Stat 643 Final Exam
December 13, 1995
Prof. Vardeman
Below are 8 questions. Write answers to exactly 5 of them. Each question will be worth 20
points. When you are finished, staple your answers together in numerical order.
1. Consider the two distributions T0 and T1 on k œ Ð • 1,1Ñ with densities with respect to
Lebesgue measure on k
00 ÐBÑ œ 12 and 01 ÐBÑ œ | B | .
a) Find a most powerful level ! œ .10 test of H0 :\ µ T0 vs H1 :\ µ T1 .
b) Consider the family of (mixture) distributions c œ ÖT) × with densities with respect to
Lebesgue measure
0) ÐBÑ œ )01 ÐBÑ € Ð1 • )Ñ00 (BÑ for ) - Ò0, 1Ó .
In this family, find a UMP level ! œ .10 test of H0 :) Ÿ .5 vs H1 :) ž .5 based on a single
observation \ . Argue carefully that the test you invent is UMP size .10 for this problem.
2. Suppose that \1 , \2 , ... , \8 are iid with probability mass function given below for ) - [0, 1Ó.
B
0
1
2
0) ÐBÑ
)2
2)Ð1 • )Ñ
Ð1 • ) Ñ 2
a) Find an explicit form for the MLE of ) and argue that the MLE is consistent.
b) Note that if ) œ 0 or ) œ 1, the MLE of ) is not asymptotically normal. Point out a basic
part of the hypotheses of Theorem 49 that is violated in the cases of ) œ 0 and ) œ 1.
c) Give explicitly two different 90% large sample confidence interval formulas for ) based on
the asymptotic distribution of the MLE when ) - Ð0, 1Ñ.
3. Suppose that for 3 œ 1, 2, ... , 8, the random vectors \3 œ Ð]3 , ^3 Ñ are iid with distribution T:
described as follows. ]1 and ^1 are independent, ]1 µ binomial (7, :Ñ and ^1 µ geometric Ð:Ñ
(i.e. with pmf 1: ÐDÑ œ :Ð1 • :ÑD•1 for D œ 1, 2, 3, ...).
a) Give the C-R lower bound on the variance of an unbiased estimator of : based on \1 , \2 ,
... , \8 .
b) Describe as explicitly as possible how you would go about testing H0 :: œ .2 vs H1 :: Á .2
based on \1 , \2 , ... , \8 with an approximate level of ! œ .1.
c) Describe how you would make an approximately 90% confidence set for : based on \1 ,
\2 , ... , \8 and the inversion of appropriate tests.
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4. Miscellaneous Short Answers
a) Consider an estimation problem with \ µ T) with density 0) with respect to Lebesgue
measure on e1 . Suppose however, that measurement precision is such that in truth one can
only observe \ “to the nearest integer." What is an appropriate
loglikelihood to use in this problem if 8 independent \3 's are to be observed?
b) Suppose that in a “regular" estimation problem with bivariate parameter ) œ Ð)1 , )2 Ñ the
2 ‚ 2 Fisher information matrix based on a single observation is M1 Ð)Ñ. Suppose further that
based on 8 independent observations one may compute the MLE, )^ 8 œ Ð)^1 8 , )^2 8 Ñ. Describe
how you would expect to compute a large sample (joint)
confidence set for ) .
c) Argue that in a simple versus simple testing problem, a test minimax with “0-1 loss"
must be an equalizer rule.
d) Continuity of risk functions is useful in showing the admissibility of Bayes decision rules.
Power functions of tests are usually continuous and are related to risk functions for 0-1 loss.
Do these facts provide the risk function continuity needed for simple admissibility arguments
for Bayes tests under 0-1 loss (when, for example, @ § e1 )? Explain.
5. Consider a Bayesian decision theoretic approach to the testing of a point null hypothesis
versus a composite alternative. For the sake of concreteness suppose that
\ µ N(), 1Ñ corresponding to H0 :) œ 0 and H1 :) Á 0, take @ œ e1 , T œ Ö0, 1× and
PÐ), +Ñ œ MÒ) œ 0ÓMÒ+ œ 1Ó € MÒ) Á 0ÓMÒ+ œ 0Ó .
a) Show that if K is a prior that is absolutely continuous with respect to Lebesgue measure on
@, 9ÐBÑ ´ 1 is Bayes with respect to K. What does this suggest about suitable priors for this
kind of problem?
b) Suppose now that K œ "# ?0 € "# A, where ?0 places mass 1 at the point Ö0× and no mass
elsewhere, and A is N(0, 52 Ñ measure. Find the form of a Bayes decision rule for this prior.
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6. Suppose that \1 , \2 , ... , \8 are independent normal random variables, each with variance 1.
Suppose further that each .3 Ð œ E\3 Ñ is either 0 or 1, so that we have for this problem a
parameter space @ œ Ö. œ Ð.1 , .2 , ... , .8 Ñ | each .3 is 0 or 1× œ Ö0, 1×8 . Based on \ œ Ð\1 ,
\2 , ... , \8 ) we will consider making decisions about all 8 means .3 . To this
end, let T œ @ œ Ö0, 1×8 and for + œ Ð+1 , +2 , ... , +8 Ñ define the (overall) loss function
PÐ., + Ñ œ ! MÒ+3 Á .3 Ó .
8
3œ1
Then, for each permutation of the first 8 positive integers 1 œ Ð1Ð 1Ñ, 1Ð2Ñ, ... , 1Ð8ÑÑ, define a
transformation 11 :e8 Ä e8 by 11 ÐCÑ œ C1 , for C œ ÐC1 , C2 , ... , C8 Ñ and C1 œ ÐC1Ð1Ñ , C1Ð2Ñ , ... ,
C1Ð8Ñ Ñ and let Z œ Ö11 | 1 is a permutation of the first 8 positive integers}.
a) Show that this decision problem is invariant under the group of transformations Z .
b) Characterize as clearly/completely as possible what it means for a nonrandomized decision
rule $ :e8 Ä T œ Ö0, 1×8 to be equivariant in this problem.
c) Equivariant decision rules in invariant decision problems have risk functions that are
constant on orbits. What are the orbits for this decision problem?
7. Consider the one parameter exponential family of distributions on k œ (0,1) with
densities with respect to Lebesgue measure
(
(Á0
exp( •1 exp( B
0( ÐBÑ œ ž
1
(œ0 .
a) Suppose that one wishes to test H0 :( œ 0 vs H1 :( Á 0 based on a single observation from
this distribution. Find a UMPU size ! œ .2 test of these hypotheses and argue that your test is
indeed UMPU.
b) For 9 your test from a), consider the test 1 • 9. What optimality properties does this test
possess? (For example, you might show that it is “best" of some specific size for some
specific pair of hypotheses.)
8. Consider a random observable \ œ ÐR ,] Ñ with distribution T) parametrized by ) œ (:,
.Ñ - Ð0, 1Ñ ‚ e1 . Suppose that R is geometric with parameter : (i.e. has pmf
1: Ð8Ñ œ :Ð1 • :Ñ8•1 for 8 œ 1, 2, 3, ...) and that the conditional distribution of ] given R is
normal with mean . and variance R •1 .
a) For testing H0 :. Ÿ .0 vs H1 :. ž .0 based on \ , : might well be termed a nuisance
parameter. Find a UMPU size ! œ .1 test of these hypotheses and argue that your test is
indeed UMPU of this size.
b) Invert your tests 9ÐBÑ to produce a 90% confidence bound for . based on \ .
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