Emory and Henry College Compute the Values Statistics Questions

Problem 1: Efficient Exponential Expectation Esti-mators
Suppose that Yı, …, Y15 ~ Exp(\) are independent, with pdf
1
хе
exp ( – 6), 2 € (0,00)
Consider the following estimators:
5
în = X, Î2
=
=
ΣΧ;, = Υ,
i=1
Part A
Find ef f(, î2)
Part B
Find ef f(al, î3)
Part C
Find eff(A2, î3)
Part D
Do any of these estimators dominate another? If so, state which ones and explain
why.
Problem 2: Consistent Uniform Estimators
Suppose X1, …, Xn Unif(0,0) are independent, and suppose that n is an odd
number. Consider the following estimators:
n+1
6 = 2X, , X(n) , Ô3 = median(X1, …, Xn)
n
Part A
Which of these estimators are consistent? Give full justification.
Part B
Suppose that instead we are interested in estimating 02, using Ôz, ôz and Ôz. Which
of these estimators converge in probability to 02? Justify your answer.
Problem 3: Sufficient Estimators and the Likeli-
hood Function
In this problem we will explore the likelihood and determine whether or not estimators
are sufficient.
Part A
Suppose X1, …, Xn Pois() are independent. Find L (2X). Is X a sufficient
estimator for 1? Justify your answer using the factorization theorem.
Part B
Suppose X1, …, Xn – N(4,0%) are independent, and o2. is known. Find L (|X, 02).
Is X a sufficient estimator for ^? Justify your answer using the factorization theorem.
(Hint: look for a way to write X inside of the likelihood…)
Part C
Suppose that we have n independent realizations from a Unif(0,0) distribution. Is
X(n) a sufficient estimator for e? You do not need to use the factorization
theorem, but you must justify your answer.
0 =
Part D
Suppose that X1, …, Xn ~ Bern(p) are independent. Find L (P|X).
Part E
So far in this course we have been treating parameters as unknown constants, and
our data as random. Here we will take a look at what happens when we treat our
parameters as random and our data as fixed. Recall L (P|X) from above. If we treat
p as random and X as fixed, what distribution does p follow? Justify your answer.
(Hint: remember where p lives, your resulting distribution should be defined over the
р
same space).