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STAT 801
Problems
These problems serve a variety of purposes. One is practice with the
ideas from class. Some problems provide counterexamples and illustrate the
regularity conditions of theorems. Some problems introduce, without much
explanation, ideas we won't have time to consider in class. The first 3 are
review intended to let me see how well you explain things you understand
and to make sure the basics are already there.
SET A
- The concentration of cadmium in a lake is measured 17 times. The
measurements average 211 parts per billion with an SD of 15 parts per
billion. Could the real concentration of cadmium be below the standard of
200 ppb?.
- Consider a population of 200 million people of whom 200 thousand have a
certain condition. A test is available with the following properties.
Assuming that a person has the condition the probability that the test
detects the condition is 0.9. Assuming that a person does not have the
condition the test detects (incorrectly) the condition with probability 0.001.
A person is picked at random from the 200 million people and the test is
administered.
- What is the chance that test detects the condition for this randomly
selected person?
- Assuming that the condition is detected by the test for this randomly
selected person what is the chance that the person has the condition?
- A mandatory testing program is contemplated. If all 200 million are
tested about how many positive results should be expected? Of these
about how many will not have the condition?
- Suppose X and Y are independent Geometric(p) random variables.
In other words
![](img1.gif)
- Let
,
and W=V-U. Express the event
U=j and W=k in terms of X and Y.
- Compute
and
and prove that the event
U=j and the event W=k are independent.
- A computer is waiting for two flags to be set. Both flags start
out not set.
For each flag the conditional probability that the flag is set next cycle
given it is not yet set is 0.5. The two flags operate independently. How
many cycles should you expect to wait before both flags are set including
the cycle on which the last flag becomes set?
SET B
- Suppose X and Y have joint density
. Prove from the
definition of density that the density of X is
.
- Suppose X is Poisson(
). After observing X a coin landing
Heads with probability p is tossed X times. Let Y be the number of
Heads and Z be the number of Tails. Find the joint and marginal distributions
of Y and Z.
- Let
be the bivariate normal density with mean 0,
unit variances and correlation
and let
be the standard
bivariate normal density. Let
.
- Show that p has normal margins but is not bivariate normal.
- Generalize the construction to show that there rv's X and Y
such that X and Y are each standard normal, X and Y are
uncorrelated but X and Y are not independent.
- Warning: This is probably hard. Don't waste too much time on
it. Suppose X and Y are independent
and
random variables. Show that
is a
random variable.
- Suppose X and Y are independent with
and
. Let Z=X+Y.
Find the distribution of Z given X and that of X given Z.
SET C
- Suppose
are iid real random variables with
density f . Let
be the X 's arranged
in increasing order.
- Find the joint density of
.
- Suppose
. Prove that
is independent of
.
- Find the density of
.
- Find the density of
.
- Suppose
are iid exponential. Let
.
- Find the joint density of
.
- Find the joint density of
.
- Suppose
are iid N(
,
).
Let
. Let
.
- Develop a recurrence relation for
and
, expressing
and
in terms of
and
.
- Find the joint density of
.
- Generate data from N(0,1). By adding
to the data for
some large values of k compare the numerical performance of these
recurrence relations to that of the one pass formula using
,
and the usual computing formulas
for the sample variance.
- Suppose X and Y are iid
.
- Show that
and
are independent.
- Show that
is uniformly
distributed on
.
- Show
is a Cauchy random variable.
SET D
- Compute the characteristic function, cumulants and central moments
for the Poisson(
) distribution.
- Compute the characteristic function, cumulants and central moments
for the Gamma distribution with shape parameter
and scale parameter
.
SET E
- Suppose
are iid
and
are iid
.
- Find complete and sufficient statistics.
- Find UMVUE's of
and
.
- Now suppose you know that
. Find UMVUE's of
and of
. (You have already found
the UMVUE for
.)
- Now suppose
and
are unknown but that you
know that
. Prove there is no UMVUE for
.
(Hint: Find the UMVUE if you knew
with a known.
Use the fact that the solution depends on a to finish the proof.)
- Why doesn't the Lehmann-Scheffé theorem apply?
- Suppose
iid Poisson(
). Find the
UMVUE for
and for
.
- Suppose
iid with
![](img75.gif)
for
. For n=1 and 2
find the UMVUE of
.
(Hint: The expected value of any function of X is a power series in
divided by
. Set this equal to
and deduce that two power series
are equal. Since this implies their coefficients are the same you can see what
the estimate must be. )
SET F
- Suppose
are independent Poisson(
)
variables. Find the UMP level
test of
versus
and evaluate the constants for the case n=3 and
.
- Suppose X has a Gamma(
) distribution with shape
parameter
known. Find the UMPU test of
and
evaluate the constants for the case
and
.
SET G
- Suppose
are independent
random variables. (This is the
usual set-up for the one-way layout.)
- Find the MLE's for
and
.
- Find the expectations and variances of these estimators.
- Let
be the error sum of squares in the ith cell in the
previous question.
- Find the joint density of the
.
- Find the best estimate of
of the form
in the sense of mean squared error.
- Do the same under the condition that the estimator must be unbiased.
