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
exp ( – 6), 2 € (0,00)
Consider the following estimators:
în = X, Î2
ΣΧ;, = Υ,
Find ef f(, î2)
Find ef f(al, î3)
Find eff(A2, î3)
Do any of these estimators dominate another? If so, state which ones and explain
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:
6 = 2X, , X(n) , Ô3 = median(X1, …, Xn)
Which of these estimators are consistent? Give full justification.
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-
In this problem we will explore the likelihood and determine whether or not estimators
Suppose X1, …, Xn Pois() are independent. Find L (2X). Is X a sufficient
estimator for 1? Justify your answer using the factorization theorem.
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…)
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.
Suppose that X1, …, Xn ~ Bern(p) are independent. Find L (P|X).
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