution.
a. Derive the MLEs for both and .
b. Show that the MLEs are jointly su cient.
c. Derive the expected values of the estimators. Are they unbiased?
d. Give a one-to-one function of the MLEs that are unbiased.
2. Suppose that we have bivariate data (x;y) as we did in the linear model, but
here Y is binary, either 0 or 1. We might think to model the probability p that
Y = 1, but it is clear a linear model should not be used, as we have constraints
on p that the linear model would not respect. Consider the model
log( pi1 pi) = xi where pi = P(Yi = 1).
a. Assuming independence of the Yi, we know they have a bernoulli(pi) distri-
bution. Write down the likelihood for .
b. Compute the probability of a ‘success’ if Xi = 0.
c. Write down the rst derivative of the log-likelihood, and realize that you
wouldn’t be able to solve it (something like Newton-Raphson would need to be
used).
3. I have a large jar of coins at home. One day I decide that I would like to
know how much money the jar contains, so I begin to count it. I count up the
total number of quarters (47), and decide that I have no interest in continuing
this.
Consider the hypothesis that:
i) change is always given optimally, i.e. the fewest number of coins, so for 32
cents in change, one quarter, one nickel and two pennies is optimal rather than
three dimes and 2 pennies
ii) the change amount on any purchase is a discrete uniform. distribution over
the possibilities (from .00 to .99)
iii) I always pay in cash only (no change)
a. Under this hypothesis, estimate the total amount of money in the jar.
b. Construct a 95 percent con dence interval for the total number of transac-
tions that produced this jar of change, as well as a CI for the total amount of money in the jar.
c. I nally count the money, and it yields 58 pennies, 25 dimes and 10 nickels.
Do a goodness of t test for the model described above in (i)-(iii).
4. The following data set of n = 20 was generated from either a standard normal
distribution or a t-distribution with unknown degrees of freedom .
-4.3 1.3 -0.4 0.8 -0.1 -3.1 0.0 0.6 -0.6 0.1
4.2 0.3 1.7 -1.3 -0.2 -0.5 1.0 -0.4 2.2 -0.8
a. Run a K-S test on the data to see if it is plausible that the data comes from
a standard normal distribution.
b. Considering what we know about the proportion of points falling within 2
standard deviations from a normal distribution, compute a goodness of t test
for the hypothesis that this data is standard normal.
c. Use the binomial distribution to compute the true p-value (the above relies
on the asymptotic distribution of 2 log , which may not be very good for n
‘small’).
d. The variance of a t-distribution with degrees of freedom can be shown to
be 2: Using this information, estimate using a frequentist approach (either
MLE or MoM, which one would better utilize the given information?).
e. Using a non-informative (improper prior) of p( ) = 1 , and recalling that the
pdf of a t-distribution is given as
f(x) = (+12 )(n )1=2 ( 2 )(1 +x2)( +1)=2;
write down the kernel of the posterior distribution.
f. Compute the asymptotic variance of the estimate in (d) using the delta
method (as its a function of a sample mean of some random variable). So that
we all have the same notation, de ne k = E(Xk).
Do one of the following options:
h. Using a plug in method (everywhere you see a in the variance, plug in
^ ), compute an approximate 95 percent con dence interval for based on your
estimate in (d). You might want to just approximate higher order moments
from a t-distribution using simulation, the calculus isn’t so easy.
i. Construct a bootstrap con dence interval for using the quantile method via
a non-parametric bootstrap (generate many ^ values from bootstrap samples
and throw away the smallest and largest ( =2)100% of them).
5. Write at most 6 sentences discussing the relative merits of frequentist vs
bayesian point estimation.