- If only
are observed what is the MLE of
?
- Find the UMVUE of
for the usual one-way layout model,
that is, the model of the last two questions.
- Exponential families: Suppose
are iid with density
![](img104.gif)
- Find minimal sufficient statistics.
- If
are the minimal sufficient statistics show
that setting
and solving gives the
likelihood equations. (Note the connection to the method of moments.)
SET H
- Find the variance stabilizing transformation for the Poisson distribution.
- Suppose X, Y and Z are independent standard exponentials. Use
numerical Fourier inversion of the characteristic function to compute
the density of
at 1. You may use the splus function
integ.romb (or any other function) found by attaching the directory
/home/math/lockhart/research/software/quadrature/.Data.
- Suppose X is an integer valued random variable. Let
be
the characteristic function of X. Show that
![](img109.gif)
- Suppose
are independent random variables such that
. Prove that
![](img112.gif)
where
is the standard normal density. You should use the previous
problem and Taylor expansion of the characteristic function around 0.
Also do the same thing using Sterling's formula.
SET I
- Suppose
are iid exponential(
).
- Find the exact confidence levels of 95% intervals based on normal
approximations to the distributions of the pivots
,
, and
for n=10, 20 and 40.
- Find the shortest exact 95% confidence interval based on
; get numerical values for n=10, 20 and 40.
- Find the exact confidence level of 95% confidence intervals based
on the chi-squared approximation to the distribution of deviance drop.
Compare the results with the previous question based on length and
coverage probabilities.
Figure out how to make a convincing comparison. Which method is better?
- Suppose
are iid with density f. Assume the median
of f is 0. In this problem you will study the asymptotic distribution of
the sample median. To simplify things you may assume that n=2m-1 is odd
so that the median is the mth order statistic.
- Let M denote the sample median and
be the number of X's
less than or equal to x. Express the event that
in terms of the number
.
- What is the exact distribution of
?
- Show that the expected value,
, of
can be
written as
.
- Show that
is asymptotically normal
with mean 0 and variance
.
- Deduce from all the preceding that
is asymptotically
normal with mean 0 and variance
.
- In question 20 take
for all i and let
. What happens to
the MLE of
?
SET J
- Suppose that
are independent random variables
and that
are the corresponding values of some covariate.
Suppose that the density of
is
![](img138.gif)
where
, and
are unknown parameters.
- Find the log-likelihood, the score function and the Fisher information.
- For the data set in /home/math/lockhart/teaching/801/data1 fit the
model and produce a contour plot of the log-likelihood surface, the profile
likelihood for
and an approximate 95% confidence interval for
.
- Consider the random effects one way layout. You have data
and a model
where the
's are iid
and the
's are iid
.
- Write down the likelihood.
- Find minimal sufficient statistics.
- Are they complete?
- Find method of moments estimates of the three parameters.
- Can you find the MLE's?
- For each of the doses
a number of animals
are treated with the corresponding dose of some drug. The
number dying at dose d is Binomial with parameter
. A common model
for
is
- Find the likelihood equations for estimating
and
.
- Find the Fisher information matrix.
- Define the parameter LD50 as the value of d for which
;
express LD50 as a function of
and
.
- Use a Taylor expansion to find large sample confidence limits for LD50.
- At each of the doses -3.204, -2.903, 2.602, -2.301 and -2.000 a sample
of 40 mice were exposed to antipneumonococcus serum. The numbers surviving
were 7, 18, 32, 35, and 38 respectively. Get numerical values for the theory
above. You can use glm or get preliminary estimates based on linear
regression of the MLE of
against dose.
- Suppose
are a sample of size n from the density
![](img161.gif)
In the following question
the digamma function
is defined by
and the trigamma
function
is the derivative of the digamma function. From
the identity
you can deduce
recurrence relations for the digamma and trigamma functions.
- For
known find the mle for
.
- When both
and
are unknown what equation must be
solved to find
, the mle of
?
- Evaluate the Fisher information matrix.
- A sample of size 20 is in the file
/home/math/lockhart/teaching/801/gamma.
Use this data in the following questions. First take
and find
the mle of
subject to this restriction.
- Now use
and
to
get method of moments estimates
and
for
the parameters.
- Do two steps of Newton Raphson to get MLEs.
- Use Fisher's scoring idea to redo the previous question.
- Compute standard errors for the MLEs and compare the difference
between the estimates in the previous 2 questions to the SEs.
- Do a likelihood ratio test of
.
SET K
- Suppose X is Bin(n,p) with p unknown. Define
for b > 0.
- Find the risk functions for these estimators.
- For the Beta
prior distribution on p
with
and
both positive find the posterior
distribution of p given X.
- Find the Bayes procedures for these priors.
- Discuss consistency and asymptotic distribution theory for the Bayes
procedures.
- Find the minimax procedure.
- Compare the risk function of the minimax procedure, the MLE and the
estimate
for n=1,100 and asymptotically. Provide
graphs.
- What happens to the posterior distribution of p as n tends to
infinity?
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Richard Lockhart
Mon Sep 23 19:51:32 PDT 1996