Stat 643 Mid Term Exam
Fall 1996
October 22, 1996
Prof. Vardeman
1. Consider a model c œ ÖT) ×ß where @ œ Ö"ß #ß $×, k
T) are specified in the table below.
B" B# B$
) œ " Þ' Þ" Þ#
) œ # Þ$ Þ" Þ#
)œ$ !
Þ# Þ%
œ ÖB" ß B# ß B$ ß B% × and the distributions
B%
Þ"
Þ%
Þ%
a) Find a minimal sufficient statistic X À k pg , where g has 3 elements. Give the distribution
of X in a tabular form similar to that used above for specifying the distribution of \Þ
Suppose that one is to make a decision about ), where T œ @ œ Ö"ß #ß $× and
PÐ)ß +Ñ œ MÒ+ Á )ÓÞ
b) Consider the behavioral decision rule 9 defined by
9B" ÐÖ"×Ñ œ Þ) 9B" ÐÖ#×Ñ œ Þ# 9B" ÐÖ$×Ñ œ !
9B# ÐÖ"×Ñ œ Þ' 9B# ÐÖ#×Ñ œ Þ$ 9B# ÐÖ$×Ñ œ Þ"
9B$ ÐÖ"×Ñ œ Þ$ 9B$ ÐÖ#×Ñ œ Þ' 9B$ ÐÖ$×Ñ œ Þ"
9B% ÐÖ"×Ñ œ ! 9B% ÐÖ#×Ñ œ Þ# 9B% ÐÖ$×Ñ œ Þ)
Find a rule that is a function of your sufficient statistic from a) and is risk equivalent to 9.
c) Argue directly from the definition of admissibility that $" ÐBÑ » " is admissible.
d) Show that
$# ÐBÑ œ œ
"
$
if B œ B" or B œ B#
if B œ B$ or B œ B%
is Bayes versus a prior K that places mass #$ on ) œ " and mass "$ on ) œ $.
e) Show that $# defined in d) is admissible.
f) Consider a randomized rule < in this decision problem defined by <ÐÖ$" ×Ñ œ
<ÐÖ$2 ×Ñ œ "# . Find a behavioral decision rule 9w that is risk equivalent to <.
"
#
and
2. Consider the family of distributions c œ ÖT) × on k œ Ò!ß_Ñ indexed by the parameter
)=()" ß )# Ñ - @ § e# , dominated by the 5-finite measure . œ ? € - for ? a unit point mass at
0 and - Lebesgue measure on k , where
0) ÐBÑ œ
.T)
ÐBÑ º expa)" MÒB œ !Ó € )# Bb .
..
a) What are the natural parameter space and "normalizing constant" OÐ) Ñ for this family of
distributions?
b) Use OÐ) Ñ to find the moment generating function for \ µ T) , E) exp=\Þ
c) Identify the UMVUE for #Ð) Ñ œ T) Ò\" œ !Ó based on \" ß \# ß ÞÞÞß \8 iid T) for ) in the
natural parameter space. Argue very carefully that your estimator is UMVU.
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3. Consider squared error loss estimation of the binomial parameter ":."
a) Write out (and to the extent possible simplify) the risk function for a linear estimator of :,
$ w ÐBÑ œ EB € FÞ
Using the result from a) it is possible to show that $ÐBÑ œ
È8
B€ #
8€È 8
has constant risk. (Don't bother
to show this here.) It is also the case that if conditional on :, \ µ Binomial Ð8ß :Ñ and : µ Beta
Ð!ß "Ñ, EÒ:l\ œ BÓ œ !€!€B
" €8 . (Again, you need not show this here.)
b) Argue carefully that $ is an admissible estimator of :Þ
c) Take the result in b) as given and prove that $ is the unique minimax rule for this estimation
problem.
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