Instructions: Solutions to problems 1 and 2 are to be submitted on Blackboard (PDF les
strongly preferred) { the deadline is 11:59pm on March 16. You are strongly encouraged to
do problems 3 through 6 but these are not to be submitted for grading.
Problems to hand in:
1. Suppose that D1; ;Dn are random directions { we can think of these random variables
as coming from a distribution on the unit circle f(x;y) : x2 + y2 = 1g and represent each
observation as an angle so that D1; ;Dn come from a distribution on [0;2 ).
(Directional or circular data, which indicate direction or cyclical time, can be of great interest
to biologists, geographers, geologists, and social scientists. The de ning characteristic of such
data is that the beginning and end of their scales meet so, for example, a direction of 5 degrees
is closer to 355 degrees than it is to 40 degrees.)
A simple family of distributions for these circular data is the von Mises distribution whose
density on [0;2 ) is
where 0 0 and 0 0 and
> 0, use the results of your simulation to estimate the value of a and . (Hint: If dVar(b n)
is a Monte Carlo estimate of the variance of the half sample mode for a sample size n, then
ln(dVar(b n)) ln(a) ln(n);
which should allow you to estimate a and .)
(c) It can be shown that Xn is independent of b n Xn where b n is the half sample mode.
(This is a property of the sample mean of the normal distribution { you will learn this in
STA452/453.) Show that we can use this fact to estimate the density function of b n from
the Monte Carlo data by the \kernel" estimator
where 2 = Var(Xi), (t) is the N(0;1) density, and D1; ;DN are the values of b n
Xn from the Monte Carlo simulation. Use the function density (with the appropriate
bandwidth) to give a density estimate for the half sample mode for n = 100.
Supplemental problems (not to hand in):
3. Suppose that X1; ;Xn are independent random variables with density or mass func-
tion f(x; ) and suppose that we estimate using the maximum likelihood estimator b ; we
estimate its standard error using the observed Fisher information estimator
where ‘0(x; );‘00(x; ) are the rst two partial derivatives of lnf(x; ) with respect to .
Alternatively, we could use the jackknife to estimate the standard error of b ; if our model
is correct then we would expect (hope) that the two estimates are similar. In order to
investigate this, we need to be able to get a good approximation to the \leave-one-out"
estimators fb ig.
(a) Show that b i satis es the equation
(b) Expand the right hand side in (a), in a Taylor series around b to show that
(You should try to think about the magnitude of the approximation error but a rigorous
proof is not required.)
(c) Use the results of part (b) to derive an approximation for the jackknife estimator of the
standard error. Comment on the di erences between the two estimators - in particular, why
is there a di erence? (Hint: What type of model { parametric or non-parametric { are we
assuming for the two standard error estimators?)
(d) For the air conditioning data considered in Assignment #1, compute the two standard
error estimates for the parameter in the Exponential model (f(x; ) = exp( x) for
x 0). Do these two estimates tell you anything about how well the Exponential model ts
the data?
4. Suppose that X1; ;Xn are independent continuous random variables with density
f(x; ) where is real-valued. We are often not able to observe the Xi’s exactly rather only
if they belong to some region Bk (k = 1; ;m); an example of this is interval censoring in
survival analysis where we are unable to observe an exact failure time but know that the
failure occurs in some nite time interval. Intuitively, we should be able to estimate more
e ciently with the actual values of fXig; in this problem, we will show that this is true (at
least) for MLEs.
Assume that B1; ;Bm are disjoint sets such that P(Xi2[mk=1Bk) = 1. De ne independent
discrete random variables Y1; ;Yn where Yi = k if Xi2Bk; the probability mass function
of Yi is
Under the standard MLE regularly conditions, the MLE of based on X1; ;Xn will have
variance approximately 1=fnIX( )g while the MLE based on Y1; ;Yn will have variance
approximately 1=fnIY ( )g.
(a) Assume the usual regularity conditions for f(x; ), in particular, that f(x; ) can be
di erentiated with respect to inside integral signs with impunity! Show that IX( ) IY ( )
and indicate under what conditions there will be strict inequality.
Hints: (i) f(x; )=p(k; ) is a density function on Bk.
(ii) For any function g,
g(x)f(x; )p(k; ) dx:
(iii) For any random variable U, E(U2) [E(U)]2 with strict inequality unless U is constant.
(b) Under what conditions on B1; ;Bm will IX( ) IY ( )?
5. In seismology, the Gutenberg-Richter law states that, in a given region, the number of
earthquakes N greater than a certain magnitude m satis es the relationship
log10(N) = a b m
for some constants a and b; the parameter b is called the b-value and characterizes the seismic
activity in a region. The Gutenberg-Richter law can be used to predict the probability of
large earthquakes although this is a very crude instrument. On Blackboard, there is a le
containing earthquakes magnitudes for 433 earthquakes in California of magnitude (rounded
to the nearest tenth) of 5.0 and greater from 1932{1992.
(a) If we have earthquakes of (exact) magnitudes M1; ;Mn greater than some known m0,
the Gutenberg-Richter law suggests that M1; ;Mn can be modeled as independent random
variables with density
f(x; ) = exp( (x m0)) for x m0.
where = b ln(10). However, if the magnitudes are rounded to the nearest then they
are e ectively discrete random variables taking values xk = m0 + =2 +k for k = 0;1;2;
with probability mass function
p(xk; ) = P(m0 +k M 0 and > 0 are hyperparameters. What is the posterior distribution of given
XAA = x1, XAa = x2, and Xaa = x